Integrand size = 20, antiderivative size = 212 \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {a^3 b^3 \arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac {a^3 b^2 \cosh (x)}{\left (a^2-b^2\right )^3}-\frac {a b^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac {a \cosh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {a \cosh ^5(x)}{5 \left (a^2-b^2\right )}-\frac {a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3}+\frac {a^2 b \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac {b \sinh ^3(x)}{3 \left (a^2-b^2\right )}-\frac {b \sinh ^5(x)}{5 \left (a^2-b^2\right )} \]
a^3*b^3*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/(a^2-b^2)^(7/2)+a^3* b^2*cosh(x)/(a^2-b^2)^3-1/3*a*b^2*cosh(x)^3/(a^2-b^2)^2-1/3*a*cosh(x)^3/(a ^2-b^2)+1/5*a*cosh(x)^5/(a^2-b^2)-a^2*b^3*sinh(x)/(a^2-b^2)^3+1/3*a^2*b*si nh(x)^3/(a^2-b^2)^2-1/3*b*sinh(x)^3/(a^2-b^2)-1/5*b*sinh(x)^5/(a^2-b^2)
Time = 1.63 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.53 \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {1}{32} \left (\frac {4 a b \left (3 a^4+10 a^2 b^2+3 b^4\right ) \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}+\frac {2 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cosh (x)}{(a-b)^3 (a+b)^3}-\frac {2 a \left (a^2+3 b^2\right ) \cosh (3 x)}{3 (a-b)^2 (a+b)^2}+\frac {2 a \cosh (5 x)}{5 (a-b) (a+b)}+\frac {2 b \left (5 a^4+10 a^2 b^2+b^4\right ) \sinh (x)}{(-a+b)^3 (a+b)^3}-3 \left (\frac {4 a b \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}+\frac {2 a \cosh (x)}{a^2-b^2}+\frac {2 b \sinh (x)}{-a^2+b^2}\right )+\frac {2 b \left (3 a^2+b^2\right ) \sinh (3 x)}{3 (a-b)^2 (a+b)^2}-\frac {2 b \sinh (5 x)}{5 (a-b) (a+b)}\right ) \]
((4*a*b*(3*a^4 + 10*a^2*b^2 + 3*b^4)*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b] *Sqrt[a + b])])/((a - b)^(7/2)*(a + b)^(7/2)) + (2*a*(a^4 + 10*a^2*b^2 + 5 *b^4)*Cosh[x])/((a - b)^3*(a + b)^3) - (2*a*(a^2 + 3*b^2)*Cosh[3*x])/(3*(a - b)^2*(a + b)^2) + (2*a*Cosh[5*x])/(5*(a - b)*(a + b)) + (2*b*(5*a^4 + 1 0*a^2*b^2 + b^4)*Sinh[x])/((-a + b)^3*(a + b)^3) - 3*((4*a*b*ArcTan[(b + a *Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/((a - b)^(3/2)*(a + b)^(3/2)) + (2 *a*Cosh[x])/(a^2 - b^2) + (2*b*Sinh[x])/(-a^2 + b^2)) + (2*b*(3*a^2 + b^2) *Sinh[3*x])/(3*(a - b)^2*(a + b)^2) - (2*b*Sinh[5*x])/(5*(a - b)*(a + b))) /32
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.
| \(\displaystyle \int \frac {\sinh ^3(x) \cosh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i x)^3 \cos (i x)^3}{a \cos (i x)-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\cos (i x)^3 \sin (i x)^3}{a \cos (i x)-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle i \left (-\frac {i b \int -\cosh ^3(x) \sinh ^2(x)dx}{a^2-b^2}+\frac {a \int -i \cosh ^2(x) \sinh ^3(x)dx}{a^2-b^2}+\frac {i a b \int -\frac {\cosh ^2(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i b \int \cosh ^3(x) \sinh ^2(x)dx}{a^2-b^2}+\frac {a \int -i \cosh ^2(x) \sinh ^3(x)dx}{a^2-b^2}-\frac {i a b \int \frac {\cosh ^2(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i b \int \cosh ^3(x) \sinh ^2(x)dx}{a^2-b^2}-\frac {i a \int \cosh ^2(x) \sinh ^3(x)dx}{a^2-b^2}-\frac {i a b \int \frac {\cosh ^2(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i b \int -\cos (i x)^3 \sin (i x)^2dx}{a^2-b^2}-\frac {i a \int i \cos (i x)^2 \sin (i x)^3dx}{a^2-b^2}-\frac {i a b \int -\frac {\cos (i x)^2 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\frac {i b \int \cos (i x)^3 \sin (i x)^2dx}{a^2-b^2}-\frac {i a \int i \cos (i x)^2 \sin (i x)^3dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i b \int \cos (i x)^3 \sin (i x)^2dx}{a^2-b^2}+\frac {a \int \cos (i x)^2 \sin (i x)^3dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle i \left (-\frac {b \int -\sinh ^2(x) \left (\sinh ^2(x)+1\right )d(i \sinh (x))}{a^2-b^2}+\frac {a \int \cos (i x)^2 \sin (i x)^3dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle i \left (-\frac {b \int \left (-\sinh ^4(x)-\sinh ^2(x)\right )d(i \sinh (x))}{a^2-b^2}+\frac {a \int \cos (i x)^2 \sin (i x)^3dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (\frac {a \int \cos (i x)^2 \sin (i x)^3dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle i \left (\frac {i a \int \cosh ^2(x) \left (1-\cosh ^2(x)\right )d\cosh (x)}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle i \left (\frac {i a \int \left (\cosh ^2(x)-\cosh ^4(x)\right )d\cosh (x)}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle i \left (\frac {i a b \left (-\frac {i b \int i \cosh ^2(x) \sinh (x)dx}{a^2-b^2}+\frac {a \int -\cosh (x) \sinh ^2(x)dx}{a^2-b^2}+\frac {i a b \int \frac {i \cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i a b \left (-\frac {i b \int i \cosh ^2(x) \sinh (x)dx}{a^2-b^2}-\frac {a \int \cosh (x) \sinh ^2(x)dx}{a^2-b^2}+\frac {i a b \int \frac {i \cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {b \int \cosh ^2(x) \sinh (x)dx}{a^2-b^2}-\frac {a \int \cosh (x) \sinh ^2(x)dx}{a^2-b^2}-\frac {a b \int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {b \int -i \cos (i x)^2 \sin (i x)dx}{a^2-b^2}-\frac {a \int -\cos (i x) \sin (i x)^2dx}{a^2-b^2}-\frac {a b \int -\frac {i \cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {b \int -i \cos (i x)^2 \sin (i x)dx}{a^2-b^2}+\frac {a \int \cos (i x) \sin (i x)^2dx}{a^2-b^2}-\frac {a b \int -\frac {i \cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \left (-\frac {i b \int \cos (i x)^2 \sin (i x)dx}{a^2-b^2}+\frac {a \int \cos (i x) \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle i \left (\frac {i a b \left (-\frac {i a \int -\sinh ^2(x)d(i \sinh (x))}{a^2-b^2}-\frac {i b \int \cos (i x)^2 \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle i \left (\frac {i a b \left (-\frac {i b \int \cos (i x)^2 \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {b \int \cosh ^2(x)d\cosh (x)}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a b \left (\frac {a \int i \sinh (x)dx}{a^2-b^2}-\frac {i b \int \cosh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a b \left (\frac {i a \int \sinh (x)dx}{a^2-b^2}-\frac {i b \int \cosh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a b \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )dx}{a^2-b^2}+\frac {i a \int -i \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a b \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )dx}{a^2-b^2}+\frac {a \int \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a b \left (\frac {a \int \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b \sinh (x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a b \left (\frac {i a b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b \sinh (x)}{a^2-b^2}+\frac {i a \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {b \left (-\frac {1}{5} i \sinh ^5(x)-\frac {1}{3} i \sinh ^3(x)\right )}{a^2-b^2}+\frac {i a \left (\frac {\cosh ^3(x)}{3}-\frac {\cosh ^5(x)}{5}\right )}{a^2-b^2}\right )\) |
3.8.14.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 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[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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]
Time = 31.60 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.25
| method | result | size |
| default | \(-\frac {16}{5 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5} \left (16 a +16 b \right )}-\frac {4}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{4} \left (8 a +8 b \right )}-\frac {a +3 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {5 a +7 b}{12 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {a \left (a +3 b \right )}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {4}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{4} \left (8 a -8 b \right )}+\frac {16}{5 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5} \left (16 a -16 b \right )}-\frac {-5 a +7 b}{12 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {a -3 b}{8 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {a \left (a -3 b \right )}{8 \left (a -b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {2 a^{3} b^{3} \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} \sqrt {a^{2}-b^{2}}}\) | \(266\) |
| risch | \(\frac {{\mathrm e}^{5 x}}{160 a +160 b}-\frac {{\mathrm e}^{3 x} a}{96 \left (a +b \right )^{2}}+\frac {{\mathrm e}^{3 x} b}{96 \left (a +b \right )^{2}}-\frac {{\mathrm e}^{x} a^{2}}{16 \left (a +b \right )^{3}}-\frac {{\mathrm e}^{x} a b}{4 \left (a +b \right )^{3}}-\frac {{\mathrm e}^{x} b^{2}}{16 \left (a +b \right )^{3}}-\frac {{\mathrm e}^{-x} a^{2}}{16 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {{\mathrm e}^{-x} a b}{4 a^{3}-12 a^{2} b +12 a \,b^{2}-4 b^{3}}-\frac {{\mathrm e}^{-x} b^{2}}{16 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {{\mathrm e}^{-3 x} a}{96 \left (a^{2}-2 a b +b^{2}\right )}-\frac {{\mathrm e}^{-3 x} b}{96 \left (a^{2}-2 a b +b^{2}\right )}+\frac {{\mathrm e}^{-5 x}}{160 a -160 b}-\frac {b^{3} a^{3} \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {b^{3} a^{3} \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}\) | \(324\) |
-16/5/(tanh(1/2*x)-1)^5/(16*a+16*b)-4/(tanh(1/2*x)-1)^4/(8*a+8*b)-1/8*(a+3 *b)/(a+b)^2/(tanh(1/2*x)-1)^2-1/12*(5*a+7*b)/(a+b)^2/(tanh(1/2*x)-1)^3+1/8 *a*(a+3*b)/(a+b)^3/(tanh(1/2*x)-1)-4/(tanh(1/2*x)+1)^4/(8*a-8*b)+16/5/(tan h(1/2*x)+1)^5/(16*a-16*b)-1/12*(-5*a+7*b)/(a-b)^2/(tanh(1/2*x)+1)^3-1/8*(a -3*b)/(a-b)^2/(tanh(1/2*x)+1)^2-1/8*a*(a-3*b)/(a-b)^3/(tanh(1/2*x)+1)+2*a^ 3*b^3/(a+b)^3/(a-b)^3/(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. 2440 vs. \(2 (196) = 392\).
Time = 0.32 (sec) , antiderivative size = 4935, normalized size of antiderivative = 23.28 \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=\text {Too large to display} \]
[1/480*(3*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a *b^6 + b^7)*cosh(x)^10 + 30*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b ^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x)*sinh(x)^9 + 3*(a^7 - a^6*b - 3*a^5*b ^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*sinh(x)^10 - 5*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cos h(x)^8 - 5*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3* a*b^6 - b^7 - 27*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2* b^5 - a*b^6 + b^7)*cosh(x)^2)*sinh(x)^8 + 40*(9*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x)^3 - (a^7 - 3*a^6 *b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x))*s inh(x)^7 + 3*a^7 + 3*a^6*b - 9*a^5*b^2 - 9*a^4*b^3 + 9*a^3*b^4 + 9*a^2*b^5 - 3*a*b^6 - 3*b^7 - 30*(a^7 + a^6*b - 9*a^5*b^2 + 7*a^4*b^3 + 7*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cosh(x)^6 - 10*(3*a^7 + 3*a^6*b - 27*a^5*b^2 + 2 1*a^4*b^3 + 21*a^3*b^4 - 27*a^2*b^5 + 3*a*b^6 + 3*b^7 - 63*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x)^4 + 1 4*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b ^7)*cosh(x)^2)*sinh(x)^6 + 4*(189*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3 *a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x)^5 - 70*(a^7 - 3*a^6*b + a^5*b^ 2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^3 - 45*(a^7 + a^6*b - 9*a^5*b^2 + 7*a^4*b^3 + 7*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*c...
Timed out. \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=\text {Exception raised: ValueError} \]
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 = 0.28 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.53 \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {2 \, a^{3} b^{3} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} - \frac {{\left (30 \, a^{2} e^{\left (4 \, x\right )} - 120 \, a b e^{\left (4 \, x\right )} + 30 \, b^{2} e^{\left (4 \, x\right )} + 5 \, a^{2} e^{\left (2 \, x\right )} - 5 \, b^{2} e^{\left (2 \, x\right )} - 3 \, a^{2} + 6 \, a b - 3 \, b^{2}\right )} e^{\left (-5 \, x\right )}}{480 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {3 \, a^{4} e^{\left (5 \, x\right )} + 12 \, a^{3} b e^{\left (5 \, x\right )} + 18 \, a^{2} b^{2} e^{\left (5 \, x\right )} + 12 \, a b^{3} e^{\left (5 \, x\right )} + 3 \, b^{4} e^{\left (5 \, x\right )} - 5 \, a^{4} e^{\left (3 \, x\right )} - 10 \, a^{3} b e^{\left (3 \, x\right )} + 10 \, a b^{3} e^{\left (3 \, x\right )} + 5 \, b^{4} e^{\left (3 \, x\right )} - 30 \, a^{4} e^{x} - 180 \, a^{3} b e^{x} - 300 \, a^{2} b^{2} e^{x} - 180 \, a b^{3} e^{x} - 30 \, b^{4} e^{x}}{480 \, {\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )}} \]
2*a^3*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)) - 1/480*(30*a^2*e^(4*x) - 120*a*b*e^(4*x) + 30*b^2*e^(4*x) + 5*a^2*e^(2*x) - 5*b^2*e^(2*x) - 3*a^2 + 6*a*b - 3*b^2)*e^ (-5*x)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/480*(3*a^4*e^(5*x) + 12*a^3*b*e ^(5*x) + 18*a^2*b^2*e^(5*x) + 12*a*b^3*e^(5*x) + 3*b^4*e^(5*x) - 5*a^4*e^( 3*x) - 10*a^3*b*e^(3*x) + 10*a*b^3*e^(3*x) + 5*b^4*e^(3*x) - 30*a^4*e^x - 180*a^3*b*e^x - 300*a^2*b^2*e^x - 180*a*b^3*e^x - 30*b^4*e^x)/(a^5 + 5*a^4 *b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)
Time = 2.81 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.75 \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^{-5\,x}}{160\,a-160\,b}+\frac {{\mathrm {e}}^{5\,x}}{160\,a+160\,b}+\frac {2\,\mathrm {atan}\left (\frac {a^3\,b^3\,{\mathrm {e}}^x\,\sqrt {a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}}}{a^7\,\sqrt {a^6\,b^6}+b^7\,\sqrt {a^6\,b^6}-3\,a^2\,b^5\,\sqrt {a^6\,b^6}+3\,a^3\,b^4\,\sqrt {a^6\,b^6}+3\,a^4\,b^3\,\sqrt {a^6\,b^6}-3\,a^5\,b^2\,\sqrt {a^6\,b^6}-a\,b^6\,\sqrt {a^6\,b^6}-a^6\,b\,\sqrt {a^6\,b^6}}\right )\,\sqrt {a^6\,b^6}}{\sqrt {a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}}}-\frac {{\mathrm {e}}^{-x}\,\left (a^2-4\,a\,b+b^2\right )}{16\,{\left (a-b\right )}^3}-\frac {{\mathrm {e}}^{-3\,x}\,\left (a+b\right )}{96\,{\left (a-b\right )}^2}-\frac {{\mathrm {e}}^{3\,x}\,\left (a-b\right )}{96\,{\left (a+b\right )}^2}-\frac {{\mathrm {e}}^x\,\left (a^2+4\,a\,b+b^2\right )}{16\,{\left (a+b\right )}^3} \]
exp(-5*x)/(160*a - 160*b) + exp(5*x)/(160*a + 160*b) + (2*atan((a^3*b^3*ex p(x)*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 2 1*a^10*b^4 - 7*a^12*b^2)^(1/2))/(a^7*(a^6*b^6)^(1/2) + b^7*(a^6*b^6)^(1/2) - 3*a^2*b^5*(a^6*b^6)^(1/2) + 3*a^3*b^4*(a^6*b^6)^(1/2) + 3*a^4*b^3*(a^6* b^6)^(1/2) - 3*a^5*b^2*(a^6*b^6)^(1/2) - a*b^6*(a^6*b^6)^(1/2) - a^6*b*(a^ 6*b^6)^(1/2)))*(a^6*b^6)^(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)*(a^2 - 4*a*b + b^2))/(16*(a - b)^3) - (exp(-3*x)*(a + b))/(96*(a - b)^2) - (exp (3*x)*(a - b))/(96*(a + b)^2) - (exp(x)*(4*a*b + a^2 + b^2))/(16*(a + b)^3 )