Integrand size = 20, antiderivative size = 261 \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {2 a^4 b \arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}-\frac {3 a^2 b^3 \arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}-\frac {4 a^2 b^2 \cosh (x)}{\left (a^2-b^2\right )^3}-\frac {a^2 \cosh (x)}{\left (a^2-b^2\right )^2}+\frac {a^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {b^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {2 a^3 b \sinh (x)}{\left (a^2-b^2\right )^3}+\frac {2 a b^3 \sinh (x)}{\left (a^2-b^2\right )^3}-\frac {2 a b \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac {a^3 b^2}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))} \]
-2*a^4*b*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/(a^2-b^2)^(7/2)-3*a ^2*b^3*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/(a^2-b^2)^(7/2)-4*a^2 *b^2*cosh(x)/(a^2-b^2)^3-a^2*cosh(x)/(a^2-b^2)^2+1/3*a^2*cosh(x)^3/(a^2-b^ 2)^2+1/3*b^2*cosh(x)^3/(a^2-b^2)^2+2*a^3*b*sinh(x)/(a^2-b^2)^3+2*a*b^3*sin h(x)/(a^2-b^2)^3-2/3*a*b*sinh(x)^3/(a^2-b^2)^2-a^3*b^2/(a^2-b^2)^3/(a*cosh (x)+b*sinh(x))
Time = 2.78 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.82 \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {1}{16} \left (-\frac {6 b \left (3 a^2+b^2\right ) \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2}}-\frac {10 b \left (5 a^4+10 a^2 b^2+b^4\right ) \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}-\frac {4 \left (a^2+b^2\right ) \cosh (x)}{(a-b)^2 (a+b)^2}-\frac {8 \left (a^4+6 a^2 b^2+b^4\right ) \cosh (x)}{(a-b)^3 (a+b)^3}+\frac {4 \left (a^2+b^2\right ) \cosh (3 x)}{3 (a-b)^2 (a+b)^2}+\frac {8 a b \sinh (x)}{(a-b)^2 (a+b)^2}+\frac {32 a b \left (a^2+b^2\right ) \sinh (x)}{(a-b)^3 (a+b)^3}-\frac {a \left (a^2+3 b^2\right )}{(a-b)^2 (a+b)^2 (a \cosh (x)+b \sinh (x))}-\frac {a \left (a^4+10 a^2 b^2+5 b^4\right )}{(a-b)^3 (a+b)^3 (a \cosh (x)+b \sinh (x))}+\frac {2 \left (a \sqrt {a-b} (a+b)+2 a b \sqrt {a+b} \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right ) \cosh (x)+2 b^2 \sqrt {a+b} \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right ) \sinh (x)\right )}{(a-b)^{3/2} (a+b)^2 (a \cosh (x)+b \sinh (x))}-\frac {8 a b \sinh (3 x)}{3 (a-b)^2 (a+b)^2}\right ) \]
((-6*b*(3*a^2 + b^2)*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/ ((a - b)^(5/2)*(a + b)^(5/2)) - (10*b*(5*a^4 + 10*a^2*b^2 + b^4)*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/((a - b)^(7/2)*(a + b)^(7/2)) - (4*(a^2 + b^2)*Cosh[x])/((a - b)^2*(a + b)^2) - (8*(a^4 + 6*a^2*b^2 + b^ 4)*Cosh[x])/((a - b)^3*(a + b)^3) + (4*(a^2 + b^2)*Cosh[3*x])/(3*(a - b)^2 *(a + b)^2) + (8*a*b*Sinh[x])/((a - b)^2*(a + b)^2) + (32*a*b*(a^2 + b^2)* Sinh[x])/((a - b)^3*(a + b)^3) - (a*(a^2 + 3*b^2))/((a - b)^2*(a + b)^2*(a *Cosh[x] + b*Sinh[x])) - (a*(a^4 + 10*a^2*b^2 + 5*b^4))/((a - b)^3*(a + b) ^3*(a*Cosh[x] + b*Sinh[x])) + (2*(a*Sqrt[a - b]*(a + b) + 2*a*b*Sqrt[a + b ]*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])]*Cosh[x] + 2*b^2*Sqrt [a + b]*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])]*Sinh[x]))/((a - b)^(3/2)*(a + b)^2*(a*Cosh[x] + b*Sinh[x])) - (8*a*b*Sinh[3*x])/(3*(a - b)^2*(a + b)^2))/16
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.
| \(\displaystyle \int \frac {\sinh ^3(x) \cosh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i x)^3 \cos (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\cos (i x)^2 \sin (i x)^3}{(a \cos (i x)-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 3590 |
| \(\displaystyle i \left (-\frac {i b \int -\frac {\cosh ^2(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}+\frac {a \int -\frac {i \cosh (x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}+\frac {i a b \int -\frac {\cosh (x) \sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i b \int \frac {\cosh ^2(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}+\frac {a \int -\frac {i \cosh (x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {i a b \int \frac {\cosh (x) \sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i b \int \frac {\cosh ^2(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {i a \int \frac {\cosh (x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {i a b \int \frac {\cosh (x) \sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i a b \int -\frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {i b \int -\frac {\cos (i x)^2 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i a \int \frac {i \cos (i x) \sin (i x)^3}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \int \frac {\cos (i x)^2 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i a \int \frac {i \cos (i x) \sin (i x)^3}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \int \frac {\cos (i x)^2 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {a \int \frac {\cos (i x) \sin (i x)^3}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (-\frac {i b \int i \cosh ^2(x) \sinh (x)dx}{a^2-b^2}+\frac {a \int -\cosh (x) \sinh ^2(x)dx}{a^2-b^2}+\frac {i a b \int \frac {i \cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (\frac {a \int -i \sinh ^3(x)dx}{a^2-b^2}-\frac {i b \int -\cosh (x) \sinh ^2(x)dx}{a^2-b^2}+\frac {i a b \int -\frac {\sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (-\frac {i b \int i \cosh ^2(x) \sinh (x)dx}{a^2-b^2}-\frac {a \int \cosh (x) \sinh ^2(x)dx}{a^2-b^2}+\frac {i a b \int \frac {i \cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (\frac {a \int -i \sinh ^3(x)dx}{a^2-b^2}+\frac {i b \int \cosh (x) \sinh ^2(x)dx}{a^2-b^2}-\frac {i a b \int \frac {\sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (\frac {b \int \cosh ^2(x) \sinh (x)dx}{a^2-b^2}-\frac {a \int \cosh (x) \sinh ^2(x)dx}{a^2-b^2}-\frac {a b \int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (-\frac {i a \int \sinh ^3(x)dx}{a^2-b^2}+\frac {i b \int \cosh (x) \sinh ^2(x)dx}{a^2-b^2}-\frac {i a b \int \frac {\sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (\frac {b \int -i \cos (i x)^2 \sin (i x)dx}{a^2-b^2}-\frac {a \int -\cos (i x) \sin (i x)^2dx}{a^2-b^2}-\frac {a b \int -\frac {i \cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (-\frac {i a \int i \sin (i x)^3dx}{a^2-b^2}+\frac {i b \int -\cos (i x) \sin (i x)^2dx}{a^2-b^2}-\frac {i a b \int -\frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (\frac {b \int -i \cos (i x)^2 \sin (i x)dx}{a^2-b^2}+\frac {a \int \cos (i x) \sin (i x)^2dx}{a^2-b^2}-\frac {a b \int -\frac {i \cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (-\frac {i a \int i \sin (i x)^3dx}{a^2-b^2}-\frac {i b \int \cos (i x) \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (-\frac {i b \int \cos (i x)^2 \sin (i x)dx}{a^2-b^2}+\frac {a \int \cos (i x) \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (\frac {a \int \sin (i x)^3dx}{a^2-b^2}-\frac {i b \int \cos (i x) \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (-\frac {i a \int -\sinh ^2(x)d(i \sinh (x))}{a^2-b^2}-\frac {i b \int \cos (i x)^2 \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (\frac {a \int \sin (i x)^3dx}{a^2-b^2}-\frac {b \int -\sinh ^2(x)d(i \sinh (x))}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (-\frac {i b \int \cos (i x)^2 \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}+\frac {a \left (\frac {a \int \sin (i x)^3dx}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {a \int \sin (i x)^3dx}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {i b \left (\frac {b \int \cosh ^2(x)d\cosh (x)}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {a \int \sin (i x)^3dx}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}+\frac {a \left (\frac {i a \int \left (1-\cosh ^2(x)\right )d\cosh (x)}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}+\frac {a \left (\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3578 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (-\frac {i b \int i \sinh (x)dx}{a^2-b^2}+\frac {a^2 \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {a \sinh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (\frac {b \int \sinh (x)dx}{a^2-b^2}+\frac {a^2 \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {a \sinh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (\frac {b \int -i \sin (i x)dx}{a^2-b^2}+\frac {a^2 \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (-\frac {i b \int \sin (i x)dx}{a^2-b^2}+\frac {a^2 \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (\frac {a^2 \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh (x)}{a^2-b^2}+\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (\frac {i a^2 \int \frac {1}{a^2-b^2-(-i b \cosh (x)-i a \sinh (x))^2}d(-i b \cosh (x)-i a \sinh (x))}{a^2-b^2}-\frac {a \sinh (x)}{a^2-b^2}+\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}+\frac {a \left (\frac {i a b \left (\frac {i a^2 \text {arctanh}\left (\frac {-i a \sinh (x)-i b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {a \sinh (x)}{a^2-b^2}+\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (\frac {i a b \left (\frac {a \int i \sinh (x)dx}{a^2-b^2}-\frac {i b \int \cosh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}+\frac {a \left (\frac {i a b \left (\frac {i a^2 \text {arctanh}\left (\frac {-i a \sinh (x)-i b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {a \sinh (x)}{a^2-b^2}+\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (\frac {i a b \left (\frac {i a \int \sinh (x)dx}{a^2-b^2}-\frac {i b \int \cosh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}+\frac {a \left (\frac {i a b \left (\frac {i a^2 \text {arctanh}\left (\frac {-i a \sinh (x)-i b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {a \sinh (x)}{a^2-b^2}+\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (\frac {i a b \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )dx}{a^2-b^2}+\frac {i a \int -i \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}+\frac {a \left (\frac {i a b \left (\frac {i a^2 \text {arctanh}\left (\frac {-i a \sinh (x)-i b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {a \sinh (x)}{a^2-b^2}+\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
3.8.20.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x _Symbol] :> Simp[-d^(-1) Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
Int[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-a)*(Sin[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1))), x] + (Simp[a^2/(a^2 + b^2) Int[Sin[c + d*x]^(m - 2)/(a *Cos[c + d*x] + b*Sin[c + d*x]), x], x] + Simp[b/(a^2 + b^2) Int[Sin[c + d*x]^(m - 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ [m, 1]
Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_. ) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b /(a^2 + b^2) Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Simp[a/(a ^2 + b^2) Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Simp[a*(b/(a^ 2 + b^2)) Int[Cos[c + d*x]^(m - 1)*(Sin[c + d*x]^(n - 1)/(a*Cos[c + d*x] + b*Sin[c + d*x])), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]
Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Sim p[b/(a^2 + b^2) Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(p + 1), x], x] + (Simp[a/(a^2 + b^2) Int[Cos[c + d*x]^( m - 1)*Sin[c + d*x]^n*(a*Cos[c + d*x] + b*Sin[c + d*x])^(p + 1), x], x] - S imp[a*(b/(a^2 + b^2)) Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^(n - 1)*(a*Co s[c + d*x] + b*Sin[c + d*x])^p, x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^ 2 + b^2, 0] && IGtQ[m, 0] && IGtQ[n, 0] && ILtQ[p, 0]
Time = 6.11 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.80
| method | result | size |
| default | \(-\frac {1}{3 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {-a +b}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {1}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{3 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {a +b}{2 \left (a -b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {4 a^{2} b \left (\frac {\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{2}+\frac {a b}{2}}{\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a}+\frac {\left (2 a^{2}+3 b^{2}\right ) \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}\) | \(208\) |
| risch | \(\frac {{\mathrm e}^{3 x}}{24 a^{2}+48 a b +24 b^{2}}-\frac {3 \,{\mathrm e}^{x} a}{8 \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {{\mathrm e}^{x} b}{8 \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 \,{\mathrm e}^{-x} a}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {{\mathrm e}^{-x} b}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {{\mathrm e}^{-3 x}}{24 a^{2}-48 a b +24 b^{2}}-\frac {2 a^{3} b^{2} {\mathrm e}^{x}}{\left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a^{2}-2 a b +b^{2}\right ) \left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right )}-\frac {2 a^{4} b \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {3 a^{2} b^{3} \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {2 a^{4} b \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {3 a^{2} b^{3} \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}\) | \(422\) |
-1/3/(a+b)^2/(tanh(1/2*x)-1)^3-1/2/(a+b)^2/(tanh(1/2*x)-1)^2-1/2/(a+b)^3*( -a+b)/(tanh(1/2*x)-1)-1/2/(a-b)^2/(tanh(1/2*x)+1)^2+1/3/(a-b)^2/(tanh(1/2* x)+1)^3-1/2*(a+b)/(a-b)^3/(tanh(1/2*x)+1)-4*a^2*b/(a+b)^3/(a-b)^3*((1/2*b^ 2*tanh(1/2*x)+1/2*a*b)/(tanh(1/2*x)^2*a+2*b*tanh(1/2*x)+a)+1/2*(2*a^2+3*b^ 2)/(a^2-b^2)^(1/2)*arctan(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. 2503 vs. \(2 (247) = 494\).
Time = 0.33 (sec) , antiderivative size = 5061, normalized size of antiderivative = 19.39 \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.27 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.19 \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {2 \, a^{3} b^{2} e^{x}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} - \frac {{\left (9 \, a e^{\left (2 \, x\right )} + 3 \, b e^{\left (2 \, x\right )} - a + b\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} - \frac {2 \, {\left (2 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {a^{4} e^{\left (3 \, x\right )} + 4 \, a^{3} b e^{\left (3 \, x\right )} + 6 \, a^{2} b^{2} e^{\left (3 \, x\right )} + 4 \, a b^{3} e^{\left (3 \, x\right )} + b^{4} e^{\left (3 \, x\right )} - 9 \, a^{4} e^{x} - 24 \, a^{3} b e^{x} - 18 \, a^{2} b^{2} e^{x} + 3 \, b^{4} e^{x}}{24 \, {\left (a^{6} + 6 \, a^{5} b + 15 \, a^{4} b^{2} + 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} + 6 \, a b^{5} + b^{6}\right )}} \]
-2*a^3*b^2*e^x/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(a*e^(2*x) + b*e^(2*x) + a - b)) - 1/24*(9*a*e^(2*x) + 3*b*e^(2*x) - a + b)*e^(-3*x)/(a^3 - 3*a^ 2*b + 3*a*b^2 - b^3) - 2*(2*a^4*b + 3*a^2*b^3)*arctan((a*e^x + b*e^x)/sqrt (a^2 - b^2))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sqrt(a^2 - b^2)) + 1/24* (a^4*e^(3*x) + 4*a^3*b*e^(3*x) + 6*a^2*b^2*e^(3*x) + 4*a*b^3*e^(3*x) + b^4 *e^(3*x) - 9*a^4*e^x - 24*a^3*b*e^x - 18*a^2*b^2*e^x + 3*b^4*e^x)/(a^6 + 6 *a^5*b + 15*a^4*b^2 + 20*a^3*b^3 + 15*a^2*b^4 + 6*a*b^5 + b^6)
Time = 2.88 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.27 \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {{\mathrm {e}}^{3\,x}}{24\,{\left (a+b\right )}^2}+\frac {{\mathrm {e}}^{-3\,x}}{24\,{\left (a-b\right )}^2}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\left (2\,a^4\,b\,\sqrt {a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}}+3\,a^2\,b^3\,\sqrt {a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}}\right )}{a^7\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}+b^7\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}-3\,a^2\,b^5\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}+3\,a^3\,b^4\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}+3\,a^4\,b^3\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}-3\,a^5\,b^2\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}-a\,b^6\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}-a^6\,b\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}}\right )\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}}{\sqrt {a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}}}-\frac {{\mathrm {e}}^x\,\left (3\,a-b\right )}{8\,{\left (a+b\right )}^3}-\frac {{\mathrm {e}}^{-x}\,\left (3\,a+b\right )}{8\,{\left (a-b\right )}^3}-\frac {2\,a^3\,b^2\,{\mathrm {e}}^x}{{\left (a+b\right )}^3\,{\left (a-b\right )}^3\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \]
exp(3*x)/(24*(a + b)^2) + exp(-3*x)/(24*(a - b)^2) - (2*atan((exp(x)*(2*a^ 4*b*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21 *a^10*b^4 - 7*a^12*b^2)^(1/2) + 3*a^2*b^3*(a^14 - b^14 + 7*a^2*b^12 - 21*a ^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)^(1/2)))/(a^7 *(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2) + b^7*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2) - 3*a^2*b^5*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2) + 3*a^3*b^4*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2) + 3*a^4*b^3*(9*a^4* b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2) - 3*a^5*b^2*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2) - a*b^6*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2) - a^6* b*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2)))*(9*a^4*b^6 + 12*a^6*b^4 + 4 *a^8*b^2)^(1/2))/(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35 *a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)^(1/2) - (exp(x)*(3*a - b))/(8*(a + b) ^3) - (exp(-x)*(3*a + b))/(8*(a - b)^3) - (2*a^3*b^2*exp(x))/((a + b)^3*(a - b)^3*(a - b + exp(2*x)*(a + b)))