3.5.36 \(\int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [436]

Optimal. Leaf size=734 \[ -\frac {2 b (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {2 i b f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 b^2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {b^2 f (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {f (e+f x) \text {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a d^2}-\frac {2 i b f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 i b f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {2 b^2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {b^2 f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^3}+\frac {f^2 \text {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a d^3} \]

[Out]

-2*b*(f*x+e)^2*arctan(exp(d*x+c))/(a^2+b^2)/d-2*(f*x+e)^2*arctanh(exp(2*d*x+2*c))/a/d+b^2*(f*x+e)^2*ln(1+exp(2
*d*x+2*c))/a/(a^2+b^2)/d-b^2*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d-b^2*(f*x+e)^2*ln(1
+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d-2*I*b*f*(f*x+e)*polylog(2,I*exp(d*x+c))/(a^2+b^2)/d^2+2*I*b*f
*(f*x+e)*polylog(2,-I*exp(d*x+c))/(a^2+b^2)/d^2+b^2*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/a/(a^2+b^2)/d^2-f*(f*
x+e)*polylog(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^2*f*(f*x+e)*polylog(2,-b*e
xp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^2-2*b^2*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/
(a^2+b^2)/d^2+2*I*b*f^2*polylog(3,I*exp(d*x+c))/(a^2+b^2)/d^3-2*I*b*f^2*polylog(3,-I*exp(d*x+c))/(a^2+b^2)/d^3
-1/2*b^2*f^2*polylog(3,-exp(2*d*x+2*c))/a/(a^2+b^2)/d^3+1/2*f^2*polylog(3,-exp(2*d*x+2*c))/a/d^3-1/2*f^2*polyl
og(3,exp(2*d*x+2*c))/a/d^3+2*b^2*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^3+2*b^2*f^2*po
lylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^3

________________________________________________________________________________________

Rubi [A]
time = 0.84, antiderivative size = 734, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5708, 5569, 4267, 2611, 2320, 6724, 5692, 5680, 2221, 6874, 4265, 3799} \begin {gather*} -\frac {2 b (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}+\frac {2 b^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^3 \left (a^2+b^2\right )}+\frac {2 b^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^3 \left (a^2+b^2\right )}-\frac {b^2 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a d^3 \left (a^2+b^2\right )}-\frac {2 i b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}+\frac {2 i b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )}+\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a d^2 \left (a^2+b^2\right )}+\frac {2 i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}-\frac {2 i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}-\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a d \left (a^2+b^2\right )}-\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a d \left (a^2+b^2\right )}+\frac {b^2 (e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{a d \left (a^2+b^2\right )}+\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) \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 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) - (2*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (b^2
*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) - (b^2*(e + f*x)^2*Log[1 + (b*E
^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) + (b^2*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b
^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*f*(e + f*x)*PolyLog[2
, I*E^(c + d*x)])/((a^2 + b^2)*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])
/(a*(a^2 + b^2)*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)
*d^2) + (b^2*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(a*(a^2 + b^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*f^2*PolyLog[3, (-I)*E^(c + d*
x)])/((a^2 + b^2)*d^3) + ((2*I)*b*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)*d^3) + (2*b^2*f^2*PolyLog[3, -((
b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^3) + (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqr
t[a^2 + b^2]))])/(a*(a^2 + b^2)*d^3) - (b^2*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)*d^3) + (f^2*Pol
yLog[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)

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 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 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 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

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 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}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {2 \int (e+f x)^2 \text {csch}(2 c+2 d x) \, dx}{a}-\frac {b \int (e+f x)^2 \text {sech}(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 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )}\\ &=\frac {b^2 (e+f x)^3}{3 a \left (a^2+b^2\right ) f}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b \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 )}-\frac {b^3 \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 )}-\frac {b^3 \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 )}-\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 {b^2 (e+f x)^3}{3 a \left (a^2+b^2\right ) f}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (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 ) d}-\frac {b^2 (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 ) d}-\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 {b \int (e+f x)^2 \text {sech}(c+d x) \, dx}{a^2+b^2}+\frac {b^2 \int (e+f x)^2 \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )}+\frac {\left (2 b^2 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 ) d}+\frac {\left (2 b^2 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 ) 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 {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (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 ) d}-\frac {b^2 (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 ) d}-\frac {2 b^2 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 ) d^2}-\frac {2 b^2 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 ) 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 {\left (2 b^2\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 )}+\frac {(2 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 {(2 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 {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 {\left (2 b^2 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 ) d^2}+\frac {\left (2 b^2 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 ) d^2}\\ &=-\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (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 ) d}-\frac {b^2 (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 ) d}+\frac {b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {2 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 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^2 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 ) d^2}-\frac {2 b^2 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 ) 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 {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 {\left (2 b^2 f\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right ) d}+\frac {\left (2 b^2 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 ) d^3}+\frac {\left (2 b^2 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 ) d^3}-\frac {\left (2 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 (2 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 {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (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 ) d}-\frac {b^2 (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 ) d}+\frac {b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {2 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 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^2 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 ) d^2}-\frac {2 b^2 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 ) d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) 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 b^2 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 ) d^3}+\frac {2 b^2 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 ) 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 {\left (2 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 (2 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^2 f^2\right ) \int \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}\\ &=-\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (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 ) d}-\frac {b^2 (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 ) d}+\frac {b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {2 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 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^2 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 ) d^2}-\frac {2 b^2 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 ) d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) 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 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 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^2 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 ) d^3}+\frac {2 b^2 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 ) 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 {\left (b^2 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 ) d^3}\\ &=-\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (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 ) d}-\frac {b^2 (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 ) d}+\frac {b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {2 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 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^2 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 ) d^2}-\frac {2 b^2 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 ) d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) 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 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 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^2 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 ) d^3}+\frac {2 b^2 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 ) d^3}-\frac {b^2 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) 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}\\ \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. \(2950\) vs. \(2(734)=1468\).
time = 22.47, size = 2950, normalized size = 4.02 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

2*((a*(-(d^3*E^c*x*(3*e^2 + 3*e*f*x + f^2*x^2)) + 3*d^2*(1 + E^c)*(e + f*x)^2*Log[1 + E^(c + d*x)] + 6*d*(1 +
E^c)*f*(e + f*x)*PolyLog[2, -E^(c + d*x)] - 6*(1 + E^c)*f^2*PolyLog[3, -E^(c + d*x)]))/(6*(a^2 + b^2)*d^3*(1 +
E^c)) + (d^2*((-I)*d*E^c*x*((-3*I)*b*e*f*x + a*(3*e^2 + 3*e*f*x + f^2*x^2)) + 3*(1 + I*E^c)*((-2*I)*b*e*f*x +
a*(e + f*x)^2)*Log[1 + I*E^(c + d*x)]) + 6*d*(1 + I*E^c)*f*((-I)*b*e + a*(e + f*x))*PolyLog[2, (-I)*E^(c + d*
x)] - (6*I)*a*(-I + E^c)*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(6*(a - I*b)*((-I)*a + b)*d^3*(-I + E^c)) - ((I/2)*
b*((-2*I)*d^2*e^2*ArcTan[E^(c + d*x)] + d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] - d^2*f^2*x^2*Log[1 + I*E^(c + d*x)
] - 2*d*f^2*x*PolyLog[2, (-I)*E^(c + d*x)] + 2*d*f^2*x*PolyLog[2, I*E^(c + d*x)] + 2*f^2*PolyLog[3, (-I)*E^(c
+ d*x)] - 2*f^2*PolyLog[3, I*E^(c + d*x)]))/((a^2 + b^2)*d^3) - ((-I)*b*d^3*e*E^(2*c)*f*x^2 + 2*a*d^2*e^2*ArcT
an[1 - (1 + I)*E^(c + d*x)] + (2*I)*a*d^2*e^2*E^(2*c)*ArcTan[1 - (1 + I)*E^(c + d*x)] + (2*I)*a*d^2*e*f*x*Log[
1 - E^(c + d*x)] - 2*a*d^2*e*E^(2*c)*f*x*Log[1 - E^(c + d*x)] + I*a*d^2*f^2*x^2*Log[1 - E^(c + d*x)] - a*d^2*E
^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] - (2*I)*a*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + 2*b*d^2*e*f*x*Log[1 - I*E^(c
+ d*x)] + 2*a*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (2*I)*b*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] - I*
a*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] + a*d^2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] + 2*d*(-I + E^(2*c))*f*(I*
b*e + a*(e + f*x))*PolyLog[2, I*E^(c + d*x)] - 2*a*d*(-I + E^(2*c))*f*(e + f*x)*PolyLog[2, E^(c + d*x)] + (2*I
)*a*f^2*PolyLog[3, I*E^(c + d*x)] - 2*a*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - (2*I)*a*f^2*PolyLog[3, E^(c +
d*x)] + 2*a*E^(2*c)*f^2*PolyLog[3, E^(c + d*x)])/(2*(a^2 + b^2)*d^3*(-I + E^(2*c))) - (b^2*(4*d^3*E^(2*c)*x*(3
*e^2 + 3*e*f*x + f^2*x^2) - 6*d^2*(-1 + E^(2*c))*(e + f*x)^2*Log[1 - E^(2*(c + d*x))] - 6*d*(-1 + E^(2*c))*f*(
e + f*x)*PolyLog[2, E^(2*(c + d*x))] + 3*(-1 + E^(2*c))*f^2*PolyLog[3, E^(2*(c + d*x))]))/(12*a*(a^2 + b^2)*d^
3*(-1 + E^(2*c))) + (b^2*((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))/(6*a*(a^2 + b^2)) - (b^2*x*(3*e^2 + 3*e*f*x + f^2*x^2)*Csch[c/2]*Sech[c/2]*Sech[c])/(24*a*(
a^2 + b^2)) + (x*Csch[c/2]*Sech[c/2]*(a^2*e^2 + b^2*e^2 - a^2*e^2*Cosh[c] - I*a^2*e^2*Sinh[c]))/(8*a*(a^2 + b^
2)*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2] + I*Sinh[c/2])) + (b^2*e*f*x^2*Cosh[2*c])/(a*(a^2 + b^2)*(-1 + Cosh[2*
c] + Sinh[2*c])*(1 + Cosh[2*c] + Sinh[2*c])) + (b^2*f^2*x^3*Cosh[2*c])/(3*a*(a^2 + b^2)*(-1 + Cosh[2*c] + Sinh
[2*c])*(1 + Cosh[2*c] + Sinh[2*c])) + (b^2*e*f*x^2*Sinh[2*c])/(a*(a^2 + b^2)*(-1 + Cosh[2*c] + Sinh[2*c])*(1 +
Cosh[2*c] + Sinh[2*c])) + (b^2*f^2*x^3*Sinh[2*c])/(3*a*(a^2 + b^2)*(-1 + Cosh[2*c] + Sinh[2*c])*(1 + Cosh[2*c
] + Sinh[2*c])) - ((1/2 - I/2)*a*e*f*x^2*Cosh[c])/((a^2 + b^2)*(-1 - (1 + I)*Cosh[c] - (2*I)*Cosh[2*c] + (1 -
I)*Cosh[3*c] + Cosh[4*c] - (1 + I)*Sinh[c] - (2*I)*Sinh[2*c] + (1 - I)*Sinh[3*c] + Sinh[4*c])) + (b*e*f*x^2*Co
sh[c])/(2*(a^2 + b^2)*(-1 - (1 + I)*Cosh[c] - (2*I)*Cosh[2*c] + (1 - I)*Cosh[3*c] + Cosh[4*c] - (1 + I)*Sinh[c
] - (2*I)*Sinh[2*c] + (1 - I)*Sinh[3*c] + Sinh[4*c])) - ((1/6 - I/6)*a*f^2*x^3*Cosh[c])/((a^2 + b^2)*(-1 - (1
+ I)*Cosh[c] - (2*I)*Cosh[2*c] + (1 - I)*Cosh[3*c] + Cosh[4*c] - (1 + I)*Sinh[c] - (2*I)*Sinh[2*c] + (1 - I)*S
inh[3*c] + Sinh[4*c])) - ((1/2 + I/2)*a*e*f*x^2*Cosh[3*c])/((a^2 + b^2)*(-1 - (1 + I)*Cosh[c] - (2*I)*Cosh[2*c
] + (1 - I)*Cosh[3*c] + Cosh[4*c] - (1 + I)*Sinh[c] - (2*I)*Sinh[2*c] + (1 - I)*Sinh[3*c] + Sinh[4*c])) - (b*e
*f*x^2*Cosh[3*c])/(2*(a^2 + b^2)*(-1 - (1 + I)*Cosh[c] - (2*I)*Cosh[2*c] + (1 - I)*Cosh[3*c] + Cosh[4*c] - (1
+ I)*Sinh[c] - (2*I)*Sinh[2*c] + (1 - I)*Sinh[3*c] + Sinh[4*c])) - ((1/6 + I/6)*a*f^2*x^3*Cosh[3*c])/((a^2 + b
^2)*(-1 - (1 + I)*Cosh[c] - (2*I)*Cosh[2*c] + (1 - I)*Cosh[3*c] + Cosh[4*c] - (1 + I)*Sinh[c] - (2*I)*Sinh[2*c
] + (1 - I)*Sinh[3*c] + Sinh[4*c])) - ((1/2 - I/2)*a*e*f*x^2*Sinh[c])/((a^2 + b^2)*(-1 - (1 + I)*Cosh[c] - (2*
I)*Cosh[2*c] + (1 - I)*Cosh[3*c] + Cosh[4*c] - (1 + I)*Sinh[c] - (2*I)*Sinh[2*c] + (1 - I)*Sinh[3*c] + Sinh[4*
c])) + (b*e*f*x^2*Sinh[c])/(2*(a^2 + b^2)*(-1 - (1 + I)*Cosh[c] - (2*I)*Cosh[2*c] + (1 - I)*Cosh[3*c] + Cosh[4
*c] - (1 + I)*Sinh[c] - (2*I)*Sinh[2*c] + (1 - I)*Sinh[3*c] + Sinh[4*c])) - ((1/6 - I/6)*a*f^2*x^3*Sinh[c])/((
a^2 + b^2)*(-1 - (1 + I)*Cosh[c] - (2*I)*Cosh[2...

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \mathrm {csch}\left (d x +c \right ) \mathrm {sech}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-(b^2*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^3 + a*b^2)*d) - 2*b*arctan(e^(-d*x - c))/((a^2 + b^2
)*d) + a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) - log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a*d)
)*e^2 + 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) + integrate(2*(b^3*f^2*x^2 + 2*b^3*f*x*e - (a*b^2*f^2*x^2
*e^c + 2*a*b^2*f*x*e^(c + 1))*e^(d*x))/(a^3*b + a*b^3 - (a^3*b*e^(2*c) + a*b^3*e^(2*c))*e^(2*d*x) - 2*(a^4*e^c
+ a^2*b^2*e^c)*e^(d*x)), x) - integrate(-2*(a*f^2*x^2 + 2*a*f*x*e - (b*f^2*x^2*e^c + 2*b*f*x*e^(c + 1))*e^(d*
x))/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*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. 2074 vs. \(2 (689) = 1378\).
time = 0.47, size = 2074, normalized size = 2.83 \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)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

(2*b^2*f^2*polylog(3, (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) + 2*b^2*f^2*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqr
t((a^2 + b^2)/b^2))/b) - 2*(a^2 + b^2)*f^2*polylog(3, cosh(d*x + c) + sinh(d*x + c)) - 2*(a^2 + b^2)*f^2*polyl
og(3, -cosh(d*x + c) - sinh(d*x + c)) - 2*(b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1))*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) - 2*(b^2*d*f^2*
x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1))*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) + 2*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*f*cosh(1) + (a^2 + b^2)*d*f
*sinh(1))*dilog(cosh(d*x + c) + sinh(d*x + c)) - 2*(a^2*d*f^2*x + I*a*b*d*f^2*x + a^2*d*f*cosh(1) + I*a*b*d*f*
cosh(1) + a^2*d*f*sinh(1) + I*a*b*d*f*sinh(1))*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) - 2*(a^2*d*f^2*x - I*a
*b*d*f^2*x + a^2*d*f*cosh(1) - I*a*b*d*f*cosh(1) + a^2*d*f*sinh(1) - I*a*b*d*f*sinh(1))*dilog(-I*cosh(d*x + c)
- I*sinh(d*x + c)) + 2*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*f*cosh(1) + (a^2 + b^2)*d*f*sinh(1))*dilog(-cosh(
d*x + c) - sinh(d*x + c)) - (b^2*c^2*f^2 - 2*b^2*c*d*f*cosh(1) + b^2*d^2*cosh(1)^2 + b^2*d^2*sinh(1)^2 - 2*(b^
2*c*d*f - b^2*d^2*cosh(1))*sinh(1))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*
a) - (b^2*c^2*f^2 - 2*b^2*c*d*f*cosh(1) + b^2*d^2*cosh(1)^2 + b^2*d^2*sinh(1)^2 - 2*(b^2*c*d*f - b^2*d^2*cosh(
1))*sinh(1))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - (b^2*d^2*f^2*x^2 -
b^2*c^2*f^2 + 2*(b^2*d^2*f*x + b^2*c*d*f)*cosh(1) + 2*(b^2*d^2*f*x + b^2*c*d*f)*sinh(1))*log(-(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) - (b^2*d^2*f^2*x^2 - b
^2*c^2*f^2 + 2*(b^2*d^2*f*x + b^2*c*d*f)*cosh(1) + 2*(b^2*d^2*f*x + b^2*c*d*f)*sinh(1))*log(-(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) + ((a^2 + b^2)*d^2*f^2*x
^2 + 2*(a^2 + b^2)*d^2*f*x*cosh(1) + (a^2 + b^2)*d^2*cosh(1)^2 + (a^2 + b^2)*d^2*sinh(1)^2 + 2*((a^2 + b^2)*d^
2*f*x + (a^2 + b^2)*d^2*cosh(1))*sinh(1))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - (a^2*c^2*f^2 + I*a*b*c^2*f^
2 - 2*a^2*c*d*f*cosh(1) - 2*I*a*b*c*d*f*cosh(1) + a^2*d^2*cosh(1)^2 + I*a*b*d^2*cosh(1)^2 + a^2*d^2*sinh(1)^2
+ I*a*b*d^2*sinh(1)^2 - 2*(a^2*c*d*f - a^2*d^2*cosh(1))*sinh(1) - 2*I*(a*b*c*d*f - a*b*d^2*cosh(1))*sinh(1))*l
og(cosh(d*x + c) + sinh(d*x + c) + I) - (a^2*c^2*f^2 - I*a*b*c^2*f^2 - 2*a^2*c*d*f*cosh(1) + 2*I*a*b*c*d*f*cos
h(1) + a^2*d^2*cosh(1)^2 - I*a*b*d^2*cosh(1)^2 + a^2*d^2*sinh(1)^2 - I*a*b*d^2*sinh(1)^2 - 2*(a^2*c*d*f - a^2*
d^2*cosh(1))*sinh(1) + 2*I*(a*b*c*d*f - a*b*d^2*cosh(1))*sinh(1))*log(cosh(d*x + c) + sinh(d*x + c) - I) + ((a
^2 + b^2)*c^2*f^2 - 2*(a^2 + b^2)*c*d*f*cosh(1) + (a^2 + b^2)*d^2*cosh(1)^2 + (a^2 + b^2)*d^2*sinh(1)^2 - 2*((
a^2 + b^2)*c*d*f - (a^2 + b^2)*d^2*cosh(1))*sinh(1))*log(cosh(d*x + c) + sinh(d*x + c) - 1) - (a^2*d^2*f^2*x^2
- I*a*b*d^2*f^2*x^2 - a^2*c^2*f^2 + I*a*b*c^2*f^2 + 2*(a^2*d^2*f*x + a^2*c*d*f)*cosh(1) - 2*I*(a*b*d^2*f*x +
a*b*c*d*f)*cosh(1) + 2*(a^2*d^2*f*x + a^2*c*d*f)*sinh(1) - 2*I*(a*b*d^2*f*x + a*b*c*d*f)*sinh(1))*log(I*cosh(d
*x + c) + I*sinh(d*x + c) + 1) - (a^2*d^2*f^2*x^2 + I*a*b*d^2*f^2*x^2 - a^2*c^2*f^2 - I*a*b*c^2*f^2 + 2*(a^2*d
^2*f*x + a^2*c*d*f)*cosh(1) + 2*I*(a*b*d^2*f*x + a*b*c*d*f)*cosh(1) + 2*(a^2*d^2*f*x + a^2*c*d*f)*sinh(1) + 2*
I*(a*b*d^2*f*x + a*b*c*d*f)*sinh(1))*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) + ((a^2 + b^2)*d^2*f^2*x^2 -
(a^2 + b^2)*c^2*f^2 + 2*((a^2 + b^2)*d^2*f*x + (a^2 + b^2)*c*d*f)*cosh(1) + 2*((a^2 + b^2)*d^2*f*x + (a^2 + b^
2)*c*d*f)*sinh(1))*log(-cosh(d*x + c) - sinh(d*x + c) + 1) + 2*(a^2*f^2 + I*a*b*f^2)*polylog(3, I*cosh(d*x + c
) + I*sinh(d*x + c)) + 2*(a^2*f^2 - I*a*b*f^2)*polylog(3, -I*cosh(d*x + c) - I*sinh(d*x + c)))/((a^3 + a*b^2)*
d^3)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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)/(a+b*sinh(d*x+c)),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3436 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not 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)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

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

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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 )\,\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)*sinh(c + d*x)*(a + b*sinh(c + d*x))),x)

[Out]

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

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