Integrand size = 20, antiderivative size = 314 \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {6 a^3 b^3 x}{\left (a^2-b^2\right )^4}-\frac {a^3 b x}{\left (a^2-b^2\right )^3}+\frac {a b^3 x}{\left (a^2-b^2\right )^3}+\frac {a b x}{4 \left (a^2-b^2\right )^2}+\frac {b^2 \cosh ^4(x)}{4 \left (a^2-b^2\right )^2}+\frac {3 a^4 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}+\frac {3 a^2 b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}+\frac {a^3 b \cosh (x) \sinh (x)}{\left (a^2-b^2\right )^3}+\frac {a b^3 \cosh (x) \sinh (x)}{\left (a^2-b^2\right )^3}+\frac {a b \cosh (x) \sinh (x)}{4 \left (a^2-b^2\right )^2}-\frac {a b \cosh ^3(x) \sinh (x)}{2 \left (a^2-b^2\right )^2}-\frac {2 a^2 b^2 \sinh ^2(x)}{\left (a^2-b^2\right )^3}+\frac {a^2 \sinh ^4(x)}{4 \left (a^2-b^2\right )^2}+\frac {a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))} \]
-6*a^3*b^3*x/(a^2-b^2)^4-a^3*b*x/(a^2-b^2)^3+a*b^3*x/(a^2-b^2)^3+1/4*a*b*x /(a^2-b^2)^2+1/4*b^2*cosh(x)^4/(a^2-b^2)^2+3*a^4*b^2*ln(a*cosh(x)+b*sinh(x ))/(a^2-b^2)^4+3*a^2*b^4*ln(a*cosh(x)+b*sinh(x))/(a^2-b^2)^4+a^3*b*cosh(x) *sinh(x)/(a^2-b^2)^3+a*b^3*cosh(x)*sinh(x)/(a^2-b^2)^3+1/4*a*b*cosh(x)*sin h(x)/(a^2-b^2)^2-1/2*a*b*cosh(x)^3*sinh(x)/(a^2-b^2)^2-2*a^2*b^2*sinh(x)^2 /(a^2-b^2)^3+1/4*a^2*sinh(x)^4/(a^2-b^2)^2+a^2*b^3*sinh(x)/(a^2-b^2)^3/(a* cosh(x)+b*sinh(x))
Time = 1.03 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {-3 a \left (a^2-b^2\right )^2 \left (a^2+3 b^2\right ) \cosh (3 x)+a^7 \cosh (5 x)-3 a^5 b^2 \cosh (5 x)+3 a^3 b^4 \cosh (5 x)-a b^6 \cosh (5 x)-4 a \cosh (x) \left (a^6+9 a^4 b^2-5 a^2 b^4-5 b^6+12 a^5 b x+72 a^3 b^3 x+12 a b^5 x-48 a^2 b^2 \left (a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))\right )+20 a^6 b \sinh (x)+84 a^4 b^3 \sinh (x)-100 a^2 b^5 \sinh (x)-4 b^7 \sinh (x)-48 a^5 b^2 x \sinh (x)-288 a^3 b^4 x \sinh (x)-48 a b^6 x \sinh (x)+192 a^4 b^3 \log (a \cosh (x)+b \sinh (x)) \sinh (x)+192 a^2 b^5 \log (a \cosh (x)+b \sinh (x)) \sinh (x)+9 a^6 b \sinh (3 x)-15 a^4 b^3 \sinh (3 x)+3 a^2 b^5 \sinh (3 x)+3 b^7 \sinh (3 x)-a^6 b \sinh (5 x)+3 a^4 b^3 \sinh (5 x)-3 a^2 b^5 \sinh (5 x)+b^7 \sinh (5 x)}{64 (a-b)^4 (a+b)^4 (a \cosh (x)+b \sinh (x))} \]
(-3*a*(a^2 - b^2)^2*(a^2 + 3*b^2)*Cosh[3*x] + a^7*Cosh[5*x] - 3*a^5*b^2*Co sh[5*x] + 3*a^3*b^4*Cosh[5*x] - a*b^6*Cosh[5*x] - 4*a*Cosh[x]*(a^6 + 9*a^4 *b^2 - 5*a^2*b^4 - 5*b^6 + 12*a^5*b*x + 72*a^3*b^3*x + 12*a*b^5*x - 48*a^2 *b^2*(a^2 + b^2)*Log[a*Cosh[x] + b*Sinh[x]]) + 20*a^6*b*Sinh[x] + 84*a^4*b ^3*Sinh[x] - 100*a^2*b^5*Sinh[x] - 4*b^7*Sinh[x] - 48*a^5*b^2*x*Sinh[x] - 288*a^3*b^4*x*Sinh[x] - 48*a*b^6*x*Sinh[x] + 192*a^4*b^3*Log[a*Cosh[x] + b *Sinh[x]]*Sinh[x] + 192*a^2*b^5*Log[a*Cosh[x] + b*Sinh[x]]*Sinh[x] + 9*a^6 *b*Sinh[3*x] - 15*a^4*b^3*Sinh[3*x] + 3*a^2*b^5*Sinh[3*x] + 3*b^7*Sinh[3*x ] - a^6*b*Sinh[5*x] + 3*a^4*b^3*Sinh[5*x] - 3*a^2*b^5*Sinh[5*x] + b^7*Sinh [5*x])/(64*(a - b)^4*(a + b)^4*(a*Cosh[x] + b*Sinh[x]))
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.
| \(\displaystyle \int \frac {\sinh ^3(x) \cosh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i x)^3 \cos (i x)^3}{(a \cos (i x)-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\cos (i x)^3 \sin (i x)^3}{(a \cos (i x)-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 3590 |
| \(\displaystyle i \left (-\frac {i b \int -\frac {\cosh ^3(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}+\frac {a \int -\frac {i \cosh ^2(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}+\frac {i a b \int -\frac {\cosh ^2(x) \sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i b \int \frac {\cosh ^3(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}+\frac {a \int -\frac {i \cosh ^2(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {i a b \int \frac {\cosh ^2(x) \sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i b \int \frac {\cosh ^3(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {i a \int \frac {\cosh ^2(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {i a b \int \frac {\cosh ^2(x) \sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i a b \int -\frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {i b \int -\frac {\cos (i x)^3 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i a \int \frac {i \cos (i x)^2 \sin (i x)^3}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \int \frac {\cos (i x)^3 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i a \int \frac {i \cos (i x)^2 \sin (i x)^3}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \int \frac {\cos (i x)^3 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {a \int \frac {\cos (i x)^2 \sin (i x)^3}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (-\frac {i b \int -\cosh ^2(x) \sinh ^2(x)dx}{a^2-b^2}+\frac {a \int -i \cosh (x) \sinh ^3(x)dx}{a^2-b^2}+\frac {i a b \int -\frac {\cosh (x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (-\frac {i b \int i \cosh ^3(x) \sinh (x)dx}{a^2-b^2}+\frac {a \int -\cosh ^2(x) \sinh ^2(x)dx}{a^2-b^2}+\frac {i a b \int \frac {i \cosh ^2(x) \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i b \int \cosh ^2(x) \sinh ^2(x)dx}{a^2-b^2}+\frac {a \int -i \cosh (x) \sinh ^3(x)dx}{a^2-b^2}-\frac {i a b \int \frac {\cosh (x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (-\frac {i b \int i \cosh ^3(x) \sinh (x)dx}{a^2-b^2}-\frac {a \int \cosh ^2(x) \sinh ^2(x)dx}{a^2-b^2}+\frac {i a b \int \frac {i \cosh ^2(x) \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i b \int \cosh ^2(x) \sinh ^2(x)dx}{a^2-b^2}-\frac {i a \int \cosh (x) \sinh ^3(x)dx}{a^2-b^2}-\frac {i a b \int \frac {\cosh (x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {b \int \cosh ^3(x) \sinh (x)dx}{a^2-b^2}-\frac {a \int \cosh ^2(x) \sinh ^2(x)dx}{a^2-b^2}-\frac {a b \int \frac {\cosh ^2(x) \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (\frac {b \int -i \cos (i x)^3 \sin (i x)dx}{a^2-b^2}-\frac {a \int -\cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}-\frac {a b \int -\frac {i \cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (\frac {i b \int -\cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}-\frac {i a \int i \cos (i x) \sin (i x)^3dx}{a^2-b^2}-\frac {i a b \int -\frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (\frac {b \int -i \cos (i x)^3 \sin (i x)dx}{a^2-b^2}+\frac {a \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}-\frac {a b \int -\frac {i \cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (-\frac {i b \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}-\frac {i a \int i \cos (i x) \sin (i x)^3dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (-\frac {i b \int \cos (i x)^3 \sin (i x)dx}{a^2-b^2}+\frac {a \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (-\frac {i b \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {a \int \cos (i x) \sin (i x)^3dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (-\frac {i b \int \cos (i x)^3 \sin (i x)dx}{a^2-b^2}+\frac {a \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (-\frac {i a \int -i \sinh ^3(x)d(i \sinh (x))}{a^2-b^2}-\frac {i b \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}-\frac {i b \left (-\frac {i b \int \cos (i x)^3 \sin (i x)dx}{a^2-b^2}+\frac {a \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (-\frac {i b \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (-\frac {i b \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {i b \left (\frac {b \int \cosh ^3(x)d\cosh (x)}{a^2-b^2}+\frac {a \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (-\frac {i b \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {i b \left (\frac {a \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b \left (\frac {1}{4} \int \cosh ^2(x)dx-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {a \left (\frac {1}{4} \int \cosh ^2(x)dx-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}+\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b \left (-\frac {1}{4} \sinh (x) \cosh ^3(x)+\frac {1}{4} \int \sin \left (i x+\frac {\pi }{2}\right )^2dx\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {a \left (-\frac {1}{4} \sinh (x) \cosh ^3(x)+\frac {1}{4} \int \sin \left (i x+\frac {\pi }{2}\right )^2dx\right )}{a^2-b^2}+\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b \left (\frac {1}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {a \left (\frac {1}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}+\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (\frac {a \int -\sinh ^2(x)dx}{a^2-b^2}-\frac {i b \int i \cosh (x) \sinh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {i \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \left (-\frac {i b \int \cosh ^2(x)dx}{a^2-b^2}+\frac {a \int i \cosh (x) \sinh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cosh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (-\frac {a \int \sinh ^2(x)dx}{a^2-b^2}-\frac {i b \int i \cosh (x) \sinh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {i \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \left (-\frac {i b \int \cosh ^2(x)dx}{a^2-b^2}+\frac {a \int i \cosh (x) \sinh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cosh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (-\frac {a \int \sinh ^2(x)dx}{a^2-b^2}+\frac {b \int \cosh (x) \sinh (x)dx}{a^2-b^2}-\frac {a b \int \frac {\sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \left (-\frac {i b \int \cosh ^2(x)dx}{a^2-b^2}+\frac {i a \int \cosh (x) \sinh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cosh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (-\frac {a \int -\sin (i x)^2dx}{a^2-b^2}+\frac {b \int -i \cos (i x) \sin (i x)dx}{a^2-b^2}-\frac {a b \int -\frac {i \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}+\frac {i a \int -i \cos (i x) \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (\frac {a \int \sin (i x)^2dx}{a^2-b^2}+\frac {b \int -i \cos (i x) \sin (i x)dx}{a^2-b^2}-\frac {a b \int -\frac {i \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}+\frac {i a \int -i \cos (i x) \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (\frac {a \int \sin (i x)^2dx}{a^2-b^2}-\frac {i b \int \cos (i x) \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}+\frac {a \int \cos (i x) \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (\frac {a \int \sin (i x)^2dx}{a^2-b^2}-\frac {b \int i \sinh (x)d(i \sinh (x))}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}-\frac {i a \int i \sinh (x)d(i \sinh (x))}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)^2}{(a \cos (i x)-i b \sin (i x))^2}dx}{a^2-b^2}+\frac {a \left (\frac {i a b \left (\frac {a \int \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {b \sinh ^2(x)}{2 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {i a b \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {i a \sinh ^2(x)}{2 \left (a^2-b^2\right )}\right )}{a^2-b^2}+\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
3.8.23.3.1 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[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[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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 = 17.21 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {1}{4 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {-a -5 b}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {a -3 b}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {3 a b \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{4 \left (a +b \right )^{4}}+\frac {1}{4 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {1}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-a -3 b}{8 \left (a -b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {-a +5 b}{8 \left (a -b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3 a b \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{4 \left (a -b \right )^{4}}+\frac {2 a^{2} b^{2} \left (\frac {b \left (a^{2}-b^{2}\right ) \tanh \left (\frac {x}{2}\right )}{\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a}+\frac {\left (3 a^{2}+3 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a \right )}{2}\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}\) | \(276\) |
| risch | \(-\frac {3 a b x}{4 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )^{2}}+\frac {{\mathrm e}^{4 x}}{64 \left (a +b \right )^{2}}-\frac {{\mathrm e}^{2 x} a}{16 \left (a +b \right )^{3}}+\frac {{\mathrm e}^{2 x} b}{16 \left (a +b \right )^{3}}-\frac {{\mathrm e}^{-2 x} a}{16 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {{\mathrm e}^{-2 x} b}{16 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {{\mathrm e}^{-4 x}}{64 a^{2}-128 a b +64 b^{2}}-\frac {6 b^{2} a^{4} x}{a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}-\frac {6 b^{4} a^{2} x}{a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}-\frac {2 a^{3} b^{3}}{\left (a -b \right )^{3} \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right )}+\frac {3 b^{2} a^{4} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}+\frac {3 b^{4} a^{2} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}\) | \(402\) |
1/4/(a+b)^2/(tanh(1/2*x)-1)^4+1/2/(a+b)^2/(tanh(1/2*x)-1)^3-1/8*(-a-5*b)/( a+b)^3/(tanh(1/2*x)-1)^2-1/8*(a-3*b)/(a+b)^3/(tanh(1/2*x)-1)+3/4*a*b/(a+b) ^4*ln(tanh(1/2*x)-1)+1/4/(a-b)^2/(tanh(1/2*x)+1)^4-1/2/(a-b)^2/(tanh(1/2*x )+1)^3-1/8*(-a-3*b)/(a-b)^3/(tanh(1/2*x)+1)-1/8*(-a+5*b)/(a-b)^3/(tanh(1/2 *x)+1)^2-3/4*a*b/(a-b)^4*ln(tanh(1/2*x)+1)+2*a^2*b^2/(a+b)^4/(a-b)^4*(b*(a ^2-b^2)*tanh(1/2*x)/(tanh(1/2*x)^2*a+2*b*tanh(1/2*x)+a)+1/2*(3*a^2+3*b^2)* ln(tanh(1/2*x)^2*a+2*b*tanh(1/2*x)+a))
Leaf count of result is larger than twice the leaf count of optimal. 4001 vs. \(2 (304) = 608\).
Time = 0.31 (sec) , antiderivative size = 4001, normalized size of antiderivative = 12.74 \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\text {Too large to display} \]
1/64*((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 + 10*(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 + (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 - 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)^8 - 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 - 15*(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 + 24*(5*(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))*sinh(x) ^7 + 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 - 4*(a^7 - 5*a^6*b + 9*a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7 + 12*(a^6*b + 5*a^5*b^2 + 10*a^4*b^3 + 10*a^3*b^4 + 5*a^2*b^ 5 + a*b^6)*x)*cosh(x)^6 - 2*(2*a^7 - 10*a^6*b + 18*a^5*b^2 - 10*a^4*b^3 - 10*a^3*b^4 + 18*a^2*b^5 - 10*a*b^6 + 2*b^7 - 105*(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 + 42*(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 + 24*(a^6*b + 5*a^5*b^2 + 10*a^4*b^3 + 10*a^3*b^4 + 5*a^2*b^5 + a*b^6) *x)*sinh(x)^6 + 12*(21*(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 - 14*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^...
Timed out. \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {3 \, a b x}{4 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )}} + \frac {3 \, {\left (a^{4} b^{2} + a^{2} b^{4}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {4 \, {\left (a + b\right )} e^{\left (-2 \, x\right )} - {\left (a - b\right )} e^{\left (-4 \, x\right )}}{64 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {a^{6} - 2 \, a^{5} b - a^{4} b^{2} + 4 \, a^{3} b^{3} - a^{2} b^{4} - 2 \, a b^{5} + b^{6} - 3 \, {\left (a^{6} - 4 \, a^{5} b + 5 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + 4 \, a b^{5} - b^{6}\right )} e^{\left (-2 \, x\right )} - 4 \, {\left (a^{6} - 6 \, a^{5} b + 15 \, a^{4} b^{2} - 52 \, a^{3} b^{3} + 15 \, a^{2} b^{4} - 6 \, a b^{5} + b^{6}\right )} e^{\left (-4 \, x\right )}}{64 \, {\left ({\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} e^{\left (-4 \, x\right )} + {\left (a^{8} - 2 \, a^{7} b - 2 \, a^{6} b^{2} + 6 \, a^{5} b^{3} - 6 \, a^{3} b^{5} + 2 \, a^{2} b^{6} + 2 \, a b^{7} - b^{8}\right )} e^{\left (-6 \, x\right )}\right )}} \]
-3/4*a*b*x/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + 3*(a^4*b^2 + a^2* b^4)*log(-(a - b)*e^(-2*x) - a - b)/(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b ^6 + b^8) - 1/64*(4*(a + b)*e^(-2*x) - (a - b)*e^(-4*x))/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/64*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a *b^5 + b^6 - 3*(a^6 - 4*a^5*b + 5*a^4*b^2 - 5*a^2*b^4 + 4*a*b^5 - b^6)*e^( -2*x) - 4*(a^6 - 6*a^5*b + 15*a^4*b^2 - 52*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*e^(-4*x))/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*e^(-4*x) + (a^8 - 2*a^7*b - 2*a^6*b^2 + 6*a^5*b^3 - 6*a^3*b^5 + 2*a^2*b^6 + 2*a*b^ 7 - b^8)*e^(-6*x))
Time = 0.27 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {3 \, a b x}{4 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )}} + \frac {{\left (36 \, a b e^{\left (4 \, x\right )} - 4 \, a^{2} e^{\left (2 \, x\right )} + 4 \, b^{2} e^{\left (2 \, x\right )} + a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-4 \, x\right )}}{64 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )}} + \frac {3 \, {\left (a^{4} b^{2} + a^{2} b^{4}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} + \frac {a^{2} e^{\left (4 \, x\right )} + 2 \, a b e^{\left (4 \, x\right )} + b^{2} e^{\left (4 \, x\right )} - 4 \, a^{2} e^{\left (2 \, x\right )} + 4 \, b^{2} e^{\left (2 \, x\right )}}{64 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )}} - \frac {3 \, a^{5} b^{2} e^{\left (2 \, x\right )} + 3 \, a^{4} b^{3} e^{\left (2 \, x\right )} + 3 \, a^{3} b^{4} e^{\left (2 \, x\right )} + 3 \, a^{2} b^{5} e^{\left (2 \, x\right )} + 3 \, a^{5} b^{2} - a^{4} b^{3} + a^{3} b^{4} - 3 \, a^{2} b^{5}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} \]
-3/4*a*b*x/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) + 1/64*(36*a*b*e^(4 *x) - 4*a^2*e^(2*x) + 4*b^2*e^(2*x) + a^2 - 2*a*b + b^2)*e^(-4*x)/(a^4 - 4 *a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) + 3*(a^4*b^2 + a^2*b^4)*log(abs(a*e^(2 *x) + b*e^(2*x) + a - b))/(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8) + 1/64*(a^2*e^(4*x) + 2*a*b*e^(4*x) + b^2*e^(4*x) - 4*a^2*e^(2*x) + 4*b^2* e^(2*x))/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) - (3*a^5*b^2*e^(2*x) + 3*a^4*b^3*e^(2*x) + 3*a^3*b^4*e^(2*x) + 3*a^2*b^5*e^(2*x) + 3*a^5*b^2 - a^4*b^3 + a^3*b^4 - 3*a^2*b^5)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(a*e^(2*x) + b*e^(2*x) + a - b))
Time = 2.76 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.55 \[ \int \frac {\cosh ^3(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {{\mathrm {e}}^{4\,x}}{64\,{\left (a+b\right )}^2}+\frac {{\mathrm {e}}^{-4\,x}}{64\,{\left (a-b\right )}^2}+\frac {\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (3\,a^4\,b^2+3\,a^2\,b^4\right )}{a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8}-\frac {{\mathrm {e}}^{-2\,x}\,\left (a+b\right )}{16\,{\left (a-b\right )}^3}-\frac {{\mathrm {e}}^{2\,x}\,\left (a-b\right )}{16\,{\left (a+b\right )}^3}-\frac {3\,a\,b\,x}{4\,{\left (a-b\right )}^4}-\frac {2\,a^3\,b^3}{{\left (a+b\right )}^4\,{\left (a-b\right )}^3\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \]
exp(4*x)/(64*(a + b)^2) + exp(-4*x)/(64*(a - b)^2) + (log(a - b + a*exp(2* x) + b*exp(2*x))*(3*a^2*b^4 + 3*a^4*b^2))/(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b ^4 - 4*a^6*b^2) - (exp(-2*x)*(a + b))/(16*(a - b)^3) - (exp(2*x)*(a - b))/ (16*(a + b)^3) - (3*a*b*x)/(4*(a - b)^4) - (2*a^3*b^3)/((a + b)^4*(a - b)^ 3*(a - b + exp(2*x)*(a + b)))