3.86 \(\int \frac{\text{csch}^3(x)}{(a+b \sinh (x))^2} \, dx\)

Optimal. Leaf size=158 \[ \frac{2 b^3 \left (4 a^2+3 b^2\right ) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^4 \left (a^2+b^2\right )^{3/2}}+\frac{b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}+\frac{\left (a^2-6 b^2\right ) \tanh ^{-1}(\cosh (x))}{2 a^4}-\frac{\left (a^2+3 b^2\right ) \coth (x) \text{csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))} \]

[Out]

((a^2 - 6*b^2)*ArcTanh[Cosh[x]])/(2*a^4) + (2*b^3*(4*a^2 + 3*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/
(a^4*(a^2 + b^2)^(3/2)) + (b*(2*a^2 + 3*b^2)*Coth[x])/(a^3*(a^2 + b^2)) - ((a^2 + 3*b^2)*Coth[x]*Csch[x])/(2*a
^2*(a^2 + b^2)) + (b^2*Coth[x]*Csch[x])/(a*(a^2 + b^2)*(a + b*Sinh[x]))

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Rubi [A]  time = 0.675246, antiderivative size = 158, 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} \[ \frac{2 b^3 \left (4 a^2+3 b^2\right ) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^4 \left (a^2+b^2\right )^{3/2}}+\frac{b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}+\frac{\left (a^2-6 b^2\right ) \tanh ^{-1}(\cosh (x))}{2 a^4}-\frac{\left (a^2+3 b^2\right ) \coth (x) \text{csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^3/(a + b*Sinh[x])^2,x]

[Out]

((a^2 - 6*b^2)*ArcTanh[Cosh[x]])/(2*a^4) + (2*b^3*(4*a^2 + 3*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/
(a^4*(a^2 + b^2)^(3/2)) + (b*(2*a^2 + 3*b^2)*Coth[x])/(a^3*(a^2 + b^2)) - ((a^2 + 3*b^2)*Coth[x]*Csch[x])/(2*a
^2*(a^2 + b^2)) + (b^2*Coth[x]*Csch[x])/(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}^3(x)}{(a+b \sinh (x))^2} \, dx &=\frac{b^2 \coth (x) \text{csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{\int \frac{\text{csch}^3(x) \left (a^2+3 b^2-a b \sinh (x)+2 b^2 \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{a \left (a^2+b^2\right )}\\ &=-\frac{\left (a^2+3 b^2\right ) \coth (x) \text{csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{i \int \frac{\text{csch}^2(x) \left (2 i b \left (2 a^2+3 b^2\right )+i a \left (a^2-b^2\right ) \sinh (x)+i b \left (a^2+3 b^2\right ) \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{2 a^2 \left (a^2+b^2\right )}\\ &=\frac{b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}-\frac{\left (a^2+3 b^2\right ) \coth (x) \text{csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac{\int \frac{\text{csch}(x) \left (a^4-5 a^2 b^2-6 b^4+a b \left (a^2+3 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)} \, dx}{2 a^3 \left (a^2+b^2\right )}\\ &=\frac{b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}-\frac{\left (a^2+3 b^2\right ) \coth (x) \text{csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac{\left (a^2-6 b^2\right ) \int \text{csch}(x) \, dx}{2 a^4}-\frac{\left (b^3 \left (4 a^2+3 b^2\right )\right ) \int \frac{1}{a+b \sinh (x)} \, dx}{a^4 \left (a^2+b^2\right )}\\ &=\frac{\left (a^2-6 b^2\right ) \tanh ^{-1}(\cosh (x))}{2 a^4}+\frac{b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}-\frac{\left (a^2+3 b^2\right ) \coth (x) \text{csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac{\left (2 b^3 \left (4 a^2+3 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^4 \left (a^2+b^2\right )}\\ &=\frac{\left (a^2-6 b^2\right ) \tanh ^{-1}(\cosh (x))}{2 a^4}+\frac{b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}-\frac{\left (a^2+3 b^2\right ) \coth (x) \text{csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{\left (4 b^3 \left (4 a^2+3 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^4 \left (a^2+b^2\right )}\\ &=\frac{\left (a^2-6 b^2\right ) \tanh ^{-1}(\cosh (x))}{2 a^4}+\frac{2 b^3 \left (4 a^2+3 b^2\right ) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^4 \left (a^2+b^2\right )^{3/2}}+\frac{b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}-\frac{\left (a^2+3 b^2\right ) \coth (x) \text{csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac{b^2 \coth (x) \text{csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\\ \end{align*}

Mathematica [A]  time = 0.70981, size = 156, normalized size = 0.99 \[ \frac{-4 \left (a^2-6 b^2\right ) \log \left (\tanh \left (\frac{x}{2}\right )\right )+\frac{16 b^3 \left (4 a^2+3 b^2\right ) \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+\frac{8 a b^4 \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}-a^2 \text{csch}^2\left (\frac{x}{2}\right )-a^2 \text{sech}^2\left (\frac{x}{2}\right )+8 a b \tanh \left (\frac{x}{2}\right )+8 a b \coth \left (\frac{x}{2}\right )}{8 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^3/(a + b*Sinh[x])^2,x]

[Out]

((16*b^3*(4*a^2 + 3*b^2)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/(-a^2 - b^2)^(3/2) + 8*a*b*Coth[x/2] - a^
2*Csch[x/2]^2 - 4*(a^2 - 6*b^2)*Log[Tanh[x/2]] - a^2*Sech[x/2]^2 + (8*a*b^4*Cosh[x])/((a^2 + b^2)*(a + b*Sinh[
x])) + 8*a*b*Tanh[x/2])/(8*a^4)

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Maple [A]  time = 0.052, size = 227, normalized size = 1.4 \begin{align*}{\frac{1}{8\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{b}{{a}^{3}}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{8\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{1}{2\,{a}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+3\,{\frac{\ln \left ( \tanh \left ( x/2 \right ) \right ){b}^{2}}{{a}^{4}}}+{\frac{b}{{a}^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-2\,{\frac{{b}^{5}\tanh \left ( x/2 \right ) }{{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 ) }}-2\,{\frac{{b}^{4}}{{a}^{3} \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 ) }}-8\,{\frac{{b}^{3}}{{a}^{2} \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 ) }-6\,{\frac{{b}^{5}}{{a}^{4} \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)^3/(a+b*sinh(x))^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.24119, size = 8699, normalized size = 55.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^3/(a+b*sinh(x))^2,x, algorithm="fricas")

[Out]

-1/2*(8*a^5*b^2 + 20*a^3*b^4 + 12*a*b^6 - 2*(a^6*b + 4*a^4*b^3 + 3*a^2*b^5)*cosh(x)^5 - 2*(a^6*b + 4*a^4*b^3 +
3*a^2*b^5)*sinh(x)^5 - 4*(a^7 - 4*a^3*b^4 - 3*a*b^6)*cosh(x)^4 - 2*(2*a^7 - 8*a^3*b^4 - 6*a*b^6 + 5*(a^6*b +
4*a^4*b^3 + 3*a^2*b^5)*cosh(x))*sinh(x)^4 + 8*(2*a^6*b + 5*a^4*b^3 + 3*a^2*b^5)*cosh(x)^3 + 4*(4*a^6*b + 10*a^
4*b^3 + 6*a^2*b^5 - 5*(a^6*b + 4*a^4*b^3 + 3*a^2*b^5)*cosh(x)^2 - 4*(a^7 - 4*a^3*b^4 - 3*a*b^6)*cosh(x))*sinh(
x)^3 - 4*(a^7 + 6*a^5*b^2 + 11*a^3*b^4 + 6*a*b^6)*cosh(x)^2 - 4*(a^7 + 6*a^5*b^2 + 11*a^3*b^4 + 6*a*b^6 + 5*(a
^6*b + 4*a^4*b^3 + 3*a^2*b^5)*cosh(x)^3 + 6*(a^7 - 4*a^3*b^4 - 3*a*b^6)*cosh(x)^2 - 6*(2*a^6*b + 5*a^4*b^3 + 3
*a^2*b^5)*cosh(x))*sinh(x)^2 + 2*((4*a^2*b^4 + 3*b^6)*cosh(x)^6 + (4*a^2*b^4 + 3*b^6)*sinh(x)^6 - 4*a^2*b^4 -
3*b^6 + 2*(4*a^3*b^3 + 3*a*b^5)*cosh(x)^5 + 2*(4*a^3*b^3 + 3*a*b^5 + 3*(4*a^2*b^4 + 3*b^6)*cosh(x))*sinh(x)^5
- 3*(4*a^2*b^4 + 3*b^6)*cosh(x)^4 - (12*a^2*b^4 + 9*b^6 - 15*(4*a^2*b^4 + 3*b^6)*cosh(x)^2 - 10*(4*a^3*b^3 + 3
*a*b^5)*cosh(x))*sinh(x)^4 - 4*(4*a^3*b^3 + 3*a*b^5)*cosh(x)^3 - 4*(4*a^3*b^3 + 3*a*b^5 - 5*(4*a^2*b^4 + 3*b^6
)*cosh(x)^3 - 5*(4*a^3*b^3 + 3*a*b^5)*cosh(x)^2 + 3*(4*a^2*b^4 + 3*b^6)*cosh(x))*sinh(x)^3 + 3*(4*a^2*b^4 + 3*
b^6)*cosh(x)^2 + (12*a^2*b^4 + 9*b^6 + 15*(4*a^2*b^4 + 3*b^6)*cosh(x)^4 + 20*(4*a^3*b^3 + 3*a*b^5)*cosh(x)^3 -
18*(4*a^2*b^4 + 3*b^6)*cosh(x)^2 - 12*(4*a^3*b^3 + 3*a*b^5)*cosh(x))*sinh(x)^2 + 2*(4*a^3*b^3 + 3*a*b^5)*cosh
(x) + 2*(4*a^3*b^3 + 3*a*b^5 + 3*(4*a^2*b^4 + 3*b^6)*cosh(x)^5 + 5*(4*a^3*b^3 + 3*a*b^5)*cosh(x)^4 - 6*(4*a^2*
b^4 + 3*b^6)*cosh(x)^3 - 6*(4*a^3*b^3 + 3*a*b^5)*cosh(x)^2 + 3*(4*a^2*b^4 + 3*b^6)*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*sq
rt(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*(7*a^6*b + 16*a^4*b^3 + 9*a^2*b^5)*cosh(x) - (a^6*b - 4*a^4*b^3 - 11*a^2*b^5 - 6*b^7 - (a^6*b - 4*
a^4*b^3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^6 - (a^6*b - 4*a^4*b^3 - 11*a^2*b^5 - 6*b^7)*sinh(x)^6 - 2*(a^7 - 4*a^5*
b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x)^5 - 2*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6 + 3*(a^6*b - 4*a^4*b^3 - 11*
a^2*b^5 - 6*b^7)*cosh(x))*sinh(x)^5 + 3*(a^6*b - 4*a^4*b^3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^4 + (3*a^6*b - 12*a^4
*b^3 - 33*a^2*b^5 - 18*b^7 - 15*(a^6*b - 4*a^4*b^3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^2 - 10*(a^7 - 4*a^5*b^2 - 11*
a^3*b^4 - 6*a*b^6)*cosh(x))*sinh(x)^4 + 4*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x)^3 + 4*(a^7 - 4*a^5*
b^2 - 11*a^3*b^4 - 6*a*b^6 - 5*(a^6*b - 4*a^4*b^3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^3 - 5*(a^7 - 4*a^5*b^2 - 11*a^
3*b^4 - 6*a*b^6)*cosh(x)^2 + 3*(a^6*b - 4*a^4*b^3 - 11*a^2*b^5 - 6*b^7)*cosh(x))*sinh(x)^3 - 3*(a^6*b - 4*a^4*
b^3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^2 - (3*a^6*b - 12*a^4*b^3 - 33*a^2*b^5 - 18*b^7 + 15*(a^6*b - 4*a^4*b^3 - 11
*a^2*b^5 - 6*b^7)*cosh(x)^4 + 20*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x)^3 - 18*(a^6*b - 4*a^4*b^3 -
11*a^2*b^5 - 6*b^7)*cosh(x)^2 - 12*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x))*sinh(x)^2 - 2*(a^7 - 4*a^
5*b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x) - 2*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6 + 3*(a^6*b - 4*a^4*b^3 - 11*
a^2*b^5 - 6*b^7)*cosh(x)^5 + 5*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x)^4 - 6*(a^6*b - 4*a^4*b^3 - 11*
a^2*b^5 - 6*b^7)*cosh(x)^3 - 6*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x)^2 + 3*(a^6*b - 4*a^4*b^3 - 11*
a^2*b^5 - 6*b^7)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) + (a^6*b - 4*a^4*b^3 - 11*a^2*b^5 - 6*b^7 - (a^6
*b - 4*a^4*b^3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^6 - (a^6*b - 4*a^4*b^3 - 11*a^2*b^5 - 6*b^7)*sinh(x)^6 - 2*(a^7 -
4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x)^5 - 2*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6 + 3*(a^6*b - 4*a^4*b^
3 - 11*a^2*b^5 - 6*b^7)*cosh(x))*sinh(x)^5 + 3*(a^6*b - 4*a^4*b^3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^4 + (3*a^6*b -
12*a^4*b^3 - 33*a^2*b^5 - 18*b^7 - 15*(a^6*b - 4*a^4*b^3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^2 - 10*(a^7 - 4*a^5*b^
2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x))*sinh(x)^4 + 4*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x)^3 + 4*(a^7 -
4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6 - 5*(a^6*b - 4*a^4*b^3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^3 - 5*(a^7 - 4*a^5*b^2
- 11*a^3*b^4 - 6*a*b^6)*cosh(x)^2 + 3*(a^6*b - 4*a^4*b^3 - 11*a^2*b^5 - 6*b^7)*cosh(x))*sinh(x)^3 - 3*(a^6*b -
4*a^4*b^3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^2 - (3*a^6*b - 12*a^4*b^3 - 33*a^2*b^5 - 18*b^7 + 15*(a^6*b - 4*a^4*b
^3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^4 + 20*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x)^3 - 18*(a^6*b - 4*a^4
*b^3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^2 - 12*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x))*sinh(x)^2 - 2*(a^7
- 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x) - 2*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6 + 3*(a^6*b - 4*a^4*b^
3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^5 + 5*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x)^4 - 6*(a^6*b - 4*a^4*b^
3 - 11*a^2*b^5 - 6*b^7)*cosh(x)^3 - 6*(a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 6*a*b^6)*cosh(x)^2 + 3*(a^6*b - 4*a^4*b^
3 - 11*a^2*b^5 - 6*b^7)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) - 1) - 2*(7*a^6*b + 16*a^4*b^3 + 9*a^2*b^5 + 5
*(a^6*b + 4*a^4*b^3 + 3*a^2*b^5)*cosh(x)^4 + 8*(a^7 - 4*a^3*b^4 - 3*a*b^6)*cosh(x)^3 - 12*(2*a^6*b + 5*a^4*b^3
+ 3*a^2*b^5)*cosh(x)^2 + 4*(a^7 + 6*a^5*b^2 + 11*a^3*b^4 + 6*a*b^6)*cosh(x))*sinh(x))/(a^8*b + 2*a^6*b^3 + a^
4*b^5 - (a^8*b + 2*a^6*b^3 + a^4*b^5)*cosh(x)^6 - (a^8*b + 2*a^6*b^3 + a^4*b^5)*sinh(x)^6 - 2*(a^9 + 2*a^7*b^2
+ a^5*b^4)*cosh(x)^5 - 2*(a^9 + 2*a^7*b^2 + a^5*b^4 + 3*(a^8*b + 2*a^6*b^3 + a^4*b^5)*cosh(x))*sinh(x)^5 + 3*
(a^8*b + 2*a^6*b^3 + a^4*b^5)*cosh(x)^4 + (3*a^8*b + 6*a^6*b^3 + 3*a^4*b^5 - 15*(a^8*b + 2*a^6*b^3 + a^4*b^5)*
cosh(x)^2 - 10*(a^9 + 2*a^7*b^2 + a^5*b^4)*cosh(x))*sinh(x)^4 + 4*(a^9 + 2*a^7*b^2 + a^5*b^4)*cosh(x)^3 + 4*(a
^9 + 2*a^7*b^2 + a^5*b^4 - 5*(a^8*b + 2*a^6*b^3 + a^4*b^5)*cosh(x)^3 - 5*(a^9 + 2*a^7*b^2 + a^5*b^4)*cosh(x)^2
+ 3*(a^8*b + 2*a^6*b^3 + a^4*b^5)*cosh(x))*sinh(x)^3 - 3*(a^8*b + 2*a^6*b^3 + a^4*b^5)*cosh(x)^2 - (3*a^8*b +
6*a^6*b^3 + 3*a^4*b^5 + 15*(a^8*b + 2*a^6*b^3 + a^4*b^5)*cosh(x)^4 + 20*(a^9 + 2*a^7*b^2 + a^5*b^4)*cosh(x)^3
- 18*(a^8*b + 2*a^6*b^3 + a^4*b^5)*cosh(x)^2 - 12*(a^9 + 2*a^7*b^2 + a^5*b^4)*cosh(x))*sinh(x)^2 - 2*(a^9 + 2
*a^7*b^2 + a^5*b^4)*cosh(x) - 2*(a^9 + 2*a^7*b^2 + a^5*b^4 + 3*(a^8*b + 2*a^6*b^3 + a^4*b^5)*cosh(x)^5 + 5*(a^
9 + 2*a^7*b^2 + a^5*b^4)*cosh(x)^4 - 6*(a^8*b + 2*a^6*b^3 + a^4*b^5)*cosh(x)^3 - 6*(a^9 + 2*a^7*b^2 + a^5*b^4)
*cosh(x)^2 + 3*(a^8*b + 2*a^6*b^3 + a^4*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)**3/(a+b*sinh(x))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.5849, size = 274, normalized size = 1.73 \begin{align*} -\frac{{\left (4 \, a^{2} b^{3} + 3 \, b^{5}\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^{6} + a^{4} b^{2}\right )} \sqrt{a^{2} + b^{2}}} - \frac{2 \,{\left (a b^{3} e^{x} - b^{4}\right )}}{{\left (a^{5} + a^{3} b^{2}\right )}{\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} + \frac{{\left (a^{2} - 6 \, b^{2}\right )} \log \left (e^{x} + 1\right )}{2 \, a^{4}} - \frac{{\left (a^{2} - 6 \, b^{2}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{2 \, a^{4}} - \frac{a e^{\left (3 \, x\right )} - 4 \, b e^{\left (2 \, x\right )} + a e^{x} + 4 \, b}{a^{3}{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^3/(a+b*sinh(x))^2,x, algorithm="giac")

[Out]

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