3.3.44 \(\int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [244]

Optimal. Leaf size=535 \[ -\frac {(e+f x)^2}{a d}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^2 \coth (c+d x)}{a d}+\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {2 b f (e+f x) \text {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {2 b f (e+f x) \text {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {2 b^2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {2 b^2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {f^2 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {2 b f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {2 b f^2 \text {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {2 b^2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {2 b^2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3} \]

[Out]

-(f*x+e)^2/a/d+2*b*(f*x+e)^2*arctanh(exp(d*x+c))/a^2/d-(f*x+e)^2*coth(d*x+c)/a/d+2*f*(f*x+e)*ln(1-exp(2*d*x+2*
c))/a/d^2+2*b*f*(f*x+e)*polylog(2,-exp(d*x+c))/a^2/d^2-2*b*f*(f*x+e)*polylog(2,exp(d*x+c))/a^2/d^2+f^2*polylog
(2,exp(2*d*x+2*c))/a/d^3-2*b*f^2*polylog(3,-exp(d*x+c))/a^2/d^3+2*b*f^2*polylog(3,exp(d*x+c))/a^2/d^3+b^2*(f*x
+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d/(a^2+b^2)^(1/2)-b^2*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2
)^(1/2)))/a^2/d/(a^2+b^2)^(1/2)+2*b^2*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d^2/(a^2+b^2)
^(1/2)-2*b^2*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/d^2/(a^2+b^2)^(1/2)-2*b^2*f^2*polylog(
3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d^3/(a^2+b^2)^(1/2)+2*b^2*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1
/2)))/a^2/d^3/(a^2+b^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.74, antiderivative size = 535, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5694, 4269, 3797, 2221, 2317, 2438, 4267, 2611, 2320, 6724, 3403, 2296} \begin {gather*} -\frac {2 b^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3 \sqrt {a^2+b^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^2 d^3 \sqrt {a^2+b^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^2 d^2 \sqrt {a^2+b^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^2 d^2 \sqrt {a^2+b^2}}+\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 d \sqrt {a^2+b^2}}-\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 d \sqrt {a^2+b^2}}-\frac {2 b f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a^2 d^3}+\frac {2 b f^2 \text {Li}_3\left (e^{c+d x}\right )}{a^2 d^3}+\frac {2 b f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac {2 b f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}+\frac {f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}+\frac {2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac {(e+f x)^2 \coth (c+d x)}{a d}-\frac {(e+f x)^2}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((e + f*x)^2/(a*d)) + (2*b*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^2*d) - ((e + f*x)^2*Coth[c + d*x])/(a*d) + (b
^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*Sqrt[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^2*Sqrt[a^2 + b^2]*d) + (2*f*(e + f*x)*Log[1 - E^(2*(c + d*x))])
/(a*d^2) + (2*b*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - (2*b*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a
^2*d^2) + (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[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^2*Sqrt[a^2 + b^2]*d^2) + (f^2*Pol
yLog[2, E^(2*(c + d*x))])/(a*d^3) - (2*b*f^2*PolyLog[3, -E^(c + d*x)])/(a^2*d^3) + (2*b*f^2*PolyLog[3, E^(c +
d*x)])/(a^2*d^3) - (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[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^2*Sqrt[a^2 + b^2]*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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 3403

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

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*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 5694

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^n, x], x] - Dist[b/a, Int[(e + f*x)^m*(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]

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]

Rubi steps

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

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Mathematica [A]
time = 13.14, size = 1011, normalized size = 1.89 \begin {gather*} \frac {-\frac {4 a d^2 e e^{2 c} f x}{-1+e^{2 c}}-\frac {2 a d^2 e^{2 c} f^2 x^2}{-1+e^{2 c}}+2 b d^2 e^2 \tanh ^{-1}\left (e^{c+d x}\right )-2 b d^2 e f x \log \left (1-e^{c+d x}\right )-b d^2 f^2 x^2 \log \left (1-e^{c+d x}\right )+2 b d^2 e f x \log \left (1+e^{c+d x}\right )+b d^2 f^2 x^2 \log \left (1+e^{c+d x}\right )+2 a d e f \log \left (1-e^{2 (c+d x)}\right )+2 a d f^2 x \log \left (1-e^{2 (c+d x)}\right )+2 b d f (e+f x) \text {PolyLog}\left (2,-e^{c+d x}\right )-2 b d f (e+f x) \text {PolyLog}\left (2,e^{c+d x}\right )+a f^2 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )-2 b f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )+2 b f^2 \text {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}+\frac {b^2 \left (-2 d^2 e^2 \sqrt {\left (a^2+b^2\right ) e^{2 c}} \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 \sqrt {a^2+b^2} d^2 e e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+\sqrt {a^2+b^2} d^2 e^c f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-2 \sqrt {a^2+b^2} d^2 e e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-\sqrt {a^2+b^2} d^2 e^c f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 \sqrt {a^2+b^2} d e^c f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-2 \sqrt {a^2+b^2} d e^c f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-2 \sqrt {a^2+b^2} e^c f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 \sqrt {a^2+b^2} e^c f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )}{a^2 \sqrt {a^2+b^2} d^3 \sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\frac {\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-e^2 \sinh \left (\frac {d x}{2}\right )-2 e f x \sinh \left (\frac {d x}{2}\right )-f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^2 \sinh \left (\frac {d x}{2}\right )+2 e f x \sinh \left (\frac {d x}{2}\right )+f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*a*d^2*e*E^(2*c)*f*x)/(-1 + E^(2*c)) - (2*a*d^2*E^(2*c)*f^2*x^2)/(-1 + E^(2*c)) + 2*b*d^2*e^2*ArcTanh[E^(c
+ d*x)] - 2*b*d^2*e*f*x*Log[1 - E^(c + d*x)] - b*d^2*f^2*x^2*Log[1 - E^(c + d*x)] + 2*b*d^2*e*f*x*Log[1 + E^(
c + d*x)] + b*d^2*f^2*x^2*Log[1 + E^(c + d*x)] + 2*a*d*e*f*Log[1 - E^(2*(c + d*x))] + 2*a*d*f^2*x*Log[1 - E^(2
*(c + d*x))] + 2*b*d*f*(e + f*x)*PolyLog[2, -E^(c + d*x)] - 2*b*d*f*(e + f*x)*PolyLog[2, E^(c + d*x)] + a*f^2*
PolyLog[2, E^(2*(c + d*x))] - 2*b*f^2*PolyLog[3, -E^(c + d*x)] + 2*b*f^2*PolyLog[3, E^(c + d*x)])/(a^2*d^3) +
(b^2*(-2*d^2*e^2*Sqrt[(a^2 + b^2)*E^(2*c)]*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*Sqrt[a^2 + b^2]*d^
2*e*E^c*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + Sqrt[a^2 + b^2]*d^2*E^c*f^2*x^2*L
og[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 2*Sqrt[a^2 + b^2]*d^2*e*E^c*f*x*Log[1 + (b*E^(
2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - Sqrt[a^2 + b^2]*d^2*E^c*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(
a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 2*Sqrt[a^2 + b^2]*d*E^c*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^
c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 2*Sqrt[a^2 + b^2]*d*E^c*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c +
Sqrt[(a^2 + b^2)*E^(2*c)]))] - 2*Sqrt[a^2 + b^2]*E^c*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 +
b^2)*E^(2*c)]))] + 2*Sqrt[a^2 + b^2]*E^c*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]
))]))/(a^2*Sqrt[a^2 + b^2]*d^3*Sqrt[(a^2 + b^2)*E^(2*c)]) + (Sech[c/2]*Sech[c/2 + (d*x)/2]*(-(e^2*Sinh[(d*x)/2
]) - 2*e*f*x*Sinh[(d*x)/2] - f^2*x^2*Sinh[(d*x)/2]))/(2*a*d) + (Csch[c/2]*Csch[c/2 + (d*x)/2]*(e^2*Sinh[(d*x)/
2] + 2*e*f*x*Sinh[(d*x)/2] + f^2*x^2*Sinh[(d*x)/2]))/(2*a*d)

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Maple [F]
time = 0.90, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \mathrm {csch}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

(b^2*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^2*d
) + b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d) + 2/((a*e^(-2*d*x - 2*c) - a)*d))*e^2 -
4*f*x*e/(a*d) - 2*(f^2*x^2 + 2*f*x*e)/(a*d*e^(2*d*x + 2*c) - a*d) + 2*f*e*log(e^(d*x + c) + 1)/(a*d^2) + 2*f*e
*log(e^(d*x + c) - 1)/(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)))*b*f^2/(a^2*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)))*b*f^2/(a^2*d^3) + 2*(b*d*f*e + a*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^2*d^3) - 2*(b*d
*f*e - a*f^2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))/(a^2*d^3) - 1/3*(b*d^3*f^2*x^3 + 3*(b*d*f*e + a
*f^2)*d^2*x^2)/(a^2*d^3) + 1/3*(b*d^3*f^2*x^3 + 3*(b*d*f*e - a*f^2)*d^2*x^2)/(a^2*d^3) + integrate(2*(b^2*f^2*
x^2*e^c + 2*b^2*f*x*e^(c + 1))*e^(d*x)/(a^2*b*e^(2*d*x + 2*c) + 2*a^3*e^(d*x + c) - a^2*b), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5257 vs. \(2 (505) = 1010\).
time = 0.45, size = 5257, normalized size = 9.83 \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)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-(2*(a^3 + a*b^2)*c^2*f^2 - 4*(a^3 + a*b^2)*c*d*f*cosh(1) + 2*(a^3 + a*b^2)*d^2*cosh(1)^2 + 2*(a^3 + a*b^2)*d^
2*sinh(1)^2 + 2*((a^3 + a*b^2)*d^2*f^2*x^2 - (a^3 + a*b^2)*c^2*f^2 + 2*((a^3 + a*b^2)*d^2*f*x + (a^3 + a*b^2)*
c*d*f)*cosh(1) + 2*((a^3 + a*b^2)*d^2*f*x + (a^3 + a*b^2)*c*d*f)*sinh(1))*cosh(d*x + c)^2 + 4*((a^3 + a*b^2)*d
^2*f^2*x^2 - (a^3 + a*b^2)*c^2*f^2 + 2*((a^3 + a*b^2)*d^2*f*x + (a^3 + a*b^2)*c*d*f)*cosh(1) + 2*((a^3 + a*b^2
)*d^2*f*x + (a^3 + a*b^2)*c*d*f)*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + 2*((a^3 + a*b^2)*d^2*f^2*x^2 - (a^3 +
a*b^2)*c^2*f^2 + 2*((a^3 + a*b^2)*d^2*f*x + (a^3 + a*b^2)*c*d*f)*cosh(1) + 2*((a^3 + a*b^2)*d^2*f*x + (a^3 + a
*b^2)*c*d*f)*sinh(1))*sinh(d*x + c)^2 + 2*(b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1) - (b^3*d*f^2*x + b^
3*d*f*cosh(1) + b^3*d*f*sinh(1))*cosh(d*x + c)^2 - 2*(b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*cosh(d*
x + c)*sinh(d*x + c) - (b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2
)*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^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1) - (b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*
cosh(d*x + c)^2 - 2*(b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*cosh(d*x + c)*sinh(d*x + c) - (b^3*d*f^2
*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*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) - (b^3*c^2*f^2 - 2*b^3*c*d*f
*cosh(1) + b^3*d^2*cosh(1)^2 + b^3*d^2*sinh(1)^2 - (b^3*c^2*f^2 - 2*b^3*c*d*f*cosh(1) + b^3*d^2*cosh(1)^2 + b^
3*d^2*sinh(1)^2 - 2*(b^3*c*d*f - b^3*d^2*cosh(1))*sinh(1))*cosh(d*x + c)^2 - 2*(b^3*c^2*f^2 - 2*b^3*c*d*f*cosh
(1) + b^3*d^2*cosh(1)^2 + b^3*d^2*sinh(1)^2 - 2*(b^3*c*d*f - b^3*d^2*cosh(1))*sinh(1))*cosh(d*x + c)*sinh(d*x
+ c) - (b^3*c^2*f^2 - 2*b^3*c*d*f*cosh(1) + b^3*d^2*cosh(1)^2 + b^3*d^2*sinh(1)^2 - 2*(b^3*c*d*f - b^3*d^2*cos
h(1))*sinh(1))*sinh(d*x + c)^2 - 2*(b^3*c*d*f - b^3*d^2*cosh(1))*sinh(1))*sqrt((a^2 + b^2)/b^2)*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^3*c^2*f^2 - 2*b^3*c*d*f*cosh(1) + b^3*d^2*
cosh(1)^2 + b^3*d^2*sinh(1)^2 - (b^3*c^2*f^2 - 2*b^3*c*d*f*cosh(1) + b^3*d^2*cosh(1)^2 + b^3*d^2*sinh(1)^2 - 2
*(b^3*c*d*f - b^3*d^2*cosh(1))*sinh(1))*cosh(d*x + c)^2 - 2*(b^3*c^2*f^2 - 2*b^3*c*d*f*cosh(1) + b^3*d^2*cosh(
1)^2 + b^3*d^2*sinh(1)^2 - 2*(b^3*c*d*f - b^3*d^2*cosh(1))*sinh(1))*cosh(d*x + c)*sinh(d*x + c) - (b^3*c^2*f^2
- 2*b^3*c*d*f*cosh(1) + b^3*d^2*cosh(1)^2 + b^3*d^2*sinh(1)^2 - 2*(b^3*c*d*f - b^3*d^2*cosh(1))*sinh(1))*sinh
(d*x + c)^2 - 2*(b^3*c*d*f - b^3*d^2*cosh(1))*sinh(1))*sqrt((a^2 + b^2)/b^2)*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^3*d^2*f^2*x^2 - b^3*c^2*f^2 - (b^3*d^2*f^2*x^2 - b^3*c^2*f^2
+ 2*(b^3*d^2*f*x + b^3*c*d*f)*cosh(1) + 2*(b^3*d^2*f*x + b^3*c*d*f)*sinh(1))*cosh(d*x + c)^2 - 2*(b^3*d^2*f^2*
x^2 - b^3*c^2*f^2 + 2*(b^3*d^2*f*x + b^3*c*d*f)*cosh(1) + 2*(b^3*d^2*f*x + b^3*c*d*f)*sinh(1))*cosh(d*x + c)*s
inh(d*x + c) - (b^3*d^2*f^2*x^2 - b^3*c^2*f^2 + 2*(b^3*d^2*f*x + b^3*c*d*f)*cosh(1) + 2*(b^3*d^2*f*x + b^3*c*d
*f)*sinh(1))*sinh(d*x + c)^2 + 2*(b^3*d^2*f*x + b^3*c*d*f)*cosh(1) + 2*(b^3*d^2*f*x + b^3*c*d*f)*sinh(1))*sqrt
((a^2 + b^2)/b^2)*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^3*d^2*f^2*x^2 - b^3*c^2*f^2 - (b^3*d^2*f^2*x^2 - b^3*c^2*f^2 + 2*(b^3*d^2*f*x + b^3*c*d*f
)*cosh(1) + 2*(b^3*d^2*f*x + b^3*c*d*f)*sinh(1))*cosh(d*x + c)^2 - 2*(b^3*d^2*f^2*x^2 - b^3*c^2*f^2 + 2*(b^3*d
^2*f*x + b^3*c*d*f)*cosh(1) + 2*(b^3*d^2*f*x + b^3*c*d*f)*sinh(1))*cosh(d*x + c)*sinh(d*x + c) - (b^3*d^2*f^2*
x^2 - b^3*c^2*f^2 + 2*(b^3*d^2*f*x + b^3*c*d*f)*cosh(1) + 2*(b^3*d^2*f*x + b^3*c*d*f)*sinh(1))*sinh(d*x + c)^2
+ 2*(b^3*d^2*f*x + b^3*c*d*f)*cosh(1) + 2*(b^3*d^2*f*x + b^3*c*d*f)*sinh(1))*sqrt((a^2 + b^2)/b^2)*log(-(a*co
sh(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) + 2*(b^3*f^2
*cosh(d*x + c)^2 + 2*b^3*f^2*cosh(d*x + c)*sinh(d*x + c) + b^3*f^2*sinh(d*x + c)^2 - b^3*f^2)*sqrt((a^2 + b^2)
/b^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^3*f^2*cosh(d*x + c)^2 + 2*b^3*f^2*cosh(d*x + c)*sinh(d*x + c) + b^3*f^2*sinh(d*x + c)^2 - b^3*f^2
)*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sq
rt((a^2 + b^2)/b^2))/b) - 2*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*f*cosh(1) + (a^2*b + b^3)*d*f*sinh(1) - (
a^3 + a*b^2)*f^2 - ((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*f*cosh(1) + (a^2*b + b^3)*d*f*sinh(1) - (a^3 + a*b
^2)*f^2)*cosh(d*x + c)^2 - 2*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*f*cosh(1) + (a^2*b + b^3)*d*f*sinh(1) -
(a^3 + a*b^2)*f^2)*cosh(d*x + c)*sinh(d*x + c) - ((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*f*cosh(1) + (a^2*b +
b^3)*d*f*sinh(1) - (a^3 + a*b^2)*f^2)*sinh(d*x...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right )^{2} \operatorname {csch}^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^2/(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 {sinh}\left (c+d\,x\right )}^2\,\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/(sinh(c + d*x)^2*(a + b*sinh(c + d*x))),x)

[Out]

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

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