3.5.45 \(\int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [445]

Optimal. Leaf size=1185 \[ \frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \text {ArcTan}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i b f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^3 f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i b f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {f (e+f x) \text {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a d^2}-\frac {2 i b^3 f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i b f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i b f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 b^4 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}-\frac {b^4 f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^3}+\frac {f^2 \text {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a d^3}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f (e+f x) \tanh (c+d x)}{a d^2}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d} \]

[Out]

-1/2*f^2*polylog(3,exp(2*d*x+2*c))/a/d^3+1/2*f^2*x^2/a/d+I*b*f^2*polylog(3,I*exp(d*x+c))/(a^2+b^2)/d^3-1/2*b^4
*f^2*polylog(3,-exp(2*d*x+2*c))/a/(a^2+b^2)^2/d^3-b*f*(f*x+e)*sech(d*x+c)/(a^2+b^2)/d^2-1/2*b^2*(f*x+e)^2*sech
(d*x+c)^2/a/(a^2+b^2)/d-1/2*b*(f*x+e)^2*sech(d*x+c)*tanh(d*x+c)/(a^2+b^2)/d-2*I*b^3*f^2*polylog(3,-I*exp(d*x+c
))/(a^2+b^2)^2/d^3-I*b*f^2*polylog(3,-I*exp(d*x+c))/(a^2+b^2)/d^3+f^2*ln(cosh(d*x+c))/a/d^3-2*(f*x+e)^2*arctan
h(exp(2*d*x+2*c))/a/d+1/2*f^2*polylog(3,-exp(2*d*x+2*c))/a/d^3-1/2*(f*x+e)^2*tanh(d*x+c)^2/a/d+2*I*b^3*f*(f*x+
e)*polylog(2,-I*exp(d*x+c))/(a^2+b^2)^2/d^2+b^2*f*(f*x+e)*tanh(d*x+c)/a/(a^2+b^2)/d^2-2*I*b^3*f*(f*x+e)*polylo
g(2,I*exp(d*x+c))/(a^2+b^2)^2/d^2-I*b*f*(f*x+e)*polylog(2,I*exp(d*x+c))/(a^2+b^2)/d^2+e*f*x/a/d-f*(f*x+e)*tanh
(d*x+c)/a/d^2-2*b^4*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d^2-2*b^4*f*(f*x+e)*p
olylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d^2+I*b*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/(a^2+b^2)
/d^2+b^4*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/a/(a^2+b^2)^2/d^2+2*I*b^3*f^2*polylog(3,I*exp(d*x+c))/(a^2+b^2)^
2/d^3+b^4*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/a/(a^2+b^2)^2/d-b^2*f^2*ln(cosh(d*x+c))/a/(a^2+b^2)/d^3-b^4*(f*x+e)^2
*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d-b^4*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a
/(a^2+b^2)^2/d+2*b^4*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d^3+2*b^4*f^2*polylog(3,-b
*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d^3-b*(f*x+e)^2*arctan(exp(d*x+c))/(a^2+b^2)/d-f*(f*x+e)*polylo
g(2,-exp(2*d*x+2*c))/a/d^2+f*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a/d^2-2*b^3*(f*x+e)^2*arctan(exp(d*x+c))/(a^2+b
^2)^2/d+b*f^2*arctan(sinh(d*x+c))/(a^2+b^2)/d^3

________________________________________________________________________________________

Rubi [A]
time = 1.73, antiderivative size = 1185, normalized size of antiderivative = 1.00, number of steps used = 57, number of rules used = 23, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.676, Rules used = {5708, 2700, 14, 5570, 6873, 12, 6874, 2631, 4267, 2611, 2320, 6724, 3801, 3556, 5692, 5680, 2221, 4265, 3799, 4271, 3855, 5559, 4269} \begin {gather*} -\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^4}{a \left (a^2+b^2\right )^2 d}-\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^4}{a \left (a^2+b^2\right )^2 d}+\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b^4}{a \left (a^2+b^2\right )^2 d}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac {f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^3}-\frac {f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right ) b^4}{2 a \left (a^2+b^2\right )^2 d^3}-\frac {2 (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}-\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^3}+\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^3}-\frac {(e+f x)^2 \text {sech}^2(c+d x) b^2}{2 a \left (a^2+b^2\right ) d}-\frac {f^2 \log (\cosh (c+d x)) b^2}{a \left (a^2+b^2\right ) d^3}+\frac {f (e+f x) \tanh (c+d x) b^2}{a \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d}+\frac {f^2 \text {ArcTan}(\sinh (c+d x)) b}{\left (a^2+b^2\right ) d^3}+\frac {i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^2}-\frac {i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^2}-\frac {i f^2 \text {Li}_3\left (-i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^3}+\frac {i f^2 \text {Li}_3\left (i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^3}-\frac {f (e+f x) \text {sech}(c+d x) b}{\left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) b}{2 \left (a^2+b^2\right ) d}+\frac {f^2 x^2}{2 a d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac {e f x}{a d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f (e+f x) \tanh (c+d x)}{a d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^2*Csch[c + d*x]*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

(e*f*x)/(a*d) + (f^2*x^2)/(2*a*d) - (2*b^3*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) - (b*(e + f*x)^2
*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) + (b*f^2*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d^3) - (2*(e + f*x)^2*ArcTa
nh[E^(2*c + 2*d*x)])/(a*d) - (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^2
*d) - (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^2*d) + (b^4*(e + f*x)^2*
Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b^2)^2*d) + (f^2*Log[Cosh[c + d*x]])/(a*d^3) - (b^2*f^2*Log[Cosh[c + d*x]]
)/(a*(a^2 + b^2)*d^3) + ((2*I)*b^3*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) + (I*b*f*(e +
f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) - ((2*I)*b^3*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^
2 + b^2)^2*d^2) - (I*b*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) - (2*b^4*f*(e + f*x)*PolyLog[2
, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^2*d^2) - (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c +
d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^2*d^2) + (b^4*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(a*(a^2
+ b^2)^2*d^2) - (f*(e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a*d^2) + (f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)]
)/(a*d^2) - ((2*I)*b^3*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^3) - (I*b*f^2*PolyLog[3, (-I)*E^(c +
d*x)])/((a^2 + b^2)*d^3) + ((2*I)*b^3*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)^2*d^3) + (I*b*f^2*PolyLog[3
, I*E^(c + d*x)])/((a^2 + b^2)*d^3) + (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2
+ b^2)^2*d^3) + (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^2*d^3) - (b^4
*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)^2*d^3) + (f^2*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a*d^3) - (f
^2*PolyLog[3, E^(2*c + 2*d*x)])/(2*a*d^3) - (b*f*(e + f*x)*Sech[c + d*x])/((a^2 + b^2)*d^2) - (b^2*(e + f*x)^2
*Sech[c + d*x]^2)/(2*a*(a^2 + b^2)*d) - (f*(e + f*x)*Tanh[c + d*x])/(a*d^2) + (b^2*f*(e + f*x)*Tanh[c + d*x])/
(a*(a^2 + b^2)*d^2) - (b*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*(a^2 + b^2)*d) - ((e + f*x)^2*Tanh[c + d*
x]^2)/(2*a*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2631

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[(a + b*x)^(m + 1)*(Log[u]/(b*(m + 1))), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[(a + b*x)^(m + 1)*(D[u, x]/u), x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

Rule 2700

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 5559

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Sim
p[(-(c + d*x)^m)*(Sech[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x]
/; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 5570

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

Rule 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 5692

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[b^2/(a^2 + b^2), Int[(e + f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Dist[1/(
a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && I
GtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]

Rule 5708

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Dis
t[b/a, Int[(e + f*x)^m*Sech[c + d*x]^p*(Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \text {csch}(c+d x) \text {sech}^3(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac {b \int (e+f x)^2 \text {sech}^3(c+d x) (a-b \sinh (c+d x)) \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )}-\frac {(2 f) \int (e+f x) \left (\frac {\log (\tanh (c+d x))}{d}-\frac {\tanh ^2(c+d x)}{2 d}\right ) \, dx}{a}\\ &=\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac {b^3 \int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a \left (a^2+b^2\right )^2}-\frac {b^5 \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )^2}-\frac {b \int \left (a (e+f x)^2 \text {sech}^3(c+d x)-b (e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)\right ) \, dx}{a \left (a^2+b^2\right )}-\frac {(2 f) \int \frac {(e+f x) \left (2 \log (\tanh (c+d x))-\tanh ^2(c+d x)\right )}{2 d} \, dx}{a}\\ &=\frac {b^4 (e+f x)^3}{3 a \left (a^2+b^2\right )^2 f}+\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac {b^3 \int \left (a (e+f x)^2 \text {sech}(c+d x)-b (e+f x)^2 \tanh (c+d x)\right ) \, dx}{a \left (a^2+b^2\right )^2}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )^2}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )^2}-\frac {b \int (e+f x)^2 \text {sech}^3(c+d x) \, dx}{a^2+b^2}+\frac {b^2 \int (e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )}-\frac {f \int (e+f x) \left (2 \log (\tanh (c+d x))-\tanh ^2(c+d x)\right ) \, dx}{a d}\\ &=\frac {b^4 (e+f x)^3}{3 a \left (a^2+b^2\right )^2 f}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac {b^3 \int (e+f x)^2 \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {b^4 \int (e+f x)^2 \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )^2}-\frac {b \int (e+f x)^2 \text {sech}(c+d x) \, dx}{2 \left (a^2+b^2\right )}-\frac {f \int \left (2 (e+f x) \log (\tanh (c+d x))-(e+f x) \tanh ^2(c+d x)\right ) \, dx}{a d}+\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}+\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}+\frac {\left (b^2 f\right ) \int (e+f x) \text {sech}^2(c+d x) \, dx}{a \left (a^2+b^2\right ) d}+\frac {\left (b f^2\right ) \int \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right ) d^2}\\ &=-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac {\left (2 b^4\right ) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{a \left (a^2+b^2\right )^2}+\frac {f \int (e+f x) \tanh ^2(c+d x) \, dx}{a d}-\frac {(2 f) \int (e+f x) \log (\tanh (c+d x)) \, dx}{a d}+\frac {\left (2 i b^3 f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac {\left (2 i b^3 f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac {(i b f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}-\frac {(i b f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}+\frac {\left (2 b^4 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^2 d^2}+\frac {\left (2 b^4 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^2 d^2}-\frac {\left (b^2 f^2\right ) \int \tanh (c+d x) \, dx}{a \left (a^2+b^2\right ) d^2}\\ &=-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f (e+f x) \tanh (c+d x)}{a d^2}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac {\int 2 d (e+f x)^2 \text {csch}(2 c+2 d x) \, dx}{a d}+\frac {f \int (e+f x) \, dx}{a d}-\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}+\frac {\left (2 b^4 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^3}+\frac {\left (2 b^4 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^3}+\frac {f^2 \int \tanh (c+d x) \, dx}{a d^2}-\frac {\left (2 i b^3 f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (2 i b^3 f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}-\frac {\left (i b f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}+\frac {\left (i b f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f (e+f x) \tanh (c+d x)}{a d^2}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac {2 \int (e+f x)^2 \text {csch}(2 c+2 d x) \, dx}{a}-\frac {\left (2 i b^3 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {\left (2 i b^3 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {\left (i b f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {\left (i b f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {\left (b^4 f^2\right ) \int \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right )^2 d^2}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 i b^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f (e+f x) \tanh (c+d x)}{a d^2}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac {(2 f) \int (e+f x) \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a d}+\frac {(2 f) \int (e+f x) \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a d}-\frac {\left (b^4 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^3}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac {2 i b^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}-\frac {b^4 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^3}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f (e+f x) \tanh (c+d x)}{a d^2}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac {f^2 \int \text {Li}_2\left (-e^{2 c+2 d x}\right ) \, dx}{a d^2}-\frac {f^2 \int \text {Li}_2\left (e^{2 c+2 d x}\right ) \, dx}{a d^2}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac {2 i b^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}-\frac {b^4 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^3}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f (e+f x) \tanh (c+d x)}{a d^2}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac {f^2 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^3}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac {2 i b^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}-\frac {b^4 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^3}+\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f (e+f x) \tanh (c+d x)}{a d^2}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3310\) vs. \(2(1185)=2370\).
time = 24.80, size = 3310, normalized size = 2.79 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)^2*Csch[c + d*x]*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

-1/6*(-12*a^3*d^3*e^2*E^(2*c)*x - 24*a*b^2*d^3*e^2*E^(2*c)*x + 12*a^3*d*E^(2*c)*f^2*x + 12*a*b^2*d*E^(2*c)*f^2
*x - 12*a^3*d^3*e*E^(2*c)*f*x^2 - 24*a*b^2*d^3*e*E^(2*c)*f*x^2 - 4*a^3*d^3*E^(2*c)*f^2*x^3 - 8*a*b^2*d^3*E^(2*
c)*f^2*x^3 + 6*a^2*b*d^2*e^2*ArcTan[E^(c + d*x)] + 18*b^3*d^2*e^2*ArcTan[E^(c + d*x)] + 6*a^2*b*d^2*e^2*E^(2*c
)*ArcTan[E^(c + d*x)] + 18*b^3*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] - 12*a^2*b*f^2*ArcTan[E^(c + d*x)] - 12*b^3
*f^2*ArcTan[E^(c + d*x)] - 12*a^2*b*E^(2*c)*f^2*ArcTan[E^(c + d*x)] - 12*b^3*E^(2*c)*f^2*ArcTan[E^(c + d*x)] +
(6*I)*a^2*b*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (18*I)*b^3*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (6*I)*a^2*b*d^2*
e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (18*I)*b^3*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (3*I)*a^2*b*d^2*f
^2*x^2*Log[1 - I*E^(c + d*x)] + (9*I)*b^3*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] + (3*I)*a^2*b*d^2*E^(2*c)*f^2*x^2
*Log[1 - I*E^(c + d*x)] + (9*I)*b^3*d^2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] - (6*I)*a^2*b*d^2*e*f*x*Log[1 +
I*E^(c + d*x)] - (18*I)*b^3*d^2*e*f*x*Log[1 + I*E^(c + d*x)] - (6*I)*a^2*b*d^2*e*E^(2*c)*f*x*Log[1 + I*E^(c +
d*x)] - (18*I)*b^3*d^2*e*E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] - (3*I)*a^2*b*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)]
- (9*I)*b^3*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] - (3*I)*a^2*b*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] - (9*I
)*b^3*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] + 6*a^3*d^2*e^2*Log[1 + E^(2*(c + d*x))] + 12*a*b^2*d^2*e^2*L
og[1 + E^(2*(c + d*x))] + 6*a^3*d^2*e^2*E^(2*c)*Log[1 + E^(2*(c + d*x))] + 12*a*b^2*d^2*e^2*E^(2*c)*Log[1 + E^
(2*(c + d*x))] - 6*a^3*f^2*Log[1 + E^(2*(c + d*x))] - 6*a*b^2*f^2*Log[1 + E^(2*(c + d*x))] - 6*a^3*E^(2*c)*f^2
*Log[1 + E^(2*(c + d*x))] - 6*a*b^2*E^(2*c)*f^2*Log[1 + E^(2*(c + d*x))] + 12*a^3*d^2*e*f*x*Log[1 + E^(2*(c +
d*x))] + 24*a*b^2*d^2*e*f*x*Log[1 + E^(2*(c + d*x))] + 12*a^3*d^2*e*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] + 24*
a*b^2*d^2*e*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] + 6*a^3*d^2*f^2*x^2*Log[1 + E^(2*(c + d*x))] + 12*a*b^2*d^2*f
^2*x^2*Log[1 + E^(2*(c + d*x))] + 6*a^3*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] + 12*a*b^2*d^2*E^(2*c)*f^
2*x^2*Log[1 + E^(2*(c + d*x))] - (6*I)*b*(a^2 + 3*b^2)*d*(1 + E^(2*c))*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)
] + (6*I)*b*(a^2 + 3*b^2)*d*(1 + E^(2*c))*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)] + 6*a^3*d*e*f*PolyLog[2, -E^(2
*(c + d*x))] + 12*a*b^2*d*e*f*PolyLog[2, -E^(2*(c + d*x))] + 6*a^3*d*e*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))]
+ 12*a*b^2*d*e*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))] + 6*a^3*d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 12*a*b^2*
d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 6*a^3*d*E^(2*c)*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 12*a*b^2*d*E^(2*c)
*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + (6*I)*a^2*b*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (18*I)*b^3*f^2*PolyLog[3,
(-I)*E^(c + d*x)] + (6*I)*a^2*b*E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (18*I)*b^3*E^(2*c)*f^2*PolyLog[3,
(-I)*E^(c + d*x)] - (6*I)*a^2*b*f^2*PolyLog[3, I*E^(c + d*x)] - (18*I)*b^3*f^2*PolyLog[3, I*E^(c + d*x)] - (6*
I)*a^2*b*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - (18*I)*b^3*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - 3*a^3*f^2*
PolyLog[3, -E^(2*(c + d*x))] - 6*a*b^2*f^2*PolyLog[3, -E^(2*(c + d*x))] - 3*a^3*E^(2*c)*f^2*PolyLog[3, -E^(2*(
c + d*x))] - 6*a*b^2*E^(2*c)*f^2*PolyLog[3, -E^(2*(c + d*x))])/((a^2 + b^2)^2*d^3*(1 + E^(2*c))) + ((-4*E^(2*c
)*x*(3*e^2 + 3*e*f*x + f^2*x^2))/(-1 + E^(2*c)) + (6*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/d + (6*f*(e + f*x)*
PolyLog[2, E^(2*(c + d*x))])/d^2 - (3*f^2*PolyLog[3, E^(2*(c + d*x))])/d^3)/(6*a) + (b^4*((2*E^(2*c)*x*(3*e^2
+ 3*e*f*x + f^2*x^2))/(-1 + E^(2*c)) - (3*(d^2*e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] + 2*d^2*e*f
*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + d^2*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E
^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 2*d^2*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])]
+ d^2*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 2*d*f*(e + f*x)*PolyLog[2, -((b
*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 2*d*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c
+ Sqrt[(a^2 + b^2)*E^(2*c)]))] - 2*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] -
2*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))]))/d^3))/(3*a*(a^2 + b^2)^2) + (Csch
[c]*Sech[c]*Sech[c + d*x]^2*(-6*a^3*e*f - 6*a*b^2*e*f + 12*a^3*d^2*e^2*x + 24*a*b^2*d^2*e^2*x - 6*a^3*f^2*x -
6*a*b^2*f^2*x + 12*a^3*d^2*e*f*x^2 + 24*a*b^2*d^2*e*f*x^2 + 4*a^3*d^2*f^2*x^3 + 8*a*b^2*d^2*f^2*x^3 + 6*a^3*e*
f*Cosh[2*c] + 6*a*b^2*e*f*Cosh[2*c] + 6*a^3*f^2*x*Cosh[2*c] + 6*a*b^2*f^2*x*Cosh[2*c] + 6*a^3*e*f*Cosh[2*d*x]
+ 6*a*b^2*e*f*Cosh[2*d*x] + 6*a^3*f^2*x*Cosh[2*d*x] + 6*a*b^2*f^2*x*Cosh[2*d*x] + 3*a^2*b*d*e^2*Cosh[c - d*x]
+ 3*b^3*d*e^2*Cosh[c - d*x] + 6*a^2*b*d*e*f*x*Cosh[c - d*x] + 6*b^3*d*e*f*x*Cosh[c - d*x] + 3*a^2*b*d*f^2*x^2*
Cosh[c - d*x] + 3*b^3*d*f^2*x^2*Cosh[c - d*x] - 3*a^2*b*d*e^2*Cosh[3*c + d*x] - 3*b^3*d*e^2*Cosh[3*c + d*x] -
6*a^2*b*d*e*f*x*Cosh[3*c + d*x] - 6*b^3*d*e*f*x...

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Maple [F]
time = 3.02, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \mathrm {csch}\left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^2*csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^2*csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-a^2*b*d^2*f^2*integrate(x^2*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^
(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - 3*b^3*d^2*f^2*integrate(x^2*e^(d*x + c)/(a^4*d^2*e^(2
*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x)
+ 2*a^3*d^2*f^2*integrate(x^2/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2
*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 4*a*b^2*d^2*f^2*integrate(x^2/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*
b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 4*a^3*d^2*f*e*int
egrate(x/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*
b^2*d^2 + b^4*d^2), x) + 8*a*b^2*d^2*f*e*integrate(x/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c)
+ b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - 2*a^2*b*d^2*f*integrate(x*e^(d*x + c + 1)
/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2
+ b^4*d^2), x) - 6*b^3*d^2*f*integrate(x*e^(d*x + c + 1)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2
*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - a^3*f^2*(2*(d*x + c)/((a^4 + 2*a^2*b^
2 + b^4)*d^3) - log(e^(2*d*x + 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d^3)) - a*b^2*f^2*(2*(d*x + c)/((a^4 + 2*a^2
*b^2 + b^4)*d^3) - log(e^(2*d*x + 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d^3)) + 2*a^2*b*f^2*arctan(e^(d*x + c))/(
(a^4 + 2*a^2*b^2 + b^4)*d^3) + 2*b^3*f^2*arctan(e^(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^3) - (b^4*log(-2*a*e^(
-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^5 + 2*a^3*b^2 + a*b^4)*d) - (a^2*b + 3*b^3)*arctan(e^(-d*x - c))/((a^4
+ 2*a^2*b^2 + b^4)*d) + (a^3 + 2*a*b^2)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d) + (b*e^(-d*x -
c) - 2*a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((a^2 + b^2 + 2*(a^2 + b^2)*e^(-2*d*x - 2*c) + (a^2 + b^2)*e^(
-4*d*x - 4*c))*d) - log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a*d))*e^2 + (2*a*f^2*x + 2*a*f*e - (b
*d*f^2*x^2*e^(3*c) + 2*b*f*e^(3*c + 1) + 2*(b*f^2*e^(3*c) + b*d*f*e^(3*c + 1))*x)*e^(3*d*x) + 2*(a*d*f^2*x^2*e
^(2*c) + a*f*e^(2*c + 1) + (a*f^2*e^(2*c) + 2*a*d*f*e^(2*c + 1))*x)*e^(2*d*x) + (b*d*f^2*x^2*e^c - 2*b*f*e^(c
+ 1) + 2*(b*d*f*e^(c + 1) - b*f^2*e^c)*x)*e^(d*x))/(a^2*d^2 + b^2*d^2 + (a^2*d^2*e^(4*c) + b^2*d^2*e^(4*c))*e^
(4*d*x) + 2*(a^2*d^2*e^(2*c) + b^2*d^2*e^(2*c))*e^(2*d*x)) + 2*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c))
)*f*e/(a*d^2) + 2*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))*f*e/(a*d^2) + (d^2*x^2*log(e^(d*x + c) + 1)
+ 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*f^2/(a*d^3) + (d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*
x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*f^2/(a*d^3) - 2/3*(d^3*f^2*x^3 + 3*d^3*f*x^2*e)/(a*d^3) + in
tegrate(2*(b^5*f^2*x^2 + 2*b^5*f*x*e - (a*b^4*f^2*x^2*e^c + 2*a*b^4*f*x*e^(c + 1))*e^(d*x))/(a^5*b + 2*a^3*b^3
+ a*b^5 - (a^5*b*e^(2*c) + 2*a^3*b^3*e^(2*c) + a*b^5*e^(2*c))*e^(2*d*x) - 2*(a^6*e^c + 2*a^4*b^2*e^c + a^2*b^
4*e^c)*e^(d*x)), x)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 25021 vs. \(2 (1119) = 2238\).
time = 0.81, size = 25021, normalized size = 21.11 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(4*((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*c*f^2)*cosh(d*x + c)^4 + 4*((a^4 + a^2*b^2)*d*f^2*x + (a^4
+ a^2*b^2)*c*f^2)*sinh(d*x + c)^4 + 4*(a^4 + a^2*b^2)*c*f^2 - 4*(a^4 + a^2*b^2)*d*f*cosh(1) + 2*((a^3*b + a*b^
3)*d^2*f^2*x^2 + 2*(a^3*b + a*b^3)*d*f^2*x + (a^3*b + a*b^3)*d^2*cosh(1)^2 + (a^3*b + a*b^3)*d^2*sinh(1)^2 + 2
*((a^3*b + a*b^3)*d^2*f*x + (a^3*b + a*b^3)*d*f)*cosh(1) + 2*((a^3*b + a*b^3)*d^2*f*x + (a^3*b + a*b^3)*d^2*co
sh(1) + (a^3*b + a*b^3)*d*f)*sinh(1))*cosh(d*x + c)^3 - 4*(a^4 + a^2*b^2)*d*f*sinh(1) + 2*((a^3*b + a*b^3)*d^2
*f^2*x^2 + 2*(a^3*b + a*b^3)*d*f^2*x + (a^3*b + a*b^3)*d^2*cosh(1)^2 + (a^3*b + a*b^3)*d^2*sinh(1)^2 + 2*((a^3
*b + a*b^3)*d^2*f*x + (a^3*b + a*b^3)*d*f)*cosh(1) + 8*((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*c*f^2)*cosh(
d*x + c) + 2*((a^3*b + a*b^3)*d^2*f*x + (a^3*b + a*b^3)*d^2*cosh(1) + (a^3*b + a*b^3)*d*f)*sinh(1))*sinh(d*x +
c)^3 - 4*((a^4 + a^2*b^2)*d^2*f^2*x^2 - (a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*d^2*cosh(1)^2 + (a^4 + a^2*
b^2)*d^2*sinh(1)^2 - 2*(a^4 + a^2*b^2)*c*f^2 + (2*(a^4 + a^2*b^2)*d^2*f*x + (a^4 + a^2*b^2)*d*f)*cosh(1) + (2*
(a^4 + a^2*b^2)*d^2*f*x + 2*(a^4 + a^2*b^2)*d^2*cosh(1) + (a^4 + a^2*b^2)*d*f)*sinh(1))*cosh(d*x + c)^2 - 2*(2
*(a^4 + a^2*b^2)*d^2*f^2*x^2 - 2*(a^4 + a^2*b^2)*d*f^2*x + 2*(a^4 + a^2*b^2)*d^2*cosh(1)^2 + 2*(a^4 + a^2*b^2)
*d^2*sinh(1)^2 - 4*(a^4 + a^2*b^2)*c*f^2 - 12*((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*c*f^2)*cosh(d*x + c)^
2 + 2*(2*(a^4 + a^2*b^2)*d^2*f*x + (a^4 + a^2*b^2)*d*f)*cosh(1) - 3*((a^3*b + a*b^3)*d^2*f^2*x^2 + 2*(a^3*b +
a*b^3)*d*f^2*x + (a^3*b + a*b^3)*d^2*cosh(1)^2 + (a^3*b + a*b^3)*d^2*sinh(1)^2 + 2*((a^3*b + a*b^3)*d^2*f*x +
(a^3*b + a*b^3)*d*f)*cosh(1) + 2*((a^3*b + a*b^3)*d^2*f*x + (a^3*b + a*b^3)*d^2*cosh(1) + (a^3*b + a*b^3)*d*f)
*sinh(1))*cosh(d*x + c) + 2*(2*(a^4 + a^2*b^2)*d^2*f*x + 2*(a^4 + a^2*b^2)*d^2*cosh(1) + (a^4 + a^2*b^2)*d*f)*
sinh(1))*sinh(d*x + c)^2 - 2*((a^3*b + a*b^3)*d^2*f^2*x^2 - 2*(a^3*b + a*b^3)*d*f^2*x + (a^3*b + a*b^3)*d^2*co
sh(1)^2 + (a^3*b + a*b^3)*d^2*sinh(1)^2 + 2*((a^3*b + a*b^3)*d^2*f*x - (a^3*b + a*b^3)*d*f)*cosh(1) + 2*((a^3*
b + a*b^3)*d^2*f*x + (a^3*b + a*b^3)*d^2*cosh(1) - (a^3*b + a*b^3)*d*f)*sinh(1))*cosh(d*x + c) + 4*(b^4*d*f^2*
x + b^4*d*f*cosh(1) + b^4*d*f*sinh(1) + (b^4*d*f^2*x + b^4*d*f*cosh(1) + b^4*d*f*sinh(1))*cosh(d*x + c)^4 + 4*
(b^4*d*f^2*x + b^4*d*f*cosh(1) + b^4*d*f*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*d*f^2*x + b^4*d*f*cosh(
1) + b^4*d*f*sinh(1))*sinh(d*x + c)^4 + 2*(b^4*d*f^2*x + b^4*d*f*cosh(1) + b^4*d*f*sinh(1))*cosh(d*x + c)^2 +
2*(b^4*d*f^2*x + b^4*d*f*cosh(1) + b^4*d*f*sinh(1) + 3*(b^4*d*f^2*x + b^4*d*f*cosh(1) + b^4*d*f*sinh(1))*cosh(
d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*d*f^2*x + b^4*d*f*cosh(1) + b^4*d*f*sinh(1))*cosh(d*x + c)^3 + (b^4*d*f^
2*x + b^4*d*f*cosh(1) + b^4*d*f*sinh(1))*cosh(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c
) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 4*(b^4*d*f^2*x + b^4*d*f*cosh(1) +
b^4*d*f*sinh(1) + (b^4*d*f^2*x + b^4*d*f*cosh(1) + b^4*d*f*sinh(1))*cosh(d*x + c)^4 + 4*(b^4*d*f^2*x + b^4*d*
f*cosh(1) + b^4*d*f*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*d*f^2*x + b^4*d*f*cosh(1) + b^4*d*f*sinh(1))
*sinh(d*x + c)^4 + 2*(b^4*d*f^2*x + b^4*d*f*cosh(1) + b^4*d*f*sinh(1))*cosh(d*x + c)^2 + 2*(b^4*d*f^2*x + b^4*
d*f*cosh(1) + b^4*d*f*sinh(1) + 3*(b^4*d*f^2*x + b^4*d*f*cosh(1) + b^4*d*f*sinh(1))*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 4*((b^4*d*f^2*x + b^4*d*f*cosh(1) + b^4*d*f*sinh(1))*cosh(d*x + c)^3 + (b^4*d*f^2*x + b^4*d*f*cosh(1)
+ b^4*d*f*sinh(1))*cosh(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c)
+ b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 4*((a^4 + 2*a^2*b^2 + b^4)*d*f^2*x + ((a^4 + 2*a^2*b^2
+ b^4)*d*f^2*x + (a^4 + 2*a^2*b^2 + b^4)*d*f*cosh(1) + (a^4 + 2*a^2*b^2 + b^4)*d*f*sinh(1))*cosh(d*x + c)^4 +
4*((a^4 + 2*a^2*b^2 + b^4)*d*f^2*x + (a^4 + 2*a^2*b^2 + b^4)*d*f*cosh(1) + (a^4 + 2*a^2*b^2 + b^4)*d*f*sinh(1)
)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + 2*a^2*b^2 + b^4)*d*f^2*x + (a^4 + 2*a^2*b^2 + b^4)*d*f*cosh(1) + (a^
4 + 2*a^2*b^2 + b^4)*d*f*sinh(1))*sinh(d*x + c)^4 + (a^4 + 2*a^2*b^2 + b^4)*d*f*cosh(1) + (a^4 + 2*a^2*b^2 + b
^4)*d*f*sinh(1) + 2*((a^4 + 2*a^2*b^2 + b^4)*d*f^2*x + (a^4 + 2*a^2*b^2 + b^4)*d*f*cosh(1) + (a^4 + 2*a^2*b^2
+ b^4)*d*f*sinh(1))*cosh(d*x + c)^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d*f^2*x + (a^4 + 2*a^2*b^2 + b^4)*d*f*cosh(1)
+ (a^4 + 2*a^2*b^2 + b^4)*d*f*sinh(1) + 3*((a^4 + 2*a^2*b^2 + b^4)*d*f^2*x + (a^4 + 2*a^2*b^2 + b^4)*d*f*cosh
(1) + (a^4 + 2*a^2*b^2 + b^4)*d*f*sinh(1))*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^4 + 2*a^2*b^2 + b^4)*d*f^
2*x + (a^4 + 2*a^2*b^2 + b^4)*d*f*cosh(1) + (a^4 + 2*a^2*b^2 + b^4)*d*f*sinh(1))*cosh(d*x + c)^3 + ((a^4 + 2*a
^2*b^2 + b^4)*d*f^2*x + (a^4 + 2*a^2*b^2 + b^4)*d*f*cosh(1) + (a^4 + 2*a^2*b^2 + b^4)*d*f*sinh(1))*cosh(d*x +
c))*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c)) + 2*(2*(a^4 + 2*a^2*b^2)*d*f^2*x + I*(a^3*b + 3*a*b^3)
*d*f^2*x + (2*(a^4 + 2*a^2*b^2)*d*f^2*x + I*(a^...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**2*csch(d*x+c)*sech(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)^2/(cosh(c + d*x)^3*sinh(c + d*x)*(a + b*sinh(c + d*x))),x)

[Out]

int((e + f*x)^2/(cosh(c + d*x)^3*sinh(c + d*x)*(a + b*sinh(c + d*x))), x)

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