Integrand size = 13, antiderivative size = 159 \[ \int \frac {\coth ^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {b \left (3 a^2+4 b^2\right ) \text {arctanh}(\cosh (x))}{a^5}-\frac {2 \sqrt {a^2+b^2} \left (a^2+4 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^5}-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))} \] Output:
b*(3*a^2+4*b^2)*arctanh(cosh(x))/a^5-2*(a^2+b^2)^(1/2)*(a^2+4*b^2)*arctanh ((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/a^5-1/3*(7*a^2+12*b^2)*coth(x)/a^4+(a^ 2+2*b^2)*coth(x)*csch(x)/a^3/b-1/3*(3+4*b^2/a^2)*coth(x)*csch(x)/b/(a+b*si nh(x))-1/3*coth(x)*csch(x)^2/a/(a+b*sinh(x))
Time = 0.59 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.48 \[ \int \frac {\coth ^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {48 \left (a^4+5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-4 a \left (4 a^2+9 b^2\right ) \coth \left (\frac {x}{2}\right )+6 a^2 b \text {csch}^2\left (\frac {x}{2}\right )+24 b \left (3 a^2+4 b^2\right ) \log \left (\cosh \left (\frac {x}{2}\right )\right )-24 b \left (3 a^2+4 b^2\right ) \log \left (\sinh \left (\frac {x}{2}\right )\right )+6 a^2 b \text {sech}^2\left (\frac {x}{2}\right )+8 a^3 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )-\frac {1}{2} a^3 \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)-\frac {24 a b \left (a^2+b^2\right ) \cosh (x)}{a+b \sinh (x)}-4 a \left (4 a^2+9 b^2\right ) \tanh \left (\frac {x}{2}\right )}{24 a^5} \] Input:
Integrate[Coth[x]^4/(a + b*Sinh[x])^2,x]
Output:
((48*(a^4 + 5*a^2*b^2 + 4*b^4)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]]) /Sqrt[-a^2 - b^2] - 4*a*(4*a^2 + 9*b^2)*Coth[x/2] + 6*a^2*b*Csch[x/2]^2 + 24*b*(3*a^2 + 4*b^2)*Log[Cosh[x/2]] - 24*b*(3*a^2 + 4*b^2)*Log[Sinh[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*Sin h[x])/2 - (24*a*b*(a^2 + b^2)*Cosh[x])/(a + b*Sinh[x]) - 4*a*(4*a^2 + 9*b^ 2)*Tanh[x/2])/(24*a^5)
Result contains complex when optimal does not.
Time = 1.48 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.25, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.615, Rules used = {3042, 3203, 26, 3042, 26, 3534, 27, 3042, 25, 3534, 27, 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.
| \(\displaystyle \int \frac {\coth ^4(x)}{(a+b \sinh (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (i x)^4 (a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 3203 |
| \(\displaystyle \frac {i \int \frac {i \text {csch}^3(x) \left (\left (3 a^2+8 b^2\right ) \sinh ^2(x)-a b \sinh (x)+6 \left (a^2+2 b^2\right )\right )}{a+b \sinh (x)}dx}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\int \frac {\text {csch}^3(x) \left (\left (3 a^2+8 b^2\right ) \sinh ^2(x)-a b \sinh (x)+6 \left (a^2+2 b^2\right )\right )}{a+b \sinh (x)}dx}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int -\frac {i \left (-\left (\left (3 a^2+8 b^2\right ) \sin (i x)^2\right )+i a b \sin (i x)+6 \left (a^2+2 b^2\right )\right )}{\sin (i x)^3 (a-i b \sin (i x))}dx}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \int \frac {-\left (\left (3 a^2+8 b^2\right ) \sin (i x)^2\right )+i a b \sin (i x)+6 \left (a^2+2 b^2\right )}{\sin (i x)^3 (a-i b \sin (i x))}dx}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {i \left (\frac {\int -\frac {2 i \text {csch}^2(x) \left (-2 a \sinh (x) b^2+3 \left (a^2+2 b^2\right ) \sinh ^2(x) b+\left (7 a^2+12 b^2\right ) b\right )}{a+b \sinh (x)}dx}{2 a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {i \left (-\frac {i \int \frac {\text {csch}^2(x) \left (-2 a \sinh (x) b^2+3 \left (a^2+2 b^2\right ) \sinh ^2(x) b+\left (7 a^2+12 b^2\right ) b\right )}{a+b \sinh (x)}dx}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {i \int -\frac {2 i a \sin (i x) b^2-3 \left (a^2+2 b^2\right ) \sin (i x)^2 b+\left (7 a^2+12 b^2\right ) b}{\sin (i x)^2 (a-i b \sin (i x))}dx}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \left (\frac {i \int \frac {2 i a \sin (i x) b^2-3 \left (a^2+2 b^2\right ) \sin (i x)^2 b+\left (7 a^2+12 b^2\right ) b}{\sin (i x)^2 (a-i b \sin (i x))}dx}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {\int \frac {3 \text {csch}(x) \left (b^2 \left (3 a^2+4 b^2\right )-a b \left (a^2+2 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)}dx}{a}+\frac {b \left (7 a^2+12 b^2\right ) \coth (x)}{a}\right )}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {3 \int \frac {\text {csch}(x) \left (b^2 \left (3 a^2+4 b^2\right )-a b \left (a^2+2 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)}dx}{a}+\frac {b \left (7 a^2+12 b^2\right ) \coth (x)}{a}\right )}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {b \left (7 a^2+12 b^2\right ) \coth (x)}{a}+\frac {3 \int \frac {i \left (\left (3 a^2+4 b^2\right ) b^2+i a \left (a^2+2 b^2\right ) \sin (i x) b\right )}{\sin (i x) (a-i b \sin (i x))}dx}{a}\right )}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {b \left (7 a^2+12 b^2\right ) \coth (x)}{a}+\frac {3 i \int \frac {\left (3 a^2+4 b^2\right ) b^2+i a \left (a^2+2 b^2\right ) \sin (i x) b}{\sin (i x) (a-i b \sin (i x))}dx}{a}\right )}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {b \left (7 a^2+12 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {i b \left (a^2+b^2\right ) \left (a^2+4 b^2\right ) \int \frac {1}{a+b \sinh (x)}dx}{a}+\frac {b^2 \left (3 a^2+4 b^2\right ) \int -i \text {csch}(x)dx}{a}\right )}{a}\right )}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {b \left (7 a^2+12 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {i b \left (a^2+b^2\right ) \left (a^2+4 b^2\right ) \int \frac {1}{a+b \sinh (x)}dx}{a}-\frac {i b^2 \left (3 a^2+4 b^2\right ) \int \text {csch}(x)dx}{a}\right )}{a}\right )}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {b \left (7 a^2+12 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {i b \left (a^2+b^2\right ) \left (a^2+4 b^2\right ) \int \frac {1}{a-i b \sin (i x)}dx}{a}-\frac {i b^2 \left (3 a^2+4 b^2\right ) \int i \csc (i x)dx}{a}\right )}{a}\right )}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {b \left (7 a^2+12 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {i b \left (a^2+b^2\right ) \left (a^2+4 b^2\right ) \int \frac {1}{a-i b \sin (i x)}dx}{a}+\frac {b^2 \left (3 a^2+4 b^2\right ) \int \csc (i x)dx}{a}\right )}{a}\right )}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {b \left (7 a^2+12 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {b^2 \left (3 a^2+4 b^2\right ) \int \csc (i x)dx}{a}+\frac {2 i b \left (a^2+b^2\right ) \left (a^2+4 b^2\right ) \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{a}\right )}{a}\right )}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {b \left (7 a^2+12 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {b^2 \left (3 a^2+4 b^2\right ) \int \csc (i x)dx}{a}-\frac {4 i b \left (a^2+b^2\right ) \left (a^2+4 b^2\right ) \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a}\right )}{a}\right )}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {b \left (7 a^2+12 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {b^2 \left (3 a^2+4 b^2\right ) \int \csc (i x)dx}{a}-\frac {2 i b \sqrt {a^2+b^2} \left (a^2+4 b^2\right ) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a}\right )}{a}\right )}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {b \left (7 a^2+12 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {i b^2 \left (3 a^2+4 b^2\right ) \text {arctanh}(\cosh (x))}{a}-\frac {2 i b \sqrt {a^2+b^2} \left (a^2+4 b^2\right ) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a}\right )}{a}\right )}{a}-\frac {3 i \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a^2 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\) |
Input:
Int[Coth[x]^4/(a + b*Sinh[x])^2,x]
Output:
((I/3)*((I*(((3*I)*((I*b^2*(3*a^2 + 4*b^2)*ArcTanh[Cosh[x]])/a - ((2*I)*b* Sqrt[a^2 + b^2]*(a^2 + 4*b^2)*ArcTanh[(2*b - 2*a*Tanh[x/2])/(2*Sqrt[a^2 + b^2])])/a))/a + (b*(7*a^2 + 12*b^2)*Coth[x])/a))/a - ((3*I)*(a^2 + 2*b^2)* Coth[x]*Csch[x])/a))/(a^2*b) - ((3 + (4*b^2)/a^2)*Coth[x]*Csch[x])/(3*b*(a + b*Sinh[x])) - (Coth[x]*Csch[x]^2)/(3*a*(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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Simp[(-Cos[e + f*x])*((a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[ e + f*x]^3)), x] + (-Simp[(3*a^2 + b^2*(m - 2))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(3*a^2*b*f*(m + 1)*Sin[e + f*x]^2)), x] - Simp[1/(3*a^2*b*( m + 1)) Int[((a + b*Sin[e + f*x])^(m + 1)/Sin[e + f*x]^3)*Simp[6*a^2 - b^ 2*(m - 1)*(m - 2) + a*b*(m + 1)*Sin[e + f*x] - (3*a^2 - b^2*m*(m - 2))*Sin[ e + f*x]^2, x], x], x]) /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && L tQ[m, -1] && IntegerQ[2*m]
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 = 8.51 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.39
| method | result | size |
| default | \(-\frac {\frac {\tanh \left (\frac {x}{2}\right )^{3} a^{2}}{3}+2 \tanh \left (\frac {x}{2}\right )^{2} a b +5 \tanh \left (\frac {x}{2}\right ) a^{2}+12 b^{2} \tanh \left (\frac {x}{2}\right )}{8 a^{4}}-\frac {1}{24 a^{2} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {5 a^{2}+12 b^{2}}{8 a^{4} \tanh \left (\frac {x}{2}\right )}+\frac {b}{4 a^{3} \tanh \left (\frac {x}{2}\right )^{2}}-\frac {b \left (3 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{5}}-\frac {2 \left (\frac {-b^{2} \left (a^{2}+b^{2}\right ) \tanh \left (\frac {x}{2}\right )-\left (a^{2}+b^{2}\right ) a b}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (a^{4}+5 a^{2} b^{2}+4 b^{4}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{a^{5}}\) | \(221\) |
| risch | \(\frac {2 a^{3} {\mathrm e}^{7 x}+4 a \,b^{2} {\mathrm e}^{7 x}-2 a^{2} b \,{\mathrm e}^{6 x}-8 b^{3} {\mathrm e}^{6 x}-14 a^{3} {\mathrm e}^{5 x}-20 a \,b^{2} {\mathrm e}^{5 x}+14 a^{2} b \,{\mathrm e}^{4 x}+24 \,{\mathrm e}^{4 x} b^{3}+14 a^{3} {\mathrm e}^{3 x}+28 a \,b^{2} {\mathrm e}^{3 x}-\frac {50 a^{2} b \,{\mathrm e}^{2 x}}{3}-24 b^{3} {\mathrm e}^{2 x}-\frac {22 a^{3} {\mathrm e}^{x}}{3}-12 a \,b^{2} {\mathrm e}^{x}+\frac {14 a^{2} b}{3}+8 b^{3}}{\left ({\mathrm e}^{2 x}-1\right )^{3} a^{4} \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}-\frac {3 b \ln \left ({\mathrm e}^{x}-1\right )}{a^{3}}-\frac {4 b^{3} \ln \left ({\mathrm e}^{x}-1\right )}{a^{5}}+\frac {3 b \ln \left ({\mathrm e}^{x}+1\right )}{a^{3}}+\frac {4 b^{3} \ln \left ({\mathrm e}^{x}+1\right )}{a^{5}}+\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{a^{3}}+\frac {4 \sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right ) b^{2}}{a^{5}}-\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{a^{3}}-\frac {4 \sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right ) b^{2}}{a^{5}}\) | \(364\) |
Input:
int(coth(x)^4/(a+b*sinh(x))^2,x,method=_RETURNVERBOSE)
Output:
-1/8/a^4*(1/3*tanh(1/2*x)^3*a^2+2*tanh(1/2*x)^2*a*b+5*tanh(1/2*x)*a^2+12*b ^2*tanh(1/2*x))-1/24/a^2/tanh(1/2*x)^3-1/8*(5*a^2+12*b^2)/a^4/tanh(1/2*x)+ 1/4/a^3*b/tanh(1/2*x)^2-1/a^5*b*(3*a^2+4*b^2)*ln(tanh(1/2*x))-2/a^5*((-b^2 *(a^2+b^2)*tanh(1/2*x)-(a^2+b^2)*a*b)/(tanh(1/2*x)^2*a-2*b*tanh(1/2*x)-a)- (a^4+5*a^2*b^2+4*b^4)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a ^2+b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 3648 vs. \(2 (149) = 298\).
Time = 0.16 (sec) , antiderivative size = 3648, normalized size of antiderivative = 22.94 \[ \int \frac {\coth ^4(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \] Input:
integrate(coth(x)^4/(a+b*sinh(x))^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\coth ^4(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\coth ^{4}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \] Input:
integrate(coth(x)**4/(a+b*sinh(x))**2,x)
Output:
Integral(coth(x)**4/(a + b*sinh(x))**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (149) = 298\).
Time = 0.13 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.13 \[ \int \frac {\coth ^4(x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 \, {\left (7 \, a^{2} b + 12 \, b^{3} + {\left (11 \, a^{3} + 18 \, a b^{2}\right )} e^{\left (-x\right )} - {\left (25 \, a^{2} b + 36 \, b^{3}\right )} e^{\left (-2 \, x\right )} - 21 \, {\left (a^{3} + 2 \, a b^{2}\right )} e^{\left (-3 \, x\right )} + 3 \, {\left (7 \, a^{2} b + 12 \, b^{3}\right )} e^{\left (-4 \, x\right )} + 3 \, {\left (7 \, a^{3} + 10 \, a b^{2}\right )} e^{\left (-5 \, x\right )} - 3 \, {\left (a^{2} b + 4 \, b^{3}\right )} e^{\left (-6 \, x\right )} - 3 \, {\left (a^{3} + 2 \, a b^{2}\right )} e^{\left (-7 \, x\right )}\right )}}{3 \, {\left (2 \, a^{5} e^{\left (-x\right )} - 4 \, a^{4} b e^{\left (-2 \, x\right )} - 6 \, a^{5} e^{\left (-3 \, x\right )} + 6 \, a^{4} b e^{\left (-4 \, x\right )} + 6 \, a^{5} e^{\left (-5 \, x\right )} - 4 \, a^{4} b e^{\left (-6 \, x\right )} - 2 \, a^{5} e^{\left (-7 \, x\right )} + a^{4} b e^{\left (-8 \, x\right )} + a^{4} b\right )}} + \frac {{\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{5}} - \frac {{\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{5}} + \frac {{\left (a^{4} + 5 \, a^{2} b^{2} + 4 \, b^{4}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{5}} \] Input:
integrate(coth(x)^4/(a+b*sinh(x))^2,x, algorithm="maxima")
Output:
-2/3*(7*a^2*b + 12*b^3 + (11*a^3 + 18*a*b^2)*e^(-x) - (25*a^2*b + 36*b^3)* e^(-2*x) - 21*(a^3 + 2*a*b^2)*e^(-3*x) + 3*(7*a^2*b + 12*b^3)*e^(-4*x) + 3 *(7*a^3 + 10*a*b^2)*e^(-5*x) - 3*(a^2*b + 4*b^3)*e^(-6*x) - 3*(a^3 + 2*a*b ^2)*e^(-7*x))/(2*a^5*e^(-x) - 4*a^4*b*e^(-2*x) - 6*a^5*e^(-3*x) + 6*a^4*b* e^(-4*x) + 6*a^5*e^(-5*x) - 4*a^4*b*e^(-6*x) - 2*a^5*e^(-7*x) + a^4*b*e^(- 8*x) + a^4*b) + (3*a^2*b + 4*b^3)*log(e^(-x) + 1)/a^5 - (3*a^2*b + 4*b^3)* log(e^(-x) - 1)/a^5 + (a^4 + 5*a^2*b^2 + 4*b^4)*log((b*e^(-x) - a - sqrt(a ^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^5)
Time = 0.14 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.52 \[ \int \frac {\coth ^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left (e^{x} + 1\right )}{a^{5}} - \frac {{\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{a^{5}} + \frac {{\left (a^{4} + 5 \, a^{2} b^{2} + 4 \, b^{4}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{5}} + \frac {2 \, {\left (a^{3} e^{x} + a b^{2} e^{x} - a^{2} b - b^{3}\right )}}{{\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )} a^{4}} + \frac {2 \, {\left (3 \, a b e^{\left (5 \, x\right )} - 6 \, a^{2} e^{\left (4 \, x\right )} - 9 \, b^{2} e^{\left (4 \, x\right )} + 6 \, a^{2} e^{\left (2 \, x\right )} + 18 \, b^{2} e^{\left (2 \, x\right )} - 3 \, a b e^{x} - 4 \, a^{2} - 9 \, b^{2}\right )}}{3 \, a^{4} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \] Input:
integrate(coth(x)^4/(a+b*sinh(x))^2,x, algorithm="giac")
Output:
(3*a^2*b + 4*b^3)*log(e^x + 1)/a^5 - (3*a^2*b + 4*b^3)*log(abs(e^x - 1))/a ^5 + (a^4 + 5*a^2*b^2 + 4*b^4)*log(abs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/ abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^5) + 2*(a^3*e^x + a*b^2*e^x - a^2*b - b^3)/((b*e^(2*x) + 2*a*e^x - b)*a^4) + 2/3*(3*a*b*e ^(5*x) - 6*a^2*e^(4*x) - 9*b^2*e^(4*x) + 6*a^2*e^(2*x) + 18*b^2*e^(2*x) - 3*a*b*e^x - 4*a^2 - 9*b^2)/(a^4*(e^(2*x) - 1)^3)
Time = 2.34 (sec) , antiderivative size = 1450, normalized size of antiderivative = 9.12 \[ \int \frac {\coth ^4(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \] Input:
int(coth(x)^4/(a + b*sinh(x))^2,x)
Output:
(3*b*log(96*a^4 + 128*b^4 + 224*a^2*b^2 + 96*a^4*exp(x) + 128*b^4*exp(x) + 224*a^2*b^2*exp(x)))/a^3 - 4/(a^2*exp(2*x) - a^2) - (6*b^2)/(a^4*exp(2*x) - a^4) - 8/(3*(3*a^2*exp(2*x) - 3*a^2*exp(4*x) + a^2*exp(6*x) - a^2)) - ( 4*a^3*b^7)/(a^5*b^7*exp(2*x) - a^7*b^5 - a^5*b^7 + a^7*b^5*exp(2*x) + 2*a^ 6*b^6*exp(x) + 2*a^8*b^4*exp(x)) - (2*a^5*b^5)/(a^5*b^7*exp(2*x) - a^7*b^5 - a^5*b^7 + a^7*b^5*exp(2*x) + 2*a^6*b^6*exp(x) + 2*a^8*b^4*exp(x)) - (3* b*log(96*a^4 + 128*b^4 + 224*a^2*b^2 - 96*a^4*exp(x) - 128*b^4*exp(x) - 22 4*a^2*b^2*exp(x)))/a^3 - 4/(a^2*exp(4*x) - 2*a^2*exp(2*x) + a^2) - (4*b^3* log(96*a^4 + 128*b^4 + 224*a^2*b^2 - 96*a^4*exp(x) - 128*b^4*exp(x) - 224* a^2*b^2*exp(x)))/a^5 + (4*b^3*log(96*a^4 + 128*b^4 + 224*a^2*b^2 + 96*a^4* exp(x) + 128*b^4*exp(x) + 224*a^2*b^2*exp(x)))/a^5 + (log(128*a^6*exp(x) - 256*a*b^5 - 64*a^5*b - 320*a^3*b^3 - 128*b^5*(a^2 + b^2)^(1/2) + 128*b^6* exp(x) - 288*a^2*b^3*(a^2 + b^2)^(1/2) + 128*a^5*exp(x)*(a^2 + b^2)^(1/2) + 672*a^2*b^4*exp(x) + 672*a^4*b^2*exp(x) - 64*a^4*b*(a^2 + b^2)^(1/2) + 3 84*a*b^4*exp(x)*(a^2 + b^2)^(1/2) + 608*a^3*b^2*exp(x)*(a^2 + b^2)^(1/2))* (a^2 + b^2)^(1/2))/a^3 - (log(128*b^5*(a^2 + b^2)^(1/2) - 256*a*b^5 - 64*a ^5*b - 320*a^3*b^3 + 128*a^6*exp(x) + 128*b^6*exp(x) + 288*a^2*b^3*(a^2 + b^2)^(1/2) - 128*a^5*exp(x)*(a^2 + b^2)^(1/2) + 672*a^2*b^4*exp(x) + 672*a ^4*b^2*exp(x) + 64*a^4*b*(a^2 + b^2)^(1/2) - 384*a*b^4*exp(x)*(a^2 + b^2)^ (1/2) - 608*a^3*b^2*exp(x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/a^3 - ...
Time = 0.16 (sec) , antiderivative size = 1552, normalized size of antiderivative = 9.76 \[ \int \frac {\coth ^4(x)}{(a+b \sinh (x))^2} \, dx =\text {Too large to display} \] Input:
int(coth(x)^4/(a+b*sinh(x))^2,x)
Output:
(6*e**(8*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a** 2*b*i + 24*e**(8*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b* *2))*b**3*i + 12*e**(7*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a** 2 + b**2))*a**3*i + 48*e**(7*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sq rt(a**2 + b**2))*a*b**2*i - 24*e**(6*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b*i - 96*e**(6*x)*sqrt(a**2 + b**2)*atan((e* *x*b*i + a*i)/sqrt(a**2 + b**2))*b**3*i - 36*e**(5*x)*sqrt(a**2 + b**2)*at an((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**3*i - 144*e**(5*x)*sqrt(a**2 + b **2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a*b**2*i + 36*e**(4*x)*sqrt( a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b*i + 144*e**(4 *x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*b**3*i + 36 *e**(3*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**3* i + 144*e**(3*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2) )*a*b**2*i - 24*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b*i - 96*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/s qrt(a**2 + b**2))*b**3*i - 12*e**x*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i) /sqrt(a**2 + b**2))*a**3*i - 48*e**x*sqrt(a**2 + b**2)*atan((e**x*b*i + a* i)/sqrt(a**2 + b**2))*a*b**2*i + 6*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i) /sqrt(a**2 + b**2))*a**2*b*i + 24*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/ sqrt(a**2 + b**2))*b**3*i - 9*e**(8*x)*log(e**x - 1)*a**2*b**2 - 12*e**...