Integrand size = 11, antiderivative size = 85 \[ \int \frac {\text {csch}(x)}{(a+b \sinh (x))^2} \, dx=-\frac {\text {arctanh}(\cosh (x))}{a^2}+\frac {2 b \left (2 a^2+b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2}}+\frac {b^2 \cosh (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))} \] Output:
-arctanh(cosh(x))/a^2+2*b*(2*a^2+b^2)*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^ (1/2))/a^2/(a^2+b^2)^(3/2)+b^2*cosh(x)/a/(a^2+b^2)/(a+b*sinh(x))
Time = 0.42 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.18 \[ \int \frac {\text {csch}(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right )+\frac {a b^2 \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}}{a^2} \] Input:
Integrate[Csch[x]/(a + b*Sinh[x])^2,x]
Output:
((2*b*(2*a^2 + b^2)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/(-a^2 - b^ 2)^(3/2) - Log[Cosh[x/2]] + Log[Sinh[x/2]] + (a*b^2*Cosh[x])/((a^2 + b^2)* (a + b*Sinh[x])))/a^2
Result contains complex when optimal does not.
Time = 0.62 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.44, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.273, Rules used = {3042, 26, 3281, 26, 3042, 26, 3480, 26, 3042, 26, 3139, 1083, 219, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\text {csch}(x)}{(a+b \sinh (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\sin (i x) (a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\sin (i x) (a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle i \left (\frac {\int -\frac {i \text {csch}(x) \left (a^2-b \sinh (x) a+b^2\right )}{a+b \sinh (x)}dx}{a \left (a^2+b^2\right )}-\frac {i b^2 \cosh (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i \int \frac {\text {csch}(x) \left (a^2-b \sinh (x) a+b^2\right )}{a+b \sinh (x)}dx}{a \left (a^2+b^2\right )}-\frac {i b^2 \cosh (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i \int \frac {i \left (a^2+i b \sin (i x) a+b^2\right )}{\sin (i x) (a-i b \sin (i x))}dx}{a \left (a^2+b^2\right )}-\frac {i b^2 \cosh (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {\int \frac {a^2+i b \sin (i x) a+b^2}{\sin (i x) (a-i b \sin (i x))}dx}{a \left (a^2+b^2\right )}-\frac {i b^2 \cosh (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle i \left (\frac {\frac {i b \left (2 a^2+b^2\right ) \int \frac {1}{a+b \sinh (x)}dx}{a}+\frac {\left (a^2+b^2\right ) \int -i \text {csch}(x)dx}{a}}{a \left (a^2+b^2\right )}-\frac {i b^2 \cosh (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {\frac {i b \left (2 a^2+b^2\right ) \int \frac {1}{a+b \sinh (x)}dx}{a}-\frac {i \left (a^2+b^2\right ) \int \text {csch}(x)dx}{a}}{a \left (a^2+b^2\right )}-\frac {i b^2 \cosh (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {\frac {i b \left (2 a^2+b^2\right ) \int \frac {1}{a-i b \sin (i x)}dx}{a}-\frac {i \left (a^2+b^2\right ) \int i \csc (i x)dx}{a}}{a \left (a^2+b^2\right )}-\frac {i b^2 \cosh (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {\frac {i b \left (2 a^2+b^2\right ) \int \frac {1}{a-i b \sin (i x)}dx}{a}+\frac {\left (a^2+b^2\right ) \int \csc (i x)dx}{a}}{a \left (a^2+b^2\right )}-\frac {i b^2 \cosh (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle i \left (\frac {\frac {\left (a^2+b^2\right ) \int \csc (i x)dx}{a}+\frac {2 i b \left (2 a^2+b^2\right ) \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{a}}{a \left (a^2+b^2\right )}-\frac {i b^2 \cosh (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle i \left (\frac {\frac {\left (a^2+b^2\right ) \int \csc (i x)dx}{a}-\frac {4 i b \left (2 a^2+b^2\right ) \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a}}{a \left (a^2+b^2\right )}-\frac {i b^2 \cosh (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle i \left (\frac {\frac {\left (a^2+b^2\right ) \int \csc (i x)dx}{a}-\frac {2 i b \left (2 a^2+b^2\right ) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}}{a \left (a^2+b^2\right )}-\frac {i b^2 \cosh (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle i \left (\frac {\frac {i \left (a^2+b^2\right ) \text {arctanh}(\cosh (x))}{a}-\frac {2 i b \left (2 a^2+b^2\right ) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}}{a \left (a^2+b^2\right )}-\frac {i b^2 \cosh (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
Input:
Int[Csch[x]/(a + b*Sinh[x])^2,x]
Output:
I*(((I*(a^2 + b^2)*ArcTanh[Cosh[x]])/a - ((2*I)*b*(2*a^2 + b^2)*ArcTanh[(2 *b - 2*a*Tanh[x/2])/(2*Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]))/(a*(a^2 + b ^2)) - (I*b^2*Cosh[x])/(a*(a^2 + b^2)*(a + b*Sinh[x])))
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-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] + Simp[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)*Si n[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])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.35
| method | result | size |
| default | \(\frac {4 b \left (\frac {-\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{2 \left (a^{2}+b^{2}\right )}-\frac {a b}{2 \left (a^{2}+b^{2}\right )}}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (2 a^{2}+b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{2}}\) | \(115\) |
| risch | \(-\frac {2 b \left ({\mathrm e}^{x} a -b \right )}{a \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{a^{2}}+\frac {2 b \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {b^{3} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{2}}-\frac {2 b \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {b^{3} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{2}}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{a^{2}}\) | \(298\) |
Input:
int(csch(x)/(a+b*sinh(x))^2,x,method=_RETURNVERBOSE)
Output:
4/a^2*b*((-1/2*b^2/(a^2+b^2)*tanh(1/2*x)-1/2*a*b/(a^2+b^2))/(tanh(1/2*x)^2 *a-2*b*tanh(1/2*x)-a)-1/2*(2*a^2+b^2)/(a^2+b^2)^(3/2)*arctanh(1/2*(2*a*tan h(1/2*x)-2*b)/(a^2+b^2)^(1/2)))+1/a^2*ln(tanh(1/2*x))
Leaf count of result is larger than twice the leaf count of optimal. 672 vs. \(2 (81) = 162\).
Time = 0.19 (sec) , antiderivative size = 672, normalized size of antiderivative = 7.91 \[ \int \frac {\text {csch}(x)}{(a+b \sinh (x))^2} \, dx =\text {Too large to display} \] Input:
integrate(csch(x)/(a+b*sinh(x))^2,x, algorithm="fricas")
Output:
-(2*a^3*b^2 + 2*a*b^4 - (2*a^2*b^2 + b^4 - (2*a^2*b^2 + b^4)*cosh(x)^2 - ( 2*a^2*b^2 + b^4)*sinh(x)^2 - 2*(2*a^3*b + a*b^3)*cosh(x) - 2*(2*a^3*b + a* b^3 + (2*a^2*b^2 + 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)*s inh(x) + 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*s inh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) - b)) - 2*(a^4*b + a^2* b^3)*cosh(x) + (a^4*b + 2*a^2*b^3 + b^5 - (a^4*b + 2*a^2*b^3 + b^5)*cosh(x )^2 - (a^4*b + 2*a^2*b^3 + b^5)*sinh(x)^2 - 2*(a^5 + 2*a^3*b^2 + a*b^4)*co sh(x) - 2*(a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^5)*cosh(x))*si nh(x))*log(cosh(x) + sinh(x) + 1) - (a^4*b + 2*a^2*b^3 + b^5 - (a^4*b + 2* a^2*b^3 + b^5)*cosh(x)^2 - (a^4*b + 2*a^2*b^3 + b^5)*sinh(x)^2 - 2*(a^5 + 2*a^3*b^2 + a*b^4)*cosh(x) - 2*(a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b ^3 + b^5)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) - 1) - 2*(a^4*b + a^2*b^ 3)*sinh(x))/(a^6*b + 2*a^4*b^3 + a^2*b^5 - (a^6*b + 2*a^4*b^3 + a^2*b^5)*c osh(x)^2 - (a^6*b + 2*a^4*b^3 + a^2*b^5)*sinh(x)^2 - 2*(a^7 + 2*a^5*b^2 + a^3*b^4)*cosh(x) - 2*(a^7 + 2*a^5*b^2 + a^3*b^4 + (a^6*b + 2*a^4*b^3 + a^2 *b^5)*cosh(x))*sinh(x))
\[ \int \frac {\text {csch}(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\operatorname {csch}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \] Input:
integrate(csch(x)/(a+b*sinh(x))**2,x)
Output:
Integral(csch(x)/(a + b*sinh(x))**2, x)
Time = 0.12 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.91 \[ \int \frac {\text {csch}(x)}{(a+b \sinh (x))^2} \, dx=-\frac {{\left (2 \, a^{2} b + b^{3}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a b e^{\left (-x\right )} + b^{2}\right )}}{a^{3} b + a b^{3} + 2 \, {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-x\right )} - {\left (a^{3} b + a b^{3}\right )} e^{\left (-2 \, x\right )}} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{a^{2}} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{a^{2}} \] Input:
integrate(csch(x)/(a+b*sinh(x))^2,x, algorithm="maxima")
Output:
-(2*a^2*b + b^3)*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt (a^2 + b^2)))/((a^4 + a^2*b^2)*sqrt(a^2 + b^2)) + 2*(a*b*e^(-x) + b^2)/(a^ 3*b + a*b^3 + 2*(a^4 + a^2*b^2)*e^(-x) - (a^3*b + a*b^3)*e^(-2*x)) - log(e ^(-x) + 1)/a^2 + log(e^(-x) - 1)/a^2
Time = 0.14 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.67 \[ \int \frac {\text {csch}(x)}{(a+b \sinh (x))^2} \, dx=-\frac {{\left (2 \, a^{2} b + b^{3}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (a b e^{x} - b^{2}\right )}}{{\left (a^{3} + a b^{2}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} - \frac {\log \left (e^{x} + 1\right )}{a^{2}} + \frac {\log \left ({\left | e^{x} - 1 \right |}\right )}{a^{2}} \] Input:
integrate(csch(x)/(a+b*sinh(x))^2,x, algorithm="giac")
Output:
-(2*a^2*b + b^3)*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^4 + a^2*b^2)*sqrt(a^2 + b^2)) - 2*(a*b*e^x - b^2)/((a^3 + a*b^2)*(b*e^(2*x) + 2*a*e^x - b)) - log(e^x + 1)/a^2 + log(a bs(e^x - 1))/a^2
Time = 4.28 (sec) , antiderivative size = 1001, normalized size of antiderivative = 11.78 \[ \int \frac {\text {csch}(x)}{(a+b \sinh (x))^2} \, dx =\text {Too large to display} \] Input:
int(1/(sinh(x)*(a + b*sinh(x))^2),x)
Output:
((2*b^5)/(a*(b^5 + a^2*b^3)) - (2*b^4*exp(x))/(b^5 + a^2*b^3))/(2*a*exp(x) - b + b*exp(2*x)) + log(exp(x) - 1)/a^2 - log(exp(x) + 1)/a^2 - (b*log((3 2*(4*a^4*b + 2*b^5 + 6*a^2*b^3 - 8*a^5*exp(x) - 3*a*b^4*exp(x) - 10*a^3*b^ 2*exp(x)))/(a*(a^2*b^5 + a^4*b^3)*(a^2*b + b^3)) + (b*((32*(2*a^6*b + 2*b^ 7 + 8*a^2*b^5 + 9*a^4*b^3 - 4*a^7*exp(x) - 3*a*b^6*exp(x) - 10*a^3*b^4*exp (x) - 11*a^5*b^2*exp(x)))/(a*b^5*(a*b^4 + a^5 + 2*a^3*b^2)) - (b*(2*a^2 + b^2)*((a^2 + b^2)^3)^(1/2)*((32*(2*a*b^3 + 4*a^3*b - 7*a^4*exp(x) - 4*a^2* b^2*exp(x)))/(b*(b^5 + a^2*b^3)) + (32*(2*a^2 + b^2)*((a^2 + b^2)^3)^(1/2) *(3*a^4*b + 2*a^2*b^3 - 4*a^5*exp(x) - 3*a^3*b^2*exp(x)))/(b^4*(a^8 + a^2* b^6 + 3*a^4*b^4 + 3*a^6*b^2))))/(a^8 + a^2*b^6 + 3*a^4*b^4 + 3*a^6*b^2))*( 2*a^2 + b^2)*((a^2 + b^2)^3)^(1/2))/(a^8 + a^2*b^6 + 3*a^4*b^4 + 3*a^6*b^2 ))*(2*a^2 + b^2)*((a^2 + b^2)^3)^(1/2))/(a^8 + a^2*b^6 + 3*a^4*b^4 + 3*a^6 *b^2) + (b*log((32*(4*a^4*b + 2*b^5 + 6*a^2*b^3 - 8*a^5*exp(x) - 3*a*b^4*e xp(x) - 10*a^3*b^2*exp(x)))/(a*(a^2*b^5 + a^4*b^3)*(a^2*b + b^3)) - (b*((3 2*(2*a^6*b + 2*b^7 + 8*a^2*b^5 + 9*a^4*b^3 - 4*a^7*exp(x) - 3*a*b^6*exp(x) - 10*a^3*b^4*exp(x) - 11*a^5*b^2*exp(x)))/(a*b^5*(a*b^4 + a^5 + 2*a^3*b^2 )) + (b*(2*a^2 + b^2)*((a^2 + b^2)^3)^(1/2)*((32*(2*a*b^3 + 4*a^3*b - 7*a^ 4*exp(x) - 4*a^2*b^2*exp(x)))/(b*(b^5 + a^2*b^3)) - (32*(2*a^2 + b^2)*((a^ 2 + b^2)^3)^(1/2)*(3*a^4*b + 2*a^2*b^3 - 4*a^5*exp(x) - 3*a^3*b^2*exp(x))) /(b^4*(a^8 + a^2*b^6 + 3*a^4*b^4 + 3*a^6*b^2))))/(a^8 + a^2*b^6 + 3*a^4...
Time = 0.17 (sec) , antiderivative size = 627, normalized size of antiderivative = 7.38 \[ \int \frac {\text {csch}(x)}{(a+b \sinh (x))^2} \, dx=\frac {-\mathrm {log}\left (e^{x}-1\right ) b^{5}+\mathrm {log}\left (e^{x}+1\right ) b^{5}+a \,b^{4}+e^{2 x} \mathrm {log}\left (e^{x}-1\right ) b^{5}-e^{2 x} \mathrm {log}\left (e^{x}+1\right ) b^{5}+e^{2 x} a^{3} b^{2}+e^{2 x} a \,b^{4}-\mathrm {log}\left (e^{x}-1\right ) a^{4} b -2 \,\mathrm {log}\left (e^{x}-1\right ) a^{2} b^{3}+\mathrm {log}\left (e^{x}+1\right ) a^{4} b +2 \,\mathrm {log}\left (e^{x}+1\right ) a^{2} b^{3}+4 e^{x} \mathrm {log}\left (e^{x}-1\right ) a^{3} b^{2}+2 e^{x} \mathrm {log}\left (e^{x}-1\right ) a \,b^{4}-4 e^{x} \mathrm {log}\left (e^{x}+1\right ) a^{3} b^{2}-2 e^{x} \mathrm {log}\left (e^{x}+1\right ) a \,b^{4}+2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} i +e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a^{4} b +2 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a^{2} b^{3}-e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a^{4} b -2 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a^{2} b^{3}+4 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} i -4 e^{2 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} i -8 e^{x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} b i -4 e^{x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{3} i -2 e^{x} \mathrm {log}\left (e^{x}+1\right ) a^{5}+a^{3} b^{2}-2 e^{2 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} i +2 e^{x} \mathrm {log}\left (e^{x}-1\right ) a^{5}}{a^{2} \left (e^{2 x} a^{4} b +2 e^{2 x} a^{2} b^{3}+e^{2 x} b^{5}+2 e^{x} a^{5}+4 e^{x} a^{3} b^{2}+2 e^{x} a \,b^{4}-a^{4} b -2 a^{2} b^{3}-b^{5}\right )} \] Input:
int(csch(x)/(a+b*sinh(x))^2,x)
Output:
( - 4*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))* a**2*b**2*i - 2*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*b**4*i - 8*e**x*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a** 2 + b**2))*a**3*b*i - 4*e**x*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt( a**2 + b**2))*a*b**3*i + 4*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a* *2 + b**2))*a**2*b**2*i + 2*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a **2 + b**2))*b**4*i + e**(2*x)*log(e**x - 1)*a**4*b + 2*e**(2*x)*log(e**x - 1)*a**2*b**3 + e**(2*x)*log(e**x - 1)*b**5 - e**(2*x)*log(e**x + 1)*a**4 *b - 2*e**(2*x)*log(e**x + 1)*a**2*b**3 - e**(2*x)*log(e**x + 1)*b**5 + e* *(2*x)*a**3*b**2 + e**(2*x)*a*b**4 + 2*e**x*log(e**x - 1)*a**5 + 4*e**x*lo g(e**x - 1)*a**3*b**2 + 2*e**x*log(e**x - 1)*a*b**4 - 2*e**x*log(e**x + 1) *a**5 - 4*e**x*log(e**x + 1)*a**3*b**2 - 2*e**x*log(e**x + 1)*a*b**4 - log (e**x - 1)*a**4*b - 2*log(e**x - 1)*a**2*b**3 - log(e**x - 1)*b**5 + log(e **x + 1)*a**4*b + 2*log(e**x + 1)*a**2*b**3 + log(e**x + 1)*b**5 + a**3*b* *2 + a*b**4)/(a**2*(e**(2*x)*a**4*b + 2*e**(2*x)*a**2*b**3 + e**(2*x)*b**5 + 2*e**x*a**5 + 4*e**x*a**3*b**2 + 2*e**x*a*b**4 - a**4*b - 2*a**2*b**3 - b**5))