3.79 \(\int \frac{\text{csch}^4(x)}{a+b \sinh (x)} \, dx\)
Optimal. Leaf size=109 \[ -\frac{2 b^4 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^4 \sqrt{a^2+b^2}}+\frac{\left (2 a^2-3 b^2\right ) \coth (x)}{3 a^3}-\frac{b \left (a^2-2 b^2\right ) \tanh ^{-1}(\cosh (x))}{2 a^4}+\frac{b \coth (x) \text{csch}(x)}{2 a^2}-\frac{\coth (x) \text{csch}^2(x)}{3 a} \]
[Out]
-(b*(a^2 - 2*b^2)*ArcTanh[Cosh[x]])/(2*a^4) - (2*b^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^4*Sqrt[a^2
+ b^2]) + ((2*a^2 - 3*b^2)*Coth[x])/(3*a^3) + (b*Coth[x]*Csch[x])/(2*a^2) - (Coth[x]*Csch[x]^2)/(3*a)
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Rubi [A] time = 0.491286, antiderivative size = 109, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used =
{2802, 3055, 3001, 3770, 2660, 618, 206} \[ -\frac{2 b^4 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^4 \sqrt{a^2+b^2}}+\frac{\left (2 a^2-3 b^2\right ) \coth (x)}{3 a^3}-\frac{b \left (a^2-2 b^2\right ) \tanh ^{-1}(\cosh (x))}{2 a^4}+\frac{b \coth (x) \text{csch}(x)}{2 a^2}-\frac{\coth (x) \text{csch}^2(x)}{3 a} \]
Antiderivative was successfully verified.
[In]
Int[Csch[x]^4/(a + b*Sinh[x]),x]
[Out]
-(b*(a^2 - 2*b^2)*ArcTanh[Cosh[x]])/(2*a^4) - (2*b^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^4*Sqrt[a^2
+ b^2]) + ((2*a^2 - 3*b^2)*Coth[x])/(3*a^3) + (b*Coth[x]*Csch[x])/(2*a^2) - (Coth[x]*Csch[x]^2)/(3*a)
Rule 2802
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 -
b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n
*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n
+ 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !
(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
Rule 3055
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))
Rule 3001
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rule 2660
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]
Rule 618
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{\text{csch}^4(x)}{a+b \sinh (x)} \, dx &=-\frac{\coth (x) \text{csch}^2(x)}{3 a}+\frac{i \int \frac{\text{csch}^3(x) \left (3 i b+2 i a \sinh (x)+2 i b \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{3 a}\\ &=\frac{b \coth (x) \text{csch}(x)}{2 a^2}-\frac{\coth (x) \text{csch}^2(x)}{3 a}-\frac{\int \frac{\text{csch}^2(x) \left (2 \left (2 a^2-3 b^2\right )+a b \sinh (x)-3 b^2 \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{6 a^2}\\ &=\frac{\left (2 a^2-3 b^2\right ) \coth (x)}{3 a^3}+\frac{b \coth (x) \text{csch}(x)}{2 a^2}-\frac{\coth (x) \text{csch}^2(x)}{3 a}-\frac{i \int \frac{\text{csch}(x) \left (3 i b \left (a^2-2 b^2\right )+3 i a b^2 \sinh (x)\right )}{a+b \sinh (x)} \, dx}{6 a^3}\\ &=\frac{\left (2 a^2-3 b^2\right ) \coth (x)}{3 a^3}+\frac{b \coth (x) \text{csch}(x)}{2 a^2}-\frac{\coth (x) \text{csch}^2(x)}{3 a}+\frac{b^4 \int \frac{1}{a+b \sinh (x)} \, dx}{a^4}+\frac{\left (b \left (a^2-2 b^2\right )\right ) \int \text{csch}(x) \, dx}{2 a^4}\\ &=-\frac{b \left (a^2-2 b^2\right ) \tanh ^{-1}(\cosh (x))}{2 a^4}+\frac{\left (2 a^2-3 b^2\right ) \coth (x)}{3 a^3}+\frac{b \coth (x) \text{csch}(x)}{2 a^2}-\frac{\coth (x) \text{csch}^2(x)}{3 a}+\frac{\left (2 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{a^4}\\ &=-\frac{b \left (a^2-2 b^2\right ) \tanh ^{-1}(\cosh (x))}{2 a^4}+\frac{\left (2 a^2-3 b^2\right ) \coth (x)}{3 a^3}+\frac{b \coth (x) \text{csch}(x)}{2 a^2}-\frac{\coth (x) \text{csch}^2(x)}{3 a}-\frac{\left (4 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac{x}{2}\right )\right )}{a^4}\\ &=-\frac{b \left (a^2-2 b^2\right ) \tanh ^{-1}(\cosh (x))}{2 a^4}-\frac{2 b^4 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^4 \sqrt{a^2+b^2}}+\frac{\left (2 a^2-3 b^2\right ) \coth (x)}{3 a^3}+\frac{b \coth (x) \text{csch}(x)}{2 a^2}-\frac{\coth (x) \text{csch}^2(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.934462, size = 186, normalized size = 1.71 \[ \frac{4 a \left (2 a^2-3 b^2\right ) \coth \left (\frac{x}{2}\right )+\frac{48 b^4 \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}+3 a^2 b \text{csch}^2\left (\frac{x}{2}\right )+3 a^2 b \text{sech}^2\left (\frac{x}{2}\right )+12 a^2 b \log \left (\tanh \left (\frac{x}{2}\right )\right )+8 a^3 \tanh \left (\frac{x}{2}\right )+8 a^3 \sinh ^4\left (\frac{x}{2}\right ) \text{csch}^3(x)-\frac{1}{2} a^3 \sinh (x) \text{csch}^4\left (\frac{x}{2}\right )-12 a b^2 \tanh \left (\frac{x}{2}\right )-24 b^3 \log \left (\tanh \left (\frac{x}{2}\right )\right )}{24 a^4} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[x]^4/(a + b*Sinh[x]),x]
[Out]
((48*b^4*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + 4*a*(2*a^2 - 3*b^2)*Coth[x/2] + 3*a^2*
b*Csch[x/2]^2 + 12*a^2*b*Log[Tanh[x/2]] - 24*b^3*Log[Tanh[x/2]] + 3*a^2*b*Sech[x/2]^2 + 8*a^3*Csch[x]^3*Sinh[x
/2]^4 - (a^3*Csch[x/2]^4*Sinh[x])/2 + 8*a^3*Tanh[x/2] - 12*a*b^2*Tanh[x/2])/(24*a^4)
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Maple [A] time = 0.031, size = 158, normalized size = 1.5 \begin{align*} -{\frac{1}{24\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{b}{8\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{3}{8\,a}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{{b}^{2}}{2\,{a}^{3}}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{24\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}+{\frac{3}{8\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{{b}^{2}}{2\,{a}^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{b}{8\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{b}{2\,{a}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{{b}^{3}}{{a}^{4}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+2\,{\frac{{b}^{4}}{{a}^{4}\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(x)^4/(a+b*sinh(x)),x)
[Out]
-1/24/a*tanh(1/2*x)^3-1/8/a^2*b*tanh(1/2*x)^2+3/8/a*tanh(1/2*x)-1/2/a^3*b^2*tanh(1/2*x)-1/24/a/tanh(1/2*x)^3+3
/8/a/tanh(1/2*x)-1/2/a^3/tanh(1/2*x)*b^2+1/8/a^2*b/tanh(1/2*x)^2+1/2/a^2*b*ln(tanh(1/2*x))-1/a^4*b^3*ln(tanh(1
/2*x))+2/a^4*b^4/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2))
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)^4/(a+b*sinh(x)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 3.06914, size = 3970, normalized size = 36.42 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)^4/(a+b*sinh(x)),x, algorithm="fricas")
[Out]
1/6*(6*(a^4*b + a^2*b^3)*cosh(x)^5 + 6*(a^4*b + a^2*b^3)*sinh(x)^5 + 8*a^5 - 4*a^3*b^2 - 12*a*b^4 - 12*(a^3*b^
2 + a*b^4)*cosh(x)^4 - 6*(2*a^3*b^2 + 2*a*b^4 - 5*(a^4*b + a^2*b^3)*cosh(x))*sinh(x)^4 + 12*(5*(a^4*b + a^2*b^
3)*cosh(x)^2 - 4*(a^3*b^2 + a*b^4)*cosh(x))*sinh(x)^3 - 24*(a^5 - a*b^4)*cosh(x)^2 - 12*(2*a^5 - 2*a*b^4 - 5*(
a^4*b + a^2*b^3)*cosh(x)^3 + 6*(a^3*b^2 + a*b^4)*cosh(x)^2)*sinh(x)^2 + 6*(b^4*cosh(x)^6 + 6*b^4*cosh(x)*sinh(
x)^5 + b^4*sinh(x)^6 - 3*b^4*cosh(x)^4 + 3*b^4*cosh(x)^2 + 3*(5*b^4*cosh(x)^2 - b^4)*sinh(x)^4 - b^4 + 4*(5*b^
4*cosh(x)^3 - 3*b^4*cosh(x))*sinh(x)^3 + 3*(5*b^4*cosh(x)^4 - 6*b^4*cosh(x)^2 + b^4)*sinh(x)^2 + 6*(b^4*cosh(x
)^5 - 2*b^4*cosh(x)^3 + b^4*cosh(x))*sinh(x))*sqrt(a^2 + b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(
x) + 2*a^2 + b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) - 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2
+ b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) - b)) - 6*(a^4*b + a^2*b^3)*cosh(x) - 3*((a^4*b - a^2
*b^3 - 2*b^5)*cosh(x)^6 + 6*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)*sinh(x)^5 + (a^4*b - a^2*b^3 - 2*b^5)*sinh(x)^6
- a^4*b + a^2*b^3 + 2*b^5 - 3*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^4 - 3*(a^4*b - a^2*b^3 - 2*b^5 - 5*(a^4*b - a^
2*b^3 - 2*b^5)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^3 - 3*(a^4*b - a^2*b^3 - 2*b^5)*c
osh(x))*sinh(x)^3 + 3*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^2 + 3*(a^4*b - a^2*b^3 - 2*b^5 + 5*(a^4*b - a^2*b^3 -
2*b^5)*cosh(x)^4 - 6*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^2)*sinh(x)^2 + 6*((a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^5 -
2*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^3 + (a^4*b - a^2*b^3 - 2*b^5)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1
) + 3*((a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^6 + 6*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)*sinh(x)^5 + (a^4*b - a^2*b^3
- 2*b^5)*sinh(x)^6 - a^4*b + a^2*b^3 + 2*b^5 - 3*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^4 - 3*(a^4*b - a^2*b^3 - 2*
b^5 - 5*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^3 - 3*(a^4*b -
a^2*b^3 - 2*b^5)*cosh(x))*sinh(x)^3 + 3*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^2 + 3*(a^4*b - a^2*b^3 - 2*b^5 + 5*
(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^4 - 6*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^2)*sinh(x)^2 + 6*((a^4*b - a^2*b^3 -
2*b^5)*cosh(x)^5 - 2*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^3 + (a^4*b - a^2*b^3 - 2*b^5)*cosh(x))*sinh(x))*log(co
sh(x) + sinh(x) - 1) - 6*(a^4*b + a^2*b^3 - 5*(a^4*b + a^2*b^3)*cosh(x)^4 + 8*(a^3*b^2 + a*b^4)*cosh(x)^3 + 8*
(a^5 - a*b^4)*cosh(x))*sinh(x))/((a^6 + a^4*b^2)*cosh(x)^6 + 6*(a^6 + a^4*b^2)*cosh(x)*sinh(x)^5 + (a^6 + a^4*
b^2)*sinh(x)^6 - a^6 - a^4*b^2 - 3*(a^6 + a^4*b^2)*cosh(x)^4 - 3*(a^6 + a^4*b^2 - 5*(a^6 + a^4*b^2)*cosh(x)^2)
*sinh(x)^4 + 4*(5*(a^6 + a^4*b^2)*cosh(x)^3 - 3*(a^6 + a^4*b^2)*cosh(x))*sinh(x)^3 + 3*(a^6 + a^4*b^2)*cosh(x)
^2 + 3*(a^6 + a^4*b^2 + 5*(a^6 + a^4*b^2)*cosh(x)^4 - 6*(a^6 + a^4*b^2)*cosh(x)^2)*sinh(x)^2 + 6*((a^6 + a^4*b
^2)*cosh(x)^5 - 2*(a^6 + a^4*b^2)*cosh(x)^3 + (a^6 + a^4*b^2)*cosh(x))*sinh(x))
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)**4/(a+b*sinh(x)),x)
[Out]
Timed out
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Giac [A] time = 1.44821, size = 231, normalized size = 2.12 \begin{align*} \frac{b^{4} \log \left (\frac{{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}} a^{4}} - \frac{{\left (a^{2} b - 2 \, b^{3}\right )} \log \left (e^{x} + 1\right )}{2 \, a^{4}} + \frac{{\left (a^{2} b - 2 \, b^{3}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{2 \, a^{4}} + \frac{3 \, a b e^{\left (5 \, x\right )} - 6 \, b^{2} e^{\left (4 \, x\right )} - 12 \, a^{2} e^{\left (2 \, x\right )} + 12 \, b^{2} e^{\left (2 \, x\right )} - 3 \, a b e^{x} + 4 \, a^{2} - 6 \, b^{2}}{3 \, a^{3}{\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)^4/(a+b*sinh(x)),x, algorithm="giac")
[Out]
b^4*log(abs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^4) -
1/2*(a^2*b - 2*b^3)*log(e^x + 1)/a^4 + 1/2*(a^2*b - 2*b^3)*log(abs(e^x - 1))/a^4 + 1/3*(3*a*b*e^(5*x) - 6*b^2
*e^(4*x) - 12*a^2*e^(2*x) + 12*b^2*e^(2*x) - 3*a*b*e^x + 4*a^2 - 6*b^2)/(a^3*(e^(2*x) - 1)^3)