Integrand size = 13, antiderivative size = 158 \[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {\left (a^2-6 b^2\right ) \text {arctanh}(\cosh (x))}{2 a^4}+\frac {2 b^3 \left (4 a^2+3 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^4 \left (a^2+b^2\right )^{3/2}}+\frac {b \left (2 a^2+3 b^2\right ) \coth (x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))} \]
1/2*(a^2-6*b^2)*arctanh(cosh(x))/a^4+2*b^3*(4*a^2+3*b^2)*arctanh((b-a*tanh (1/2*x))/(a^2+b^2)^(1/2))/a^4/(a^2+b^2)^(3/2)+b*(2*a^2+3*b^2)*coth(x)/a^3/ (a^2+b^2)-1/2*(a^2+3*b^2)*coth(x)*csch(x)/a^2/(a^2+b^2)+b^2*coth(x)*csch(x )/a/(a^2+b^2)/(a+b*sinh(x))
Time = 0.96 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.10 \[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {16 b^3 \left (4 a^2+3 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+8 a b \coth \left (\frac {x}{2}\right )-a^2 \text {csch}^2\left (\frac {x}{2}\right )+4 \left (a^2-6 b^2\right ) \log \left (\cosh \left (\frac {x}{2}\right )\right )-4 \left (a^2-6 b^2\right ) \log \left (\sinh \left (\frac {x}{2}\right )\right )-a^2 \text {sech}^2\left (\frac {x}{2}\right )+\frac {8 a b^4 \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+8 a b \tanh \left (\frac {x}{2}\right )}{8 a^4} \]
((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[Co sh[x/2]] - 4*(a^2 - 6*b^2)*Log[Sinh[x/2]] - a^2*Sech[x/2]^2 + (8*a*b^4*Cos h[x])/((a^2 + b^2)*(a + b*Sinh[x])) + 8*a*b*Tanh[x/2])/(8*a^4)
Result contains complex when optimal does not.
Time = 1.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.24, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.692, Rules used = {3042, 26, 3281, 26, 3042, 26, 3534, 26, 3042, 25, 3534, 25, 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.
| \(\displaystyle \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\sin (i x)^3 (a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\sin (i x)^3 (a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle -i \left (\frac {\int \frac {i \text {csch}^3(x) \left (a^2-b \sinh (x) a+3 b^2+2 b^2 \sinh ^2(x)\right )}{a+b \sinh (x)}dx}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i \int \frac {\text {csch}^3(x) \left (a^2-b \sinh (x) a+3 b^2+2 b^2 \sinh ^2(x)\right )}{a+b \sinh (x)}dx}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i \int -\frac {i \left (a^2+i b \sin (i x) a+3 b^2-2 b^2 \sin (i x)^2\right )}{\sin (i x)^3 (a-i b \sin (i x))}dx}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {\int \frac {a^2+i b \sin (i x) a+3 b^2-2 b^2 \sin (i x)^2}{\sin (i x)^3 (a-i b \sin (i x))}dx}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -i \left (\frac {\frac {\int -\frac {i \text {csch}^2(x) \left (b \left (a^2+3 b^2\right ) \sinh ^2(x)+a \left (a^2-b^2\right ) \sinh (x)+2 b \left (2 a^2+3 b^2\right )\right )}{a+b \sinh (x)}dx}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {-\frac {i \int \frac {\text {csch}^2(x) \left (b \left (a^2+3 b^2\right ) \sinh ^2(x)+a \left (a^2-b^2\right ) \sinh (x)+2 b \left (2 a^2+3 b^2\right )\right )}{a+b \sinh (x)}dx}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {-\frac {i \int -\frac {-b \left (a^2+3 b^2\right ) \sin (i x)^2-i a \left (a^2-b^2\right ) \sin (i x)+2 b \left (2 a^2+3 b^2\right )}{\sin (i x)^2 (a-i b \sin (i x))}dx}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (\frac {\frac {i \int \frac {-b \left (a^2+3 b^2\right ) \sin (i x)^2-i a \left (a^2-b^2\right ) \sin (i x)+2 b \left (2 a^2+3 b^2\right )}{\sin (i x)^2 (a-i b \sin (i x))}dx}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -i \left (\frac {\frac {i \left (\frac {\int -\frac {\text {csch}(x) \left (\left (a^2-6 b^2\right ) \left (a^2+b^2\right )+a b \left (a^2+3 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)}dx}{a}+\frac {2 b \left (2 a^2+3 b^2\right ) \coth (x)}{a}\right )}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (\frac {\frac {i \left (\frac {2 b \left (2 a^2+3 b^2\right ) \coth (x)}{a}-\frac {\int \frac {\text {csch}(x) \left (\left (a^2-6 b^2\right ) \left (a^2+b^2\right )+a b \left (a^2+3 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)}dx}{a}\right )}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {\frac {i \left (\frac {2 b \left (2 a^2+3 b^2\right ) \coth (x)}{a}-\frac {\int \frac {i \left (\left (a^2-6 b^2\right ) \left (a^2+b^2\right )-i a b \left (a^2+3 b^2\right ) \sin (i x)\right )}{\sin (i x) (a-i b \sin (i x))}dx}{a}\right )}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {\frac {i \left (\frac {2 b \left (2 a^2+3 b^2\right ) \coth (x)}{a}-\frac {i \int \frac {\left (a^2-6 b^2\right ) \left (a^2+b^2\right )-i a b \left (a^2+3 b^2\right ) \sin (i x)}{\sin (i x) (a-i b \sin (i x))}dx}{a}\right )}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle -i \left (\frac {\frac {i \left (\frac {2 b \left (2 a^2+3 b^2\right ) \coth (x)}{a}-\frac {i \left (\frac {\left (a^2-6 b^2\right ) \left (a^2+b^2\right ) \int -i \text {csch}(x)dx}{a}-\frac {2 i b^3 \left (4 a^2+3 b^2\right ) \int \frac {1}{a+b \sinh (x)}dx}{a}\right )}{a}\right )}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {\frac {i \left (\frac {2 b \left (2 a^2+3 b^2\right ) \coth (x)}{a}-\frac {i \left (-\frac {i \left (a^2-6 b^2\right ) \left (a^2+b^2\right ) \int \text {csch}(x)dx}{a}-\frac {2 i b^3 \left (4 a^2+3 b^2\right ) \int \frac {1}{a+b \sinh (x)}dx}{a}\right )}{a}\right )}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {\frac {i \left (\frac {2 b \left (2 a^2+3 b^2\right ) \coth (x)}{a}-\frac {i \left (-\frac {i \left (a^2-6 b^2\right ) \left (a^2+b^2\right ) \int i \csc (i x)dx}{a}-\frac {2 i b^3 \left (4 a^2+3 b^2\right ) \int \frac {1}{a-i b \sin (i x)}dx}{a}\right )}{a}\right )}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {\frac {i \left (\frac {2 b \left (2 a^2+3 b^2\right ) \coth (x)}{a}-\frac {i \left (\frac {\left (a^2-6 b^2\right ) \left (a^2+b^2\right ) \int \csc (i x)dx}{a}-\frac {2 i b^3 \left (4 a^2+3 b^2\right ) \int \frac {1}{a-i b \sin (i x)}dx}{a}\right )}{a}\right )}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -i \left (\frac {\frac {i \left (\frac {2 b \left (2 a^2+3 b^2\right ) \coth (x)}{a}-\frac {i \left (\frac {\left (a^2-6 b^2\right ) \left (a^2+b^2\right ) \int \csc (i x)dx}{a}-\frac {4 i b^3 \left (4 a^2+3 b^2\right ) \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{a}\right )}{a}\right )}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -i \left (\frac {\frac {i \left (\frac {2 b \left (2 a^2+3 b^2\right ) \coth (x)}{a}-\frac {i \left (\frac {\left (a^2-6 b^2\right ) \left (a^2+b^2\right ) \int \csc (i x)dx}{a}+\frac {8 i b^3 \left (4 a^2+3 b^2\right ) \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a}\right )}{a}\right )}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -i \left (\frac {\frac {i \left (\frac {2 b \left (2 a^2+3 b^2\right ) \coth (x)}{a}-\frac {i \left (\frac {\left (a^2-6 b^2\right ) \left (a^2+b^2\right ) \int \csc (i x)dx}{a}+\frac {4 i b^3 \left (4 a^2+3 b^2\right ) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}\right )}{a}\right )}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -i \left (\frac {\frac {i \left (\frac {2 b \left (2 a^2+3 b^2\right ) \coth (x)}{a}-\frac {i \left (\frac {i \left (a^2-6 b^2\right ) \left (a^2+b^2\right ) \text {arctanh}(\cosh (x))}{a}+\frac {4 i b^3 \left (4 a^2+3 b^2\right ) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}\right )}{a}\right )}{2 a}-\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 a}}{a \left (a^2+b^2\right )}+\frac {i b^2 \coth (x) \text {csch}(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\right )\) |
(-I)*((((I/2)*(((-I)*((I*(a^2 - 6*b^2)*(a^2 + b^2)*ArcTanh[Cosh[x]])/a + ( (4*I)*b^3*(4*a^2 + 3*b^2)*ArcTanh[(2*b - 2*a*Tanh[x/2])/(2*Sqrt[a^2 + b^2] )])/(a*Sqrt[a^2 + b^2])))/a + (2*b*(2*a^2 + 3*b^2)*Coth[x])/a))/a - ((I/2) *(a^2 + 3*b^2)*Coth[x]*Csch[x])/a)/(a*(a^2 + b^2)) + (I*b^2*Coth[x]*Csch[x ])/(a*(a^2 + b^2)*(a + b*Sinh[x])))
3.1.86.3.1 Defintions of rubi rules used
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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(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] + 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[(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] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.97 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.11
| method | result | size |
| default | \(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{2} a}{2}+4 b \tanh \left (\frac {x}{2}\right )}{4 a^{3}}+\frac {4 b^{3} \left (\frac {-\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{2 \left (a^{2}+b^{2}\right )}-\frac {a b}{2 \left (a^{2}+b^{2}\right )}}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (4 a^{2}+3 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{a^{4}}-\frac {1}{8 a^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (-2 a^{2}+12 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tanh \left (\frac {x}{2}\right )}\) | \(175\) |
| risch | \(-\frac {a^{3} b \,{\mathrm e}^{5 x}+3 a \,b^{3} {\mathrm e}^{5 x}+2 a^{4} {\mathrm e}^{4 x}-2 a^{2} b^{2} {\mathrm e}^{4 x}-6 b^{4} {\mathrm e}^{4 x}-8 a^{3} b \,{\mathrm e}^{3 x}-12 a \,b^{3} {\mathrm e}^{3 x}+2 a^{4} {\mathrm e}^{2 x}+10 a^{2} b^{2} {\mathrm e}^{2 x}+12 b^{4} {\mathrm e}^{2 x}+7 a^{3} b \,{\mathrm e}^{x}+9 b^{3} {\mathrm e}^{x} a -4 a^{2} b^{2}-6 b^{4}}{a^{3} \left ({\mathrm e}^{2 x}-1\right )^{2} \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}+\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2 a^{2}}-\frac {3 \ln \left ({\mathrm e}^{x}+1\right ) b^{2}}{a^{4}}+\frac {4 b^{3} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{2}}+\frac {3 b^{5} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{4}}-\frac {4 b^{3} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{2}}-\frac {3 b^{5} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{4}}-\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2 a^{2}}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right ) b^{2}}{a^{4}}\) | \(464\) |
1/4/a^3*(1/2*tanh(1/2*x)^2*a+4*b*tanh(1/2*x))+4/a^4*b^3*((-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* (4*a^2+3*b^2)/(a^2+b^2)^(3/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^ (1/2)))-1/8/a^2/tanh(1/2*x)^2+1/4/a^4*(-2*a^2+12*b^2)*ln(tanh(1/2*x))+1/a^ 3*b/tanh(1/2*x)
Leaf count of result is larger than twice the leaf count of optimal. 3754 vs. \(2 (150) = 300\).
Time = 0.51 (sec) , antiderivative size = 3754, normalized size of antiderivative = 23.76 \[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \]
-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)*c osh(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))*sin h(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...
\[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (150) = 300\).
Time = 0.29 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.30 \[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=-\frac {{\left (4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{6} + a^{4} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {4 \, a^{2} b^{2} + 6 \, b^{4} + {\left (7 \, a^{3} b + 9 \, a b^{3}\right )} e^{\left (-x\right )} - 2 \, {\left (a^{4} + 5 \, a^{2} b^{2} + 6 \, b^{4}\right )} e^{\left (-2 \, x\right )} - 4 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} e^{\left (-3 \, x\right )} - 2 \, {\left (a^{4} - a^{2} b^{2} - 3 \, b^{4}\right )} e^{\left (-4 \, x\right )} + {\left (a^{3} b + 3 \, a b^{3}\right )} e^{\left (-5 \, x\right )}}{a^{5} b + a^{3} b^{3} + 2 \, {\left (a^{6} + a^{4} b^{2}\right )} e^{\left (-x\right )} - 3 \, {\left (a^{5} b + a^{3} b^{3}\right )} e^{\left (-2 \, x\right )} - 4 \, {\left (a^{6} + a^{4} b^{2}\right )} e^{\left (-3 \, x\right )} + 3 \, {\left (a^{5} b + a^{3} b^{3}\right )} e^{\left (-4 \, x\right )} + 2 \, {\left (a^{6} + a^{4} b^{2}\right )} e^{\left (-5 \, x\right )} - {\left (a^{5} b + a^{3} b^{3}\right )} e^{\left (-6 \, x\right )}} + \frac {{\left (a^{2} - 6 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, a^{4}} - \frac {{\left (a^{2} - 6 \, b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{2 \, a^{4}} \]
-(4*a^2*b^3 + 3*b^5)*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/((a^6 + a^4*b^2)*sqrt(a^2 + b^2)) + (4*a^2*b^2 + 6*b^4 + (7*a^3*b + 9*a*b^3)*e^(-x) - 2*(a^4 + 5*a^2*b^2 + 6*b^4)*e^(-2*x) - 4*(2* a^3*b + 3*a*b^3)*e^(-3*x) - 2*(a^4 - a^2*b^2 - 3*b^4)*e^(-4*x) + (a^3*b + 3*a*b^3)*e^(-5*x))/(a^5*b + a^3*b^3 + 2*(a^6 + a^4*b^2)*e^(-x) - 3*(a^5*b + a^3*b^3)*e^(-2*x) - 4*(a^6 + a^4*b^2)*e^(-3*x) + 3*(a^5*b + a^3*b^3)*e^( -4*x) + 2*(a^6 + a^4*b^2)*e^(-5*x) - (a^5*b + a^3*b^3)*e^(-6*x)) + 1/2*(a^ 2 - 6*b^2)*log(e^(-x) + 1)/a^4 - 1/2*(a^2 - 6*b^2)*log(e^(-x) - 1)/a^4
Time = 0.27 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.28 \[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=-\frac {{\left (4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + a^{4} b^{2}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (a b^{3} e^{x} - b^{4}\right )}}{{\left (a^{5} + a^{3} b^{2}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} + \frac {{\left (a^{2} - 6 \, b^{2}\right )} \log \left (e^{x} + 1\right )}{2 \, a^{4}} - \frac {{\left (a^{2} - 6 \, b^{2}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{2 \, a^{4}} - \frac {a e^{\left (3 \, x\right )} - 4 \, b e^{\left (2 \, x\right )} + a e^{x} + 4 \, b}{a^{3} {\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \]
-(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)
Time = 4.44 (sec) , antiderivative size = 977, normalized size of antiderivative = 6.18 \[ \int \frac {\text {csch}^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {4\,b}{a^3}-\frac {{\mathrm {e}}^x}{a^2}}{{\mathrm {e}}^{2\,x}-1}+\frac {\frac {2\,b^7}{a^3\,\left (a^2\,b^3+b^5\right )}-\frac {2\,b^6\,{\mathrm {e}}^x}{a^2\,\left (a^2\,b^3+b^5\right )}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}-\frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (a^2-6\,b^2\right )}{2\,a^4}+\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (a^2-6\,b^2\right )}{2\,a^4}-\frac {2\,{\mathrm {e}}^x}{a^2\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}+\frac {b^3\,\ln \left (\frac {8\,\left (4\,a^2+3\,b^2\right )\,\left (20\,a^9\,b^5-72\,b^{11}\,\sqrt {{\left (a^2+b^2\right )}^3}-9\,a^3\,b^{11}-30\,a^5\,b^9-18\,a^7\,b^7-2\,a^{13}\,b+15\,a^{11}\,b^3+4\,a^{14}\,{\mathrm {e}}^x-192\,a^2\,b^9\,\sqrt {{\left (a^2+b^2\right )}^3}-128\,a^4\,b^7\,\sqrt {{\left (a^2+b^2\right )}^3}+27\,a^4\,b^{10}\,{\mathrm {e}}^x+72\,a^6\,b^8\,{\mathrm {e}}^x+30\,a^8\,b^6\,{\mathrm {e}}^x-48\,a^{10}\,b^4\,{\mathrm {e}}^x-29\,a^{12}\,b^2\,{\mathrm {e}}^x+312\,a^3\,b^8\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+206\,a^5\,b^6\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+8\,a\,b^4\,{\mathrm {e}}^x\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}+118\,a\,b^{10}\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}\right )}{a^9\,b^2\,\sqrt {{\left (a^2+b^2\right )}^3}\,{\left (a^2+b^2\right )}^4}-\frac {8\,\left (-4\,a^4+21\,a^2\,b^2+18\,b^4\right )\,\left (-4\,{\mathrm {e}}^x\,a^5+2\,a^4\,b+19\,{\mathrm {e}}^x\,a^3\,b^2-10\,a^2\,b^3+21\,{\mathrm {e}}^x\,a\,b^4-12\,b^5\right )}{a^9\,b^2\,{\left (a^2+b^2\right )}^2}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}\,\left (4\,a^2+3\,b^2\right )}{a^{10}+3\,a^8\,b^2+3\,a^6\,b^4+a^4\,b^6}-\frac {b^3\,\ln \left (\frac {8\,\left (4\,a^2+3\,b^2\right )\,\left (2\,a^{13}\,b-72\,b^{11}\,\sqrt {{\left (a^2+b^2\right )}^3}+9\,a^3\,b^{11}+30\,a^5\,b^9+18\,a^7\,b^7-20\,a^9\,b^5-15\,a^{11}\,b^3-4\,a^{14}\,{\mathrm {e}}^x-192\,a^2\,b^9\,\sqrt {{\left (a^2+b^2\right )}^3}-128\,a^4\,b^7\,\sqrt {{\left (a^2+b^2\right )}^3}-27\,a^4\,b^{10}\,{\mathrm {e}}^x-72\,a^6\,b^8\,{\mathrm {e}}^x-30\,a^8\,b^6\,{\mathrm {e}}^x+48\,a^{10}\,b^4\,{\mathrm {e}}^x+29\,a^{12}\,b^2\,{\mathrm {e}}^x+312\,a^3\,b^8\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+206\,a^5\,b^6\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+8\,a\,b^4\,{\mathrm {e}}^x\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}+118\,a\,b^{10}\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}\right )}{a^9\,b^2\,\sqrt {{\left (a^2+b^2\right )}^3}\,{\left (a^2+b^2\right )}^4}-\frac {8\,\left (-4\,a^4+21\,a^2\,b^2+18\,b^4\right )\,\left (-4\,{\mathrm {e}}^x\,a^5+2\,a^4\,b+19\,{\mathrm {e}}^x\,a^3\,b^2-10\,a^2\,b^3+21\,{\mathrm {e}}^x\,a\,b^4-12\,b^5\right )}{a^9\,b^2\,{\left (a^2+b^2\right )}^2}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}\,\left (4\,a^2+3\,b^2\right )}{a^{10}+3\,a^8\,b^2+3\,a^6\,b^4+a^4\,b^6} \]
((4*b)/a^3 - exp(x)/a^2)/(exp(2*x) - 1) + ((2*b^7)/(a^3*(b^5 + a^2*b^3)) - (2*b^6*exp(x))/(a^2*(b^5 + a^2*b^3)))/(2*a*exp(x) - b + b*exp(2*x)) - (lo g(exp(x) - 1)*(a^2 - 6*b^2))/(2*a^4) + (log(exp(x) + 1)*(a^2 - 6*b^2))/(2* a^4) - (2*exp(x))/(a^2*(exp(4*x) - 2*exp(2*x) + 1)) + (b^3*log((8*(4*a^2 + 3*b^2)*(20*a^9*b^5 - 72*b^11*((a^2 + b^2)^3)^(1/2) - 9*a^3*b^11 - 30*a^5* b^9 - 18*a^7*b^7 - 2*a^13*b + 15*a^11*b^3 + 4*a^14*exp(x) - 192*a^2*b^9*(( a^2 + b^2)^3)^(1/2) - 128*a^4*b^7*((a^2 + b^2)^3)^(1/2) + 27*a^4*b^10*exp( x) + 72*a^6*b^8*exp(x) + 30*a^8*b^6*exp(x) - 48*a^10*b^4*exp(x) - 29*a^12* b^2*exp(x) + 312*a^3*b^8*exp(x)*((a^2 + b^2)^3)^(1/2) + 206*a^5*b^6*exp(x) *((a^2 + b^2)^3)^(1/2) + 8*a*b^4*exp(x)*((a^2 + b^2)^3)^(3/2) + 118*a*b^10 *exp(x)*((a^2 + b^2)^3)^(1/2)))/(a^9*b^2*((a^2 + b^2)^3)^(1/2)*(a^2 + b^2) ^4) - (8*(18*b^4 - 4*a^4 + 21*a^2*b^2)*(2*a^4*b - 12*b^5 - 10*a^2*b^3 - 4* a^5*exp(x) + 21*a*b^4*exp(x) + 19*a^3*b^2*exp(x)))/(a^9*b^2*(a^2 + b^2)^2) )*((a^2 + b^2)^3)^(1/2)*(4*a^2 + 3*b^2))/(a^10 + a^4*b^6 + 3*a^6*b^4 + 3*a ^8*b^2) - (b^3*log((8*(4*a^2 + 3*b^2)*(2*a^13*b - 72*b^11*((a^2 + b^2)^3)^ (1/2) + 9*a^3*b^11 + 30*a^5*b^9 + 18*a^7*b^7 - 20*a^9*b^5 - 15*a^11*b^3 - 4*a^14*exp(x) - 192*a^2*b^9*((a^2 + b^2)^3)^(1/2) - 128*a^4*b^7*((a^2 + b^ 2)^3)^(1/2) - 27*a^4*b^10*exp(x) - 72*a^6*b^8*exp(x) - 30*a^8*b^6*exp(x) + 48*a^10*b^4*exp(x) + 29*a^12*b^2*exp(x) + 312*a^3*b^8*exp(x)*((a^2 + b^2) ^3)^(1/2) + 206*a^5*b^6*exp(x)*((a^2 + b^2)^3)^(1/2) + 8*a*b^4*exp(x)*(...