Integrand size = 13, antiderivative size = 198 \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=-\frac {b \left (a^2-4 b^2\right ) \text {arctanh}(\cosh (x))}{a^5}-\frac {2 b^4 \left (5 a^2+4 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^5 \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac {b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))} \] Output:
-b*(a^2-4*b^2)*arctanh(cosh(x))/a^5-2*b^4*(5*a^2+4*b^2)*arctanh((b-a*tanh( 1/2*x))/(a^2+b^2)^(1/2))/a^5/(a^2+b^2)^(3/2)+1/3*(2*a^4-7*a^2*b^2-12*b^4)* coth(x)/a^4/(a^2+b^2)+b*(a^2+2*b^2)*coth(x)*csch(x)/a^3/(a^2+b^2)-1/3*(a^2 +4*b^2)*coth(x)*csch(x)^2/a^2/(a^2+b^2)+b^2*coth(x)*csch(x)^2/a/(a^2+b^2)/ (a+b*sinh(x))
Time = 1.06 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.17 \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {-\frac {48 b^4 \left (5 a^2+4 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+4 a \left (2 a^2-9 b^2\right ) \coth \left (\frac {x}{2}\right )+6 a^2 b \text {csch}^2\left (\frac {x}{2}\right )-24 b \left (a^2-4 b^2\right ) \log \left (\cosh \left (\frac {x}{2}\right )\right )+24 b \left (a^2-4 b^2\right ) \log \left (\sinh \left (\frac {x}{2}\right )\right )+6 a^2 b \text {sech}^2\left (\frac {x}{2}\right )+8 a^3 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )-\frac {1}{2} a^3 \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)-\frac {24 a b^5 \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+4 a \left (2 a^2-9 b^2\right ) \tanh \left (\frac {x}{2}\right )}{24 a^5} \] Input:
Integrate[Csch[x]^4/(a + b*Sinh[x])^2,x]
Output:
((-48*b^4*(5*a^2 + 4*b^2)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/(-a^ 2 - b^2)^(3/2) + 4*a*(2*a^2 - 9*b^2)*Coth[x/2] + 6*a^2*b*Csch[x/2]^2 - 24* b*(a^2 - 4*b^2)*Log[Cosh[x/2]] + 24*b*(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*Sinh[x])/2 - (24*a*b^5*Cosh[x])/((a^2 + b^2)*(a + b*Sinh[x])) + 4*a*(2*a^2 - 9*b^2)* Tanh[x/2])/(24*a^5)
Result contains complex when optimal does not.
Time = 1.87 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.15, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.769, Rules used = {3042, 3281, 3042, 3534, 25, 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.
| \(\displaystyle \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (i x)^4 (a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle \frac {\int \frac {\text {csch}^4(x) \left (a^2-b \sinh (x) a+4 b^2+3 b^2 \sinh ^2(x)\right )}{a+b \sinh (x)}dx}{a \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\int \frac {a^2+i b \sin (i x) a+4 b^2-3 b^2 \sin (i x)^2}{\sin (i x)^4 (a-i b \sin (i x))}dx}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int -\frac {\text {csch}^3(x) \left (2 b \left (a^2+4 b^2\right ) \sinh ^2(x)+a \left (2 a^2-b^2\right ) \sinh (x)+6 b \left (a^2+2 b^2\right )\right )}{a+b \sinh (x)}dx}{3 a}-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}}{a \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {\text {csch}^3(x) \left (2 b \left (a^2+4 b^2\right ) \sinh ^2(x)+a \left (2 a^2-b^2\right ) \sinh (x)+6 b \left (a^2+2 b^2\right )\right )}{a+b \sinh (x)}dx}{3 a}-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}}{a \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}-\frac {\int -\frac {i \left (-2 b \left (a^2+4 b^2\right ) \sin (i x)^2-i a \left (2 a^2-b^2\right ) \sin (i x)+6 b \left (a^2+2 b^2\right )\right )}{\sin (i x)^3 (a-i b \sin (i x))}dx}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \int \frac {-2 b \left (a^2+4 b^2\right ) \sin (i x)^2-i a \left (2 a^2-b^2\right ) \sin (i x)+6 b \left (a^2+2 b^2\right )}{\sin (i x)^3 (a-i b \sin (i x))}dx}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (\frac {\int \frac {2 i \text {csch}^2(x) \left (2 a^4-7 b^2 a^2-b \left (a^2-2 b^2\right ) \sinh (x) a-12 b^4-3 b^2 \left (a^2+2 b^2\right ) \sinh ^2(x)\right )}{a+b \sinh (x)}dx}{2 a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (\frac {i \int \frac {\text {csch}^2(x) \left (2 a^4-7 b^2 a^2-b \left (a^2-2 b^2\right ) \sinh (x) a-12 b^4-3 b^2 \left (a^2+2 b^2\right ) \sinh ^2(x)\right )}{a+b \sinh (x)}dx}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (\frac {i \int -\frac {2 a^4-7 b^2 a^2+i b \left (a^2-2 b^2\right ) \sin (i x) a-12 b^4+3 b^2 \left (a^2+2 b^2\right ) \sin (i x)^2}{\sin (i x)^2 (a-i b \sin (i x))}dx}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {i \int \frac {2 a^4-7 b^2 a^2+i b \left (a^2-2 b^2\right ) \sin (i x) a-12 b^4+3 b^2 \left (a^2+2 b^2\right ) \sin (i x)^2}{\sin (i x)^2 (a-i b \sin (i x))}dx}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {i \left (\frac {\int \frac {3 \text {csch}(x) \left (a \left (a^2+2 b^2\right ) \sinh (x) b^2+\left (a^4-3 b^2 a^2-4 b^4\right ) b\right )}{a+b \sinh (x)}dx}{a}+\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{a}\right )}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {i \left (\frac {3 \int \frac {\text {csch}(x) \left (a \left (a^2+2 b^2\right ) \sinh (x) b^2+\left (a^4-3 b^2 a^2-4 b^4\right ) b\right )}{a+b \sinh (x)}dx}{a}+\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{a}\right )}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {i \left (\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{a}+\frac {3 \int \frac {i \left (b \left (a^4-3 b^2 a^2-4 b^4\right )-i a b^2 \left (a^2+2 b^2\right ) \sin (i x)\right )}{\sin (i x) (a-i b \sin (i x))}dx}{a}\right )}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {i \left (\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{a}+\frac {3 i \int \frac {b \left (a^4-3 b^2 a^2-4 b^4\right )-i a b^2 \left (a^2+2 b^2\right ) \sin (i x)}{\sin (i x) (a-i b \sin (i x))}dx}{a}\right )}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {i \left (\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{a}+\frac {3 i \left (\frac {b \left (a^4-3 a^2 b^2-4 b^4\right ) \int -i \text {csch}(x)dx}{a}-\frac {i b^4 \left (5 a^2+4 b^2\right ) \int \frac {1}{a+b \sinh (x)}dx}{a}\right )}{a}\right )}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {i \left (\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{a}+\frac {3 i \left (-\frac {i b^4 \left (5 a^2+4 b^2\right ) \int \frac {1}{a+b \sinh (x)}dx}{a}-\frac {i b \left (a^4-3 a^2 b^2-4 b^4\right ) \int \text {csch}(x)dx}{a}\right )}{a}\right )}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {i \left (\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{a}+\frac {3 i \left (-\frac {i b^4 \left (5 a^2+4 b^2\right ) \int \frac {1}{a-i b \sin (i x)}dx}{a}-\frac {i b \left (a^4-3 a^2 b^2-4 b^4\right ) \int i \csc (i x)dx}{a}\right )}{a}\right )}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {i \left (\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{a}+\frac {3 i \left (\frac {b \left (a^4-3 a^2 b^2-4 b^4\right ) \int \csc (i x)dx}{a}-\frac {i b^4 \left (5 a^2+4 b^2\right ) \int \frac {1}{a-i b \sin (i x)}dx}{a}\right )}{a}\right )}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {i \left (\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{a}+\frac {3 i \left (\frac {b \left (a^4-3 a^2 b^2-4 b^4\right ) \int \csc (i x)dx}{a}-\frac {2 i b^4 \left (5 a^2+4 b^2\right ) \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{a}\right )}{a}\right )}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {i \left (\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{a}+\frac {3 i \left (\frac {4 i b^4 \left (5 a^2+4 b^2\right ) \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a}+\frac {b \left (a^4-3 a^2 b^2-4 b^4\right ) \int \csc (i x)dx}{a}\right )}{a}\right )}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {i \left (\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{a}+\frac {3 i \left (\frac {b \left (a^4-3 a^2 b^2-4 b^4\right ) \int \csc (i x)dx}{a}+\frac {2 i b^4 \left (5 a^2+4 b^2\right ) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}\right )}{a}\right )}{a}-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {3 i b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a}-\frac {i \left (\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{a}+\frac {3 i \left (\frac {2 i b^4 \left (5 a^2+4 b^2\right ) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}+\frac {i b \left (a^4-3 a^2 b^2-4 b^4\right ) \text {arctanh}(\cosh (x))}{a}\right )}{a}\right )}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}\) |
Input:
Int[Csch[x]^4/(a + b*Sinh[x])^2,x]
Output:
(-1/3*((a^2 + 4*b^2)*Coth[x]*Csch[x]^2)/a + ((I/3)*(((-I)*(((3*I)*((I*b*(a ^4 - 3*a^2*b^2 - 4*b^4)*ArcTanh[Cosh[x]])/a + ((2*I)*b^4*(5*a^2 + 4*b^2)*A rcTanh[(2*b - 2*a*Tanh[x/2])/(2*Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2])))/a + ((2*a^4 - 7*a^2*b^2 - 12*b^4)*Coth[x])/a))/a - ((3*I)*b*(a^2 + 2*b^2)*C oth[x]*Csch[x])/a))/a)/(a*(a^2 + b^2)) + (b^2*Coth[x]*Csch[x]^2)/(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_)*(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_)*((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 = 0.50 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.11
| method | result | size |
| default | \(-\frac {\frac {\tanh \left (\frac {x}{2}\right )^{3} a^{2}}{3}+2 \tanh \left (\frac {x}{2}\right )^{2} a b -3 \tanh \left (\frac {x}{2}\right ) a^{2}+12 b^{2} \tanh \left (\frac {x}{2}\right )}{8 a^{4}}-\frac {1}{24 a^{2} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {-3 a^{2}+12 b^{2}}{8 a^{4} \tanh \left (\frac {x}{2}\right )}+\frac {b}{4 a^{3} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {b \left (a^{2}-4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{5}}-\frac {2 b^{4} \left (\frac {-\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{a^{2}+b^{2}}-\frac {a b}{a^{2}+b^{2}}}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (5 a^{2}+4 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{a^{5}}\) | \(219\) |
| risch | \(-\frac {2 \left (-3 a^{3} b^{2} {\mathrm e}^{7 x}-6 a \,b^{4} {\mathrm e}^{7 x}-6 a^{4} b \,{\mathrm e}^{6 x}+3 a^{2} b^{3} {\mathrm e}^{6 x}+12 b^{5} {\mathrm e}^{6 x}+21 a^{3} b^{2} {\mathrm e}^{5 x}+30 a \,b^{4} {\mathrm e}^{5 x}+6 a^{4} b \,{\mathrm e}^{4 x}-21 a^{2} b^{3} {\mathrm e}^{4 x}-36 b^{5} {\mathrm e}^{4 x}+12 a^{5} {\mathrm e}^{3 x}-21 a^{3} b^{2} {\mathrm e}^{3 x}-42 a \,b^{4} {\mathrm e}^{3 x}-2 a^{4} b \,{\mathrm e}^{2 x}+25 a^{2} b^{3} {\mathrm e}^{2 x}+36 b^{5} {\mathrm e}^{2 x}-4 a^{5} {\mathrm e}^{x}+11 a^{3} b^{2} {\mathrm e}^{x}+18 b^{4} {\mathrm e}^{x} a +2 a^{4} b -7 a^{2} b^{3}-12 b^{5}\right )}{3 a^{4} \left ({\mathrm e}^{2 x}-1\right )^{3} \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}-\frac {b \ln \left ({\mathrm e}^{x}+1\right )}{a^{3}}+\frac {4 b^{3} \ln \left ({\mathrm e}^{x}+1\right )}{a^{5}}+\frac {b \ln \left ({\mathrm e}^{x}-1\right )}{a^{3}}-\frac {4 b^{3} \ln \left ({\mathrm e}^{x}-1\right )}{a^{5}}+\frac {5 b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{3}}+\frac {4 b^{6} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{5}}-\frac {5 b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{3}}-\frac {4 b^{6} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{5}}\) | \(549\) |
Input:
int(csch(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-3*tanh(1/2*x)*a^2+12*b ^2*tanh(1/2*x))-1/24/a^2/tanh(1/2*x)^3-1/8/a^4*(-3*a^2+12*b^2)/tanh(1/2*x) +1/4/a^3*b/tanh(1/2*x)^2+1/a^5*b*(a^2-4*b^2)*ln(tanh(1/2*x))-2*b^4/a^5*((- b^2/(a^2+b^2)*tanh(1/2*x)-a*b/(a^2+b^2))/(tanh(1/2*x)^2*a-2*b*tanh(1/2*x)- a)-(5*a^2+4*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)))
Leaf count of result is larger than twice the leaf count of optimal. 6430 vs. \(2 (190) = 380\).
Time = 0.34 (sec) , antiderivative size = 6430, normalized size of antiderivative = 32.47 \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \] Input:
integrate(csch(x)^4/(a+b*sinh(x))^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\operatorname {csch}^{4}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \] Input:
integrate(csch(x)**4/(a+b*sinh(x))**2,x)
Output:
Integral(csch(x)**4/(a + b*sinh(x))**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (190) = 380\).
Time = 0.13 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.41 \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (5 \, a^{2} b^{4} + 4 \, b^{6}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{7} + a^{5} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (2 \, a^{4} b - 7 \, a^{2} b^{3} - 12 \, b^{5} + {\left (4 \, a^{5} - 11 \, a^{3} b^{2} - 18 \, a b^{4}\right )} e^{\left (-x\right )} - {\left (2 \, a^{4} b - 25 \, a^{2} b^{3} - 36 \, b^{5}\right )} e^{\left (-2 \, x\right )} - 3 \, {\left (4 \, a^{5} - 7 \, a^{3} b^{2} - 14 \, a b^{4}\right )} e^{\left (-3 \, x\right )} + 3 \, {\left (2 \, a^{4} b - 7 \, a^{2} b^{3} - 12 \, b^{5}\right )} e^{\left (-4 \, x\right )} - 3 \, {\left (7 \, a^{3} b^{2} + 10 \, a b^{4}\right )} e^{\left (-5 \, x\right )} - 3 \, {\left (2 \, a^{4} b - a^{2} b^{3} - 4 \, b^{5}\right )} e^{\left (-6 \, x\right )} + 3 \, {\left (a^{3} b^{2} + 2 \, a b^{4}\right )} e^{\left (-7 \, x\right )}\right )}}{3 \, {\left (a^{6} b + a^{4} b^{3} + 2 \, {\left (a^{7} + a^{5} b^{2}\right )} e^{\left (-x\right )} - 4 \, {\left (a^{6} b + a^{4} b^{3}\right )} e^{\left (-2 \, x\right )} - 6 \, {\left (a^{7} + a^{5} b^{2}\right )} e^{\left (-3 \, x\right )} + 6 \, {\left (a^{6} b + a^{4} b^{3}\right )} e^{\left (-4 \, x\right )} + 6 \, {\left (a^{7} + a^{5} b^{2}\right )} e^{\left (-5 \, x\right )} - 4 \, {\left (a^{6} b + a^{4} b^{3}\right )} e^{\left (-6 \, x\right )} - 2 \, {\left (a^{7} + a^{5} b^{2}\right )} e^{\left (-7 \, x\right )} + {\left (a^{6} b + a^{4} b^{3}\right )} e^{\left (-8 \, x\right )}\right )}} - \frac {{\left (a^{2} b - 4 \, b^{3}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{5}} + \frac {{\left (a^{2} b - 4 \, b^{3}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{5}} \] Input:
integrate(csch(x)^4/(a+b*sinh(x))^2,x, algorithm="maxima")
Output:
(5*a^2*b^4 + 4*b^6)*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + s qrt(a^2 + b^2)))/((a^7 + a^5*b^2)*sqrt(a^2 + b^2)) + 2/3*(2*a^4*b - 7*a^2* b^3 - 12*b^5 + (4*a^5 - 11*a^3*b^2 - 18*a*b^4)*e^(-x) - (2*a^4*b - 25*a^2* b^3 - 36*b^5)*e^(-2*x) - 3*(4*a^5 - 7*a^3*b^2 - 14*a*b^4)*e^(-3*x) + 3*(2* a^4*b - 7*a^2*b^3 - 12*b^5)*e^(-4*x) - 3*(7*a^3*b^2 + 10*a*b^4)*e^(-5*x) - 3*(2*a^4*b - a^2*b^3 - 4*b^5)*e^(-6*x) + 3*(a^3*b^2 + 2*a*b^4)*e^(-7*x))/ (a^6*b + a^4*b^3 + 2*(a^7 + a^5*b^2)*e^(-x) - 4*(a^6*b + a^4*b^3)*e^(-2*x) - 6*(a^7 + a^5*b^2)*e^(-3*x) + 6*(a^6*b + a^4*b^3)*e^(-4*x) + 6*(a^7 + a^ 5*b^2)*e^(-5*x) - 4*(a^6*b + a^4*b^3)*e^(-6*x) - 2*(a^7 + a^5*b^2)*e^(-7*x ) + (a^6*b + a^4*b^3)*e^(-8*x)) - (a^2*b - 4*b^3)*log(e^(-x) + 1)/a^5 + (a ^2*b - 4*b^3)*log(e^(-x) - 1)/a^5
Time = 0.13 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.19 \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (5 \, a^{2} b^{4} + 4 \, b^{6}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{7} + a^{5} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a b^{4} e^{x} - b^{5}\right )}}{{\left (a^{6} + a^{4} b^{2}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} - \frac {{\left (a^{2} b - 4 \, b^{3}\right )} \log \left (e^{x} + 1\right )}{a^{5}} + \frac {{\left (a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{a^{5}} + \frac {2 \, {\left (3 \, a b e^{\left (5 \, x\right )} - 9 \, b^{2} e^{\left (4 \, x\right )} - 6 \, a^{2} e^{\left (2 \, x\right )} + 18 \, b^{2} e^{\left (2 \, x\right )} - 3 \, a b e^{x} + 2 \, a^{2} - 9 \, b^{2}\right )}}{3 \, a^{4} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \] Input:
integrate(csch(x)^4/(a+b*sinh(x))^2,x, algorithm="giac")
Output:
(5*a^2*b^4 + 4*b^6)*log(abs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/((a^7 + a^5*b^2)*sqrt(a^2 + b^2)) + 2*(a*b^4* e^x - b^5)/((a^6 + a^4*b^2)*(b*e^(2*x) + 2*a*e^x - b)) - (a^2*b - 4*b^3)*l og(e^x + 1)/a^5 + (a^2*b - 4*b^3)*log(abs(e^x - 1))/a^5 + 2/3*(3*a*b*e^(5* x) - 9*b^2*e^(4*x) - 6*a^2*e^(2*x) + 18*b^2*e^(2*x) - 3*a*b*e^x + 2*a^2 - 9*b^2)/(a^4*(e^(2*x) - 1)^3)
Time = 4.69 (sec) , antiderivative size = 975, normalized size of antiderivative = 4.92 \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx =\text {Too large to display} \] Input:
int(1/(sinh(x)^4*(a + b*sinh(x))^2),x)
Output:
(log(exp(x) - 1)*(a^2*b - 4*b^3))/a^5 - 8/(3*a^2*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (4/a^2 - (4*b*exp(x))/a^3)/(exp(4*x) - 2*exp(2*x) + 1) - ((6*b^2)/a^4 - (2*b*exp(x))/a^3)/(exp(2*x) - 1) - ((2*b^8)/(a^4*(b^5 + a ^2*b^3)) - (2*b^7*exp(x))/(a^3*(b^5 + a^2*b^3)))/(2*a*exp(x) - b + b*exp(2 *x)) - (log(exp(x) + 1)*(a^2*b - 4*b^3))/a^5 - (b^4*log((32*b*(16*b^4 - 5* a^4 + 16*a^2*b^2)*(2*a^4*b - 8*b^5 - 6*a^2*b^3 - 4*a^5*exp(x) + 14*a*b^4*e xp(x) + 11*a^3*b^2*exp(x)))/(a^12*(a^2 + b^2)^2) - (32*b*(5*a^2 + 4*b^2)*( 5*a^5*b^9 - 32*b^11*((a^2 + b^2)^3)^(1/2) - 2*a^13*b + 20*a^7*b^7 + 24*a^9 *b^5 + 7*a^11*b^3 + 4*a^14*exp(x) - 80*a^2*b^9*((a^2 + b^2)^3)^(1/2) - 50* a^4*b^7*((a^2 + b^2)^3)^(1/2) - 15*a^6*b^8*exp(x) - 50*a^8*b^6*exp(x) - 52 *a^10*b^4*exp(x) - 13*a^12*b^2*exp(x) + 127*a^3*b^8*exp(x)*((a^2 + b^2)^3) ^(1/2) + 79*a^5*b^6*exp(x)*((a^2 + b^2)^3)^(1/2) + 5*a*b^4*exp(x)*((a^2 + b^2)^3)^(3/2) + 51*a*b^10*exp(x)*((a^2 + b^2)^3)^(1/2)))/(a^12*((a^2 + b^2 )^3)^(1/2)*(a^2 + b^2)^4))*((a^2 + b^2)^3)^(1/2)*(5*a^2 + 4*b^2))/(a^11 + a^5*b^6 + 3*a^7*b^4 + 3*a^9*b^2) + (b^4*log((32*b*(16*b^4 - 5*a^4 + 16*a^2 *b^2)*(2*a^4*b - 8*b^5 - 6*a^2*b^3 - 4*a^5*exp(x) + 14*a*b^4*exp(x) + 11*a ^3*b^2*exp(x)))/(a^12*(a^2 + b^2)^2) - (32*b*(5*a^2 + 4*b^2)*(2*a^13*b - 3 2*b^11*((a^2 + b^2)^3)^(1/2) - 5*a^5*b^9 - 20*a^7*b^7 - 24*a^9*b^5 - 7*a^1 1*b^3 - 4*a^14*exp(x) - 80*a^2*b^9*((a^2 + b^2)^3)^(1/2) - 50*a^4*b^7*((a^ 2 + b^2)^3)^(1/2) + 15*a^6*b^8*exp(x) + 50*a^8*b^6*exp(x) + 52*a^10*b^4...
Time = 0.17 (sec) , antiderivative size = 2608, normalized size of antiderivative = 13.17 \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx =\text {Too large to display} \] Input:
int(csch(x)^4/(a+b*sinh(x))^2,x)
Output:
(30*e**(8*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a* *2*b**5*i + 24*e**(8*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*b**7*i + 60*e**(7*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt (a**2 + b**2))*a**3*b**4*i + 48*e**(7*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a*b**6*i - 120*e**(6*x)*sqrt(a**2 + b**2)*atan(( e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**5*i - 96*e**(6*x)*sqrt(a**2 + b **2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*b**7*i - 180*e**(5*x)*sqrt(a **2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**3*b**4*i - 144*e** (5*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a*b**6*i + 180*e**(4*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))* a**2*b**5*i + 144*e**(4*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a* *2 + b**2))*b**7*i + 180*e**(3*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/ sqrt(a**2 + b**2))*a**3*b**4*i + 144*e**(3*x)*sqrt(a**2 + b**2)*atan((e**x *b*i + a*i)/sqrt(a**2 + b**2))*a*b**6*i - 120*e**(2*x)*sqrt(a**2 + b**2)*a tan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**5*i - 96*e**(2*x)*sqrt(a** 2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*b**7*i - 60*e**x*sqrt(a **2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**3*b**4*i - 48*e**x *sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a*b**6*i + 30* sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**5*i + 2 4*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*b**7*i + 3...