3.87 \(\int \frac{\text{csch}^4(x)}{(a+b \sinh (x))^2} \, dx\)

Optimal. Leaf size=198 \[ -\frac{2 b^4 \left (5 a^2+4 b^2\right ) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^5 \left (a^2+b^2\right )^{3/2}}+\frac{\left (-7 a^2 b^2+2 a^4-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}-\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac{\left (a^2+4 b^2\right ) \coth (x) \text{csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac{b \left (a^2+2 b^2\right ) \coth (x) \text{csch}(x)}{a^3 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))} \]

[Out]

-((b*(a^2 - 4*b^2)*ArcTanh[Cosh[x]])/a^5) - (2*b^4*(5*a^2 + 4*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])
/(a^5*(a^2 + b^2)^(3/2)) + ((2*a^4 - 7*a^2*b^2 - 12*b^4)*Coth[x])/(3*a^4*(a^2 + b^2)) + (b*(a^2 + 2*b^2)*Coth[
x]*Csch[x])/(a^3*(a^2 + b^2)) - ((a^2 + 4*b^2)*Coth[x]*Csch[x]^2)/(3*a^2*(a^2 + b^2)) + (b^2*Coth[x]*Csch[x]^2
)/(a*(a^2 + b^2)*(a + b*Sinh[x]))

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Rubi [A]  time = 0.883786, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{2 b^4 \left (5 a^2+4 b^2\right ) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^5 \left (a^2+b^2\right )^{3/2}}+\frac{\left (-7 a^2 b^2+2 a^4-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}-\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac{\left (a^2+4 b^2\right ) \coth (x) \text{csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac{b \left (a^2+2 b^2\right ) \coth (x) \text{csch}(x)}{a^3 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^4/(a + b*Sinh[x])^2,x]

[Out]

-((b*(a^2 - 4*b^2)*ArcTanh[Cosh[x]])/a^5) - (2*b^4*(5*a^2 + 4*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])
/(a^5*(a^2 + b^2)^(3/2)) + ((2*a^4 - 7*a^2*b^2 - 12*b^4)*Coth[x])/(3*a^4*(a^2 + b^2)) + (b*(a^2 + 2*b^2)*Coth[
x]*Csch[x])/(a^3*(a^2 + b^2)) - ((a^2 + 4*b^2)*Coth[x]*Csch[x]^2)/(3*a^2*(a^2 + b^2)) + (b^2*Coth[x]*Csch[x]^2
)/(a*(a^2 + b^2)*(a + b*Sinh[x]))

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))^2} \, dx &=\frac{b^2 \coth (x) \text{csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{\int \frac{\text{csch}^4(x) \left (a^2+4 b^2-a b \sinh (x)+3 b^2 \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{a \left (a^2+b^2\right )}\\ &=-\frac{\left (a^2+4 b^2\right ) \coth (x) \text{csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{i \int \frac{\text{csch}^3(x) \left (6 i b \left (a^2+2 b^2\right )+i a \left (2 a^2-b^2\right ) \sinh (x)+2 i b \left (a^2+4 b^2\right ) \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{3 a^2 \left (a^2+b^2\right )}\\ &=\frac{b \left (a^2+2 b^2\right ) \coth (x) \text{csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac{\left (a^2+4 b^2\right ) \coth (x) \text{csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac{\int \frac{\text{csch}^2(x) \left (2 \left (2 a^4-7 a^2 b^2-12 b^4\right )-2 a b \left (a^2-2 b^2\right ) \sinh (x)-6 b^2 \left (a^2+2 b^2\right ) \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{6 a^3 \left (a^2+b^2\right )}\\ &=\frac{\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac{b \left (a^2+2 b^2\right ) \coth (x) \text{csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac{\left (a^2+4 b^2\right ) \coth (x) \text{csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac{i \int \frac{\text{csch}(x) \left (6 i b \left (a^4-3 a^2 b^2-4 b^4\right )+6 i a b^2 \left (a^2+2 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)} \, dx}{6 a^4 \left (a^2+b^2\right )}\\ &=\frac{\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac{b \left (a^2+2 b^2\right ) \coth (x) \text{csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac{\left (a^2+4 b^2\right ) \coth (x) \text{csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{\left (b \left (a^2-4 b^2\right )\right ) \int \text{csch}(x) \, dx}{a^5}+\frac{\left (b^4 \left (5 a^2+4 b^2\right )\right ) \int \frac{1}{a+b \sinh (x)} \, dx}{a^5 \left (a^2+b^2\right )}\\ &=-\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}+\frac{\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac{b \left (a^2+2 b^2\right ) \coth (x) \text{csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac{\left (a^2+4 b^2\right ) \coth (x) \text{csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{\left (2 b^4 \left (5 a^2+4 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{a^5 \left (a^2+b^2\right )}\\ &=-\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}+\frac{\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac{b \left (a^2+2 b^2\right ) \coth (x) \text{csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac{\left (a^2+4 b^2\right ) \coth (x) \text{csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac{\left (4 b^4 \left (5 a^2+4 b^2\right )\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^5 \left (a^2+b^2\right )}\\ &=-\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac{2 b^4 \left (5 a^2+4 b^2\right ) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^5 \left (a^2+b^2\right )^{3/2}}+\frac{\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac{b \left (a^2+2 b^2\right ) \coth (x) \text{csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac{\left (a^2+4 b^2\right ) \coth (x) \text{csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\\ \end{align*}

Mathematica [A]  time = 0.955536, size = 214, normalized size = 1.08 \[ \frac{4 a \left (2 a^2-9 b^2\right ) \tanh \left (\frac{x}{2}\right )+4 a \left (2 a^2-9 b^2\right ) \coth \left (\frac{x}{2}\right )-\frac{48 b^4 \left (5 a^2+4 b^2\right ) \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}-\frac{24 a b^5 \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+6 a^2 b \text{csch}^2\left (\frac{x}{2}\right )+6 a^2 b \text{sech}^2\left (\frac{x}{2}\right )+8 a^3 \sinh ^4\left (\frac{x}{2}\right ) \text{csch}^3(x)-\frac{1}{2} a^3 \sinh (x) \text{csch}^4\left (\frac{x}{2}\right )+24 b (a-2 b) (a+2 b) \log \left (\tanh \left (\frac{x}{2}\right )\right )}{24 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^4/(a + b*Sinh[x])^2,x]

[Out]

((-48*b^4*(5*a^2 + 4*b^2)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/(-a^2 - b^2)^(3/2) + 4*a*(2*a^2 - 9*b^2)
*Coth[x/2] + 6*a^2*b*Csch[x/2]^2 + 24*(a - 2*b)*b*(a + 2*b)*Log[Tanh[x/2]] + 6*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 - (24*a*b^5*Cosh[x])/((a^2 + b^2)*(a + b*Sinh[x])) + 4*a*(2*a^2
- 9*b^2)*Tanh[x/2])/(24*a^5)

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Maple [A]  time = 0.052, size = 277, normalized size = 1.4 \begin{align*} -{\frac{1}{24\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{b}{4\,{a}^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{3}{8\,{a}^{2}}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{3\,{b}^{2}}{2\,{a}^{4}}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{24\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}+{\frac{3}{8\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{3\,{b}^{2}}{2\,{a}^{4}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{b}{4\,{a}^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{b}{{a}^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-4\,{\frac{{b}^{3}\ln \left ( \tanh \left ( x/2 \right ) \right ) }{{a}^{5}}}+2\,{\frac{{b}^{6}\tanh \left ( x/2 \right ) }{{a}^{5} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) \left ({a}^{2}+{b}^{2} \right ) }}+2\,{\frac{{b}^{5}}{{a}^{4} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) \left ({a}^{2}+{b}^{2} \right ) }}+10\,{\frac{{b}^{4}}{{a}^{3} \left ({a}^{2}+{b}^{2} \right ) ^{3/2}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }+8\,{\frac{{b}^{6}}{{a}^{5} \left ({a}^{2}+{b}^{2} \right ) ^{3/2}}{\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))^2,x)

[Out]

-1/24/a^2*tanh(1/2*x)^3-1/4/a^3*b*tanh(1/2*x)^2+3/8/a^2*tanh(1/2*x)-3/2/a^4*b^2*tanh(1/2*x)-1/24/a^2/tanh(1/2*
x)^3+3/8/a^2/tanh(1/2*x)-3/2/a^4/tanh(1/2*x)*b^2+1/4/a^3*b/tanh(1/2*x)^2+1/a^3*b*ln(tanh(1/2*x))-4/a^5*b^3*ln(
tanh(1/2*x))+2/a^5*b^6/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)/(a^2+b^2)*tanh(1/2*x)+2/a^4*b^5/(a*tanh(1/2*x)^2-2*
tanh(1/2*x)*b-a)/(a^2+b^2)+10/a^3*b^4/(a^2+b^2)^(3/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2))+8/a^5
*b^6/(a^2+b^2)^(3/2)*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))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.53278, size = 14660, normalized size = 74.04 \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))^2,x, algorithm="fricas")

[Out]

-1/3*(4*a^7*b - 10*a^5*b^3 - 38*a^3*b^5 - 24*a*b^7 - 6*(a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6)*cosh(x)^7 - 6*(a^6*b^
2 + 3*a^4*b^4 + 2*a^2*b^6)*sinh(x)^7 - 6*(2*a^7*b + a^5*b^3 - 5*a^3*b^5 - 4*a*b^7)*cosh(x)^6 - 6*(2*a^7*b + a^
5*b^3 - 5*a^3*b^5 - 4*a*b^7 + 7*(a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6)*cosh(x))*sinh(x)^6 + 6*(7*a^6*b^2 + 17*a^4*b
^4 + 10*a^2*b^6)*cosh(x)^5 + 6*(7*a^6*b^2 + 17*a^4*b^4 + 10*a^2*b^6 - 21*(a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6)*cos
h(x)^2 - 6*(2*a^7*b + a^5*b^3 - 5*a^3*b^5 - 4*a*b^7)*cosh(x))*sinh(x)^5 + 6*(2*a^7*b - 5*a^5*b^3 - 19*a^3*b^5
- 12*a*b^7)*cosh(x)^4 + 6*(2*a^7*b - 5*a^5*b^3 - 19*a^3*b^5 - 12*a*b^7 - 35*(a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6)*
cosh(x)^3 - 15*(2*a^7*b + a^5*b^3 - 5*a^3*b^5 - 4*a*b^7)*cosh(x)^2 + 5*(7*a^6*b^2 + 17*a^4*b^4 + 10*a^2*b^6)*c
osh(x))*sinh(x)^4 + 6*(4*a^8 - 3*a^6*b^2 - 21*a^4*b^4 - 14*a^2*b^6)*cosh(x)^3 + 6*(4*a^8 - 3*a^6*b^2 - 21*a^4*
b^4 - 14*a^2*b^6 - 35*(a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6)*cosh(x)^4 - 20*(2*a^7*b + a^5*b^3 - 5*a^3*b^5 - 4*a*b^
7)*cosh(x)^3 + 10*(7*a^6*b^2 + 17*a^4*b^4 + 10*a^2*b^6)*cosh(x)^2 + 4*(2*a^7*b - 5*a^5*b^3 - 19*a^3*b^5 - 12*a
*b^7)*cosh(x))*sinh(x)^3 - 2*(2*a^7*b - 23*a^5*b^3 - 61*a^3*b^5 - 36*a*b^7)*cosh(x)^2 - 2*(2*a^7*b - 23*a^5*b^
3 - 61*a^3*b^5 - 36*a*b^7 + 63*(a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6)*cosh(x)^5 + 45*(2*a^7*b + a^5*b^3 - 5*a^3*b^5
- 4*a*b^7)*cosh(x)^4 - 30*(7*a^6*b^2 + 17*a^4*b^4 + 10*a^2*b^6)*cosh(x)^3 - 18*(2*a^7*b - 5*a^5*b^3 - 19*a^3*
b^5 - 12*a*b^7)*cosh(x)^2 - 9*(4*a^8 - 3*a^6*b^2 - 21*a^4*b^4 - 14*a^2*b^6)*cosh(x))*sinh(x)^2 - 3*((5*a^2*b^5
+ 4*b^7)*cosh(x)^8 + (5*a^2*b^5 + 4*b^7)*sinh(x)^8 + 2*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^7 + 2*(5*a^3*b^4 + 4*a*b
^6 + 4*(5*a^2*b^5 + 4*b^7)*cosh(x))*sinh(x)^7 + 5*a^2*b^5 + 4*b^7 - 4*(5*a^2*b^5 + 4*b^7)*cosh(x)^6 - 2*(10*a^
2*b^5 + 8*b^7 - 14*(5*a^2*b^5 + 4*b^7)*cosh(x)^2 - 7*(5*a^3*b^4 + 4*a*b^6)*cosh(x))*sinh(x)^6 - 6*(5*a^3*b^4 +
4*a*b^6)*cosh(x)^5 - 2*(15*a^3*b^4 + 12*a*b^6 - 28*(5*a^2*b^5 + 4*b^7)*cosh(x)^3 - 21*(5*a^3*b^4 + 4*a*b^6)*c
osh(x)^2 + 12*(5*a^2*b^5 + 4*b^7)*cosh(x))*sinh(x)^5 + 6*(5*a^2*b^5 + 4*b^7)*cosh(x)^4 + 2*(15*a^2*b^5 + 12*b^
7 + 35*(5*a^2*b^5 + 4*b^7)*cosh(x)^4 + 35*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^3 - 30*(5*a^2*b^5 + 4*b^7)*cosh(x)^2 -
15*(5*a^3*b^4 + 4*a*b^6)*cosh(x))*sinh(x)^4 + 6*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^3 + 2*(15*a^3*b^4 + 12*a*b^6 +
28*(5*a^2*b^5 + 4*b^7)*cosh(x)^5 + 35*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^4 - 40*(5*a^2*b^5 + 4*b^7)*cosh(x)^3 - 30*
(5*a^3*b^4 + 4*a*b^6)*cosh(x)^2 + 12*(5*a^2*b^5 + 4*b^7)*cosh(x))*sinh(x)^3 - 4*(5*a^2*b^5 + 4*b^7)*cosh(x)^2
- 2*(10*a^2*b^5 + 8*b^7 - 14*(5*a^2*b^5 + 4*b^7)*cosh(x)^6 - 21*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^5 + 30*(5*a^2*b^
5 + 4*b^7)*cosh(x)^4 + 30*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^3 - 18*(5*a^2*b^5 + 4*b^7)*cosh(x)^2 - 9*(5*a^3*b^4 +
4*a*b^6)*cosh(x))*sinh(x)^2 - 2*(5*a^3*b^4 + 4*a*b^6)*cosh(x) + 2*(4*(5*a^2*b^5 + 4*b^7)*cosh(x)^7 - 5*a^3*b^4
- 4*a*b^6 + 7*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^6 - 12*(5*a^2*b^5 + 4*b^7)*cosh(x)^5 - 15*(5*a^3*b^4 + 4*a*b^6)*c
osh(x)^4 + 12*(5*a^2*b^5 + 4*b^7)*cosh(x)^3 + 9*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^2 - 4*(5*a^2*b^5 + 4*b^7)*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)) - 2*(4*a^8 - 7*a^6*b^2 - 29*a^4*b^4 - 18*a^2*b^6)*cosh(x) + 3*((a^6*b^2 - 2*a^4*b
^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^8 + (a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*sinh(x)^8 + a^6*b^2 - 2*a^4*b^4
- 7*a^2*b^6 - 4*b^8 + 2*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^7 + 2*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5
- 4*a*b^7 + 4*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x))*sinh(x)^7 - 4*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b
^6 - 4*b^8)*cosh(x)^6 - 2*(2*a^6*b^2 - 4*a^4*b^4 - 14*a^2*b^6 - 8*b^8 - 14*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 -
4*b^8)*cosh(x)^2 - 7*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x))*sinh(x)^6 - 6*(a^7*b - 2*a^5*b^3 - 7*a
^3*b^5 - 4*a*b^7)*cosh(x)^5 - 2*(3*a^7*b - 6*a^5*b^3 - 21*a^3*b^5 - 12*a*b^7 - 28*(a^6*b^2 - 2*a^4*b^4 - 7*a^2
*b^6 - 4*b^8)*cosh(x)^3 - 21*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^2 + 12*(a^6*b^2 - 2*a^4*b^4 - 7
*a^2*b^6 - 4*b^8)*cosh(x))*sinh(x)^5 + 6*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^4 + 2*(3*a^6*b^2 -
6*a^4*b^4 - 21*a^2*b^6 - 12*b^8 + 35*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^4 + 35*(a^7*b - 2*a^5*b
^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^3 - 30*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^2 - 15*(a^7*b - 2*a
^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x))*sinh(x)^4 + 6*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^3 + 2*(
3*a^7*b - 6*a^5*b^3 - 21*a^3*b^5 - 12*a*b^7 + 28*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^5 + 35*(a^7
*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^4 - 40*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^3 - 30*
(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^2 + 12*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x))*si
nh(x)^3 - 4*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^2 - 2*(2*a^6*b^2 - 4*a^4*b^4 - 14*a^2*b^6 - 8*b^
8 - 14*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^6 - 21*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh
(x)^5 + 30*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^4 + 30*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*
cosh(x)^3 - 18*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^2 - 9*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^
7)*cosh(x))*sinh(x)^2 - 2*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x) - 2*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5
- 4*a*b^7 - 4*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^7 - 7*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^
7)*cosh(x)^6 + 12*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^5 + 15*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*
a*b^7)*cosh(x)^4 - 12*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^3 - 9*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 -
4*a*b^7)*cosh(x)^2 + 4*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1)
- 3*((a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^8 + (a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*sinh(x)^
8 + a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8 + 2*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^7 + 2*(a^7*b
- 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7 + 4*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x))*sinh(x)^7 - 4*(a^6*b
^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^6 - 2*(2*a^6*b^2 - 4*a^4*b^4 - 14*a^2*b^6 - 8*b^8 - 14*(a^6*b^2 -
2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^2 - 7*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x))*sinh(x)^6 - 6*
(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^5 - 2*(3*a^7*b - 6*a^5*b^3 - 21*a^3*b^5 - 12*a*b^7 - 28*(a^6
*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^3 - 21*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^2 + 12*
(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x))*sinh(x)^5 + 6*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cos
h(x)^4 + 2*(3*a^6*b^2 - 6*a^4*b^4 - 21*a^2*b^6 - 12*b^8 + 35*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)
^4 + 35*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^3 - 30*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cos
h(x)^2 - 15*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x))*sinh(x)^4 + 6*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 -
4*a*b^7)*cosh(x)^3 + 2*(3*a^7*b - 6*a^5*b^3 - 21*a^3*b^5 - 12*a*b^7 + 28*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*
b^8)*cosh(x)^5 + 35*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^4 - 40*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6
- 4*b^8)*cosh(x)^3 - 30*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^2 + 12*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*
b^6 - 4*b^8)*cosh(x))*sinh(x)^3 - 4*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^2 - 2*(2*a^6*b^2 - 4*a^4
*b^4 - 14*a^2*b^6 - 8*b^8 - 14*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^6 - 21*(a^7*b - 2*a^5*b^3 - 7
*a^3*b^5 - 4*a*b^7)*cosh(x)^5 + 30*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^4 + 30*(a^7*b - 2*a^5*b^3
- 7*a^3*b^5 - 4*a*b^7)*cosh(x)^3 - 18*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^2 - 9*(a^7*b - 2*a^5*
b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x))*sinh(x)^2 - 2*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x) - 2*(a^7*b
- 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7 - 4*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^7 - 7*(a^7*b - 2*a^5*
b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^6 + 12*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^5 + 15*(a^7*b - 2*
a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^4 - 12*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^3 - 9*(a^7*b -
2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^2 + 4*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x))*sinh(x))*lo
g(cosh(x) + sinh(x) - 1) - 2*(4*a^8 - 7*a^6*b^2 - 29*a^4*b^4 - 18*a^2*b^6 + 21*(a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^
6)*cosh(x)^6 + 18*(2*a^7*b + a^5*b^3 - 5*a^3*b^5 - 4*a*b^7)*cosh(x)^5 - 15*(7*a^6*b^2 + 17*a^4*b^4 + 10*a^2*b^
6)*cosh(x)^4 - 12*(2*a^7*b - 5*a^5*b^3 - 19*a^3*b^5 - 12*a*b^7)*cosh(x)^3 - 9*(4*a^8 - 3*a^6*b^2 - 21*a^4*b^4
- 14*a^2*b^6)*cosh(x)^2 + 2*(2*a^7*b - 23*a^5*b^3 - 61*a^3*b^5 - 36*a*b^7)*cosh(x))*sinh(x))/(a^9*b + 2*a^7*b^
3 + a^5*b^5 + (a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x)^8 + (a^9*b + 2*a^7*b^3 + a^5*b^5)*sinh(x)^8 + 2*(a^10 + 2*
a^8*b^2 + a^6*b^4)*cosh(x)^7 + 2*(a^10 + 2*a^8*b^2 + a^6*b^4 + 4*(a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x))*sinh(x
)^7 - 4*(a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x)^6 - 2*(2*a^9*b + 4*a^7*b^3 + 2*a^5*b^5 - 14*(a^9*b + 2*a^7*b^3 +
a^5*b^5)*cosh(x)^2 - 7*(a^10 + 2*a^8*b^2 + a^6*b^4)*cosh(x))*sinh(x)^6 - 6*(a^10 + 2*a^8*b^2 + a^6*b^4)*cosh(
x)^5 - 2*(3*a^10 + 6*a^8*b^2 + 3*a^6*b^4 - 28*(a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x)^3 - 21*(a^10 + 2*a^8*b^2 +
a^6*b^4)*cosh(x)^2 + 12*(a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x))*sinh(x)^5 + 6*(a^9*b + 2*a^7*b^3 + a^5*b^5)*co
sh(x)^4 + 2*(3*a^9*b + 6*a^7*b^3 + 3*a^5*b^5 + 35*(a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x)^4 + 35*(a^10 + 2*a^8*b
^2 + a^6*b^4)*cosh(x)^3 - 30*(a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x)^2 - 15*(a^10 + 2*a^8*b^2 + a^6*b^4)*cosh(x)
)*sinh(x)^4 + 6*(a^10 + 2*a^8*b^2 + a^6*b^4)*cosh(x)^3 + 2*(3*a^10 + 6*a^8*b^2 + 3*a^6*b^4 + 28*(a^9*b + 2*a^7
*b^3 + a^5*b^5)*cosh(x)^5 + 35*(a^10 + 2*a^8*b^2 + a^6*b^4)*cosh(x)^4 - 40*(a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(
x)^3 - 30*(a^10 + 2*a^8*b^2 + a^6*b^4)*cosh(x)^2 + 12*(a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x))*sinh(x)^3 - 4*(a^
9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x)^2 - 2*(2*a^9*b + 4*a^7*b^3 + 2*a^5*b^5 - 14*(a^9*b + 2*a^7*b^3 + a^5*b^5)*c
osh(x)^6 - 21*(a^10 + 2*a^8*b^2 + a^6*b^4)*cosh(x)^5 + 30*(a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x)^4 + 30*(a^10 +
2*a^8*b^2 + a^6*b^4)*cosh(x)^3 - 18*(a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x)^2 - 9*(a^10 + 2*a^8*b^2 + a^6*b^4)*
cosh(x))*sinh(x)^2 - 2*(a^10 + 2*a^8*b^2 + a^6*b^4)*cosh(x) - 2*(a^10 + 2*a^8*b^2 + a^6*b^4 - 4*(a^9*b + 2*a^7
*b^3 + a^5*b^5)*cosh(x)^7 - 7*(a^10 + 2*a^8*b^2 + a^6*b^4)*cosh(x)^6 + 12*(a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x
)^5 + 15*(a^10 + 2*a^8*b^2 + a^6*b^4)*cosh(x)^4 - 12*(a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x)^3 - 9*(a^10 + 2*a^8
*b^2 + a^6*b^4)*cosh(x)^2 + 4*(a^9*b + 2*a^7*b^3 + a^5*b^5)*cosh(x))*sinh(x))

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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))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.85886, size = 319, normalized size = 1.61 \begin{align*} \frac{{\left (5 \, a^{2} b^{4} + 4 \, b^{6}\right )} \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 )}{{\left (a^{7} + a^{5} b^{2}\right )} \sqrt{a^{2} + b^{2}}} + \frac{2 \,{\left (a b^{4} e^{x} - b^{5}\right )}}{{\left (a^{6} + a^{4} b^{2}\right )}{\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} - \frac{{\left (a^{2} b - 4 \, b^{3}\right )} \log \left (e^{x} + 1\right )}{a^{5}} + \frac{{\left (a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{a^{5}} + \frac{2 \,{\left (3 \, a b e^{\left (5 \, x\right )} - 9 \, b^{2} e^{\left (4 \, x\right )} - 6 \, a^{2} e^{\left (2 \, x\right )} + 18 \, b^{2} e^{\left (2 \, x\right )} - 3 \, a b e^{x} + 2 \, a^{2} - 9 \, b^{2}\right )}}{3 \, a^{4}{\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))^2,x, algorithm="giac")

[Out]

(5*a^2*b^4 + 4*b^6)*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)))/((a^7 +
a^5*b^2)*sqrt(a^2 + b^2)) + 2*(a*b^4*e^x - b^5)/((a^6 + a^4*b^2)*(b*e^(2*x) + 2*a*e^x - b)) - (a^2*b - 4*b^3)
*log(e^x + 1)/a^5 + (a^2*b - 4*b^3)*log(abs(e^x - 1))/a^5 + 2/3*(3*a*b*e^(5*x) - 9*b^2*e^(4*x) - 6*a^2*e^(2*x)
+ 18*b^2*e^(2*x) - 3*a*b*e^x + 2*a^2 - 9*b^2)/(a^4*(e^(2*x) - 1)^3)