Integrand size = 36, antiderivative size = 649 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a f^2 x}{4 b^2 d^2}-\frac {a^3 (e+f x)^3}{3 b^4 f}-\frac {a (e+f x)^3}{6 b^2 f}+\frac {2 a^2 f^2 \cosh (c+d x)}{b^3 d^3}+\frac {4 f^2 \cosh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \cosh (c+d x)}{b^3 d}+\frac {a f (e+f x) \cosh ^2(c+d x)}{2 b^2 d^2}+\frac {2 f^2 \cosh ^3(c+d x)}{27 b d^3}+\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^2 \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {2 a^2 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a^2 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}-\frac {2 a^2 f (e+f x) \sinh (c+d x)}{b^3 d^2}-\frac {4 f (e+f x) \sinh (c+d x)}{9 b d^2}-\frac {a f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {2 f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b d^2} \]
-1/4*a*f^2*x/b^2/d^2-1/3*a^3*(f*x+e)^3/b^4/f-1/6*a*(f*x+e)^3/b^2/f+2*a^2*f ^2*cosh(d*x+c)/b^3/d^3+4/9*f^2*cosh(d*x+c)/b/d^3+a^2*(f*x+e)^2*cosh(d*x+c) /b^3/d+1/2*a*f*(f*x+e)*cosh(d*x+c)^2/b^2/d^2+2/27*f^2*cosh(d*x+c)^3/b/d^3+ 1/3*(f*x+e)^2*cosh(d*x+c)^3/b/d-2*a^2*f*(f*x+e)*sinh(d*x+c)/b^3/d^2-4/9*f* (f*x+e)*sinh(d*x+c)/b/d^2-1/4*a*f^2*cosh(d*x+c)*sinh(d*x+c)/b^2/d^3-1/2*a* (f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/b^2/d-2/9*f*(f*x+e)*cosh(d*x+c)^2*sinh(d *x+c)/b/d^2+a^2*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2) ^(1/2)/b^4/d-a^2*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2 )^(1/2)/b^4/d+2*a^2*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2))) *(a^2+b^2)^(1/2)/b^4/d^2-2*a^2*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b ^2)^(1/2)))*(a^2+b^2)^(1/2)/b^4/d^2-2*a^2*f^2*polylog(3,-b*exp(d*x+c)/(a-( a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^4/d^3+2*a^2*f^2*polylog(3,-b*exp(d*x+c) /(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^4/d^3
Time = 2.97 (sec) , antiderivative size = 966, normalized size of antiderivative = 1.49 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {216 a^3 d^3 e^2 x+108 a b^2 d^3 e^2 x+216 a^3 d^3 e f x^2+108 a b^2 d^3 e f x^2+72 a^3 d^3 f^2 x^3+36 a b^2 d^3 f^2 x^3+432 a^2 \sqrt {a^2+b^2} d^2 e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-216 a^2 b d^2 e^2 \cosh (c+d x)-54 b^3 d^2 e^2 \cosh (c+d x)-432 a^2 b f^2 \cosh (c+d x)-108 b^3 f^2 \cosh (c+d x)-432 a^2 b d^2 e f x \cosh (c+d x)-108 b^3 d^2 e f x \cosh (c+d x)-216 a^2 b d^2 f^2 x^2 \cosh (c+d x)-54 b^3 d^2 f^2 x^2 \cosh (c+d x)-54 a b^2 d e f \cosh (2 (c+d x))-54 a b^2 d f^2 x \cosh (2 (c+d x))-18 b^3 d^2 e^2 \cosh (3 (c+d x))-4 b^3 f^2 \cosh (3 (c+d x))-36 b^3 d^2 e f x \cosh (3 (c+d x))-18 b^3 d^2 f^2 x^2 \cosh (3 (c+d x))-432 a^2 \sqrt {a^2+b^2} d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-216 a^2 \sqrt {a^2+b^2} d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+432 a^2 \sqrt {a^2+b^2} d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+216 a^2 \sqrt {a^2+b^2} d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-432 a^2 \sqrt {a^2+b^2} d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+432 a^2 \sqrt {a^2+b^2} d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+432 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-432 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+432 a^2 b d e f \sinh (c+d x)+108 b^3 d e f \sinh (c+d x)+432 a^2 b d f^2 x \sinh (c+d x)+108 b^3 d f^2 x \sinh (c+d x)+54 a b^2 d^2 e^2 \sinh (2 (c+d x))+27 a b^2 f^2 \sinh (2 (c+d x))+108 a b^2 d^2 e f x \sinh (2 (c+d x))+54 a b^2 d^2 f^2 x^2 \sinh (2 (c+d x))+12 b^3 d e f \sinh (3 (c+d x))+12 b^3 d f^2 x \sinh (3 (c+d x))}{216 b^4 d^3} \]
-1/216*(216*a^3*d^3*e^2*x + 108*a*b^2*d^3*e^2*x + 216*a^3*d^3*e*f*x^2 + 10 8*a*b^2*d^3*e*f*x^2 + 72*a^3*d^3*f^2*x^3 + 36*a*b^2*d^3*f^2*x^3 + 432*a^2* Sqrt[a^2 + b^2]*d^2*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 216 *a^2*b*d^2*e^2*Cosh[c + d*x] - 54*b^3*d^2*e^2*Cosh[c + d*x] - 432*a^2*b*f^ 2*Cosh[c + d*x] - 108*b^3*f^2*Cosh[c + d*x] - 432*a^2*b*d^2*e*f*x*Cosh[c + d*x] - 108*b^3*d^2*e*f*x*Cosh[c + d*x] - 216*a^2*b*d^2*f^2*x^2*Cosh[c + d *x] - 54*b^3*d^2*f^2*x^2*Cosh[c + d*x] - 54*a*b^2*d*e*f*Cosh[2*(c + d*x)] - 54*a*b^2*d*f^2*x*Cosh[2*(c + d*x)] - 18*b^3*d^2*e^2*Cosh[3*(c + d*x)] - 4*b^3*f^2*Cosh[3*(c + d*x)] - 36*b^3*d^2*e*f*x*Cosh[3*(c + d*x)] - 18*b^3* d^2*f^2*x^2*Cosh[3*(c + d*x)] - 432*a^2*Sqrt[a^2 + b^2]*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 216*a^2*Sqrt[a^2 + b^2]*d^2*f^2*x ^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 432*a^2*Sqrt[a^2 + b^2 ]*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 216*a^2*Sqrt[ a^2 + b^2]*d^2*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 43 2*a^2*Sqrt[a^2 + b^2]*d*f*(e + f*x)*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[ a^2 + b^2])] + 432*a^2*Sqrt[a^2 + b^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 432*a^2*Sqrt[a^2 + b^2]*f^2*PolyLog[3, ( b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 432*a^2*Sqrt[a^2 + b^2]*f^2*PolyL og[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 432*a^2*b*d*e*f*Sinh[c + d*x] + 108*b^3*d*e*f*Sinh[c + d*x] + 432*a^2*b*d*f^2*x*Sinh[c + d*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.
| \(\displaystyle \int \frac {(e+f x)^2 \sinh ^2(c+d x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5970 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \int (e+f x) \cosh ^3(c+d x)dx}{3 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d}}{b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \int (e+f x) \cosh (c+d x)dx-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\int (e+f x)^2 \cosh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f^2 \int \cosh ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^2dx-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f^2 \int \cosh ^2(c+d x)dx}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {f^2 \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f^2 \left (\frac {\int 1dx}{2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2dx}{b^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}+\frac {\int -i (e+f x)^2 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \int (e+f x)^2 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {2 \left (a^2+b^2\right ) \int -\frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{b}\right )}{b}\) |
3.4.68.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, 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[((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[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( -I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^m*(Cosh[a + b*x]^(n + 1)/(b*(n + 1 ))), x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. )*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-a/b^2 Int[(e + f*x)^m*Cos h[c + d*x]^(n - 2), x], x] + (Simp[1/b Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)*Sinh[c + d*x], x], x] + Simp[(a^2 + b^2)/b^2 Int[(e + f*x)^m*(Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/b Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x] - S imp[a/b Int[(e + f*x)^m*Cosh[c + d*x]^p*(Sinh[c + d*x]^(n - 1)/(a + b*Sin h[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[ n, 0] && IGtQ[p, 0]
\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 4311 vs. \(2 (595) = 1190\).
Time = 0.33 (sec) , antiderivative size = 4311, normalized size of antiderivative = 6.64 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
1/432*(18*b^3*d^2*f^2*x^2 + 18*b^3*d^2*e^2 + 2*(9*b^3*d^2*f^2*x^2 + 9*b^3* d^2*e^2 - 6*b^3*d*e*f + 2*b^3*f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh( d*x + c)^6 + 2*(9*b^3*d^2*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3*f^ 2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*sinh(d*x + c)^6 + 12*b^3*d*e*f - 27*( 2*a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e^2 - 2*a*b^2*d*e*f + a*b^2*f^2 + 2*(2*a *b^2*d^2*e*f - a*b^2*d*f^2)*x)*cosh(d*x + c)^5 - 3*(18*a*b^2*d^2*f^2*x^2 + 18*a*b^2*d^2*e^2 - 18*a*b^2*d*e*f + 9*a*b^2*f^2 + 18*(2*a*b^2*d^2*e*f - a *b^2*d*f^2)*x - 4*(9*b^3*d^2*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3 *f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 4 *b^3*f^2 + 54*((4*a^2*b + b^3)*d^2*f^2*x^2 + (4*a^2*b + b^3)*d^2*e^2 - 2*( 4*a^2*b + b^3)*d*e*f + 2*(4*a^2*b + b^3)*f^2 + 2*((4*a^2*b + b^3)*d^2*e*f - (4*a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c)^4 + 3*(18*(4*a^2*b + b^3)*d^2*f^ 2*x^2 + 18*(4*a^2*b + b^3)*d^2*e^2 - 36*(4*a^2*b + b^3)*d*e*f + 36*(4*a^2* b + b^3)*f^2 + 10*(9*b^3*d^2*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3 *f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh(d*x + c)^2 + 36*((4*a^2*b + b ^3)*d^2*e*f - (4*a^2*b + b^3)*d*f^2)*x - 45*(2*a*b^2*d^2*f^2*x^2 + 2*a*b^2 *d^2*e^2 - 2*a*b^2*d*e*f + a*b^2*f^2 + 2*(2*a*b^2*d^2*e*f - a*b^2*d*f^2)*x )*cosh(d*x + c))*sinh(d*x + c)^4 - 72*((2*a^3 + a*b^2)*d^3*f^2*x^3 + 3*(2* a^3 + a*b^2)*d^3*e*f*x^2 + 3*(2*a^3 + a*b^2)*d^3*e^2*x)*cosh(d*x + c)^3 - 2*(36*(2*a^3 + a*b^2)*d^3*f^2*x^3 + 108*(2*a^3 + a*b^2)*d^3*e*f*x^2 + 1...
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
1/24*e^2*(24*sqrt(a^2 + b^2)*a^2*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2) )/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(b^4*d) - (3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 + b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) - 12*(2*a^ 3 + a*b^2)*(d*x + c)/(b^4*d) + (3*a*b*e^(-2*d*x - 2*c) + b^2*e^(-3*d*x - 3 *c) + 3*(4*a^2 + b^2)*e^(-d*x - c))/(b^3*d)) - 1/432*(72*(2*a^3*d^3*f^2*e^ (3*c) + a*b^2*d^3*f^2*e^(3*c))*x^3 + 216*(2*a^3*d^3*e*f*e^(3*c) + a*b^2*d^ 3*e*f*e^(3*c))*x^2 - 2*(9*b^3*d^2*f^2*x^2*e^(6*c) + 6*(3*d^2*e*f - d*f^2)* b^3*x*e^(6*c) - 2*(3*d*e*f - f^2)*b^3*e^(6*c))*e^(3*d*x) + 27*(2*a*b^2*d^2 *f^2*x^2*e^(5*c) + 2*(2*d^2*e*f - d*f^2)*a*b^2*x*e^(5*c) - (2*d*e*f - f^2) *a*b^2*e^(5*c))*e^(2*d*x) + 54*(8*(d*e*f - f^2)*a^2*b*e^(4*c) + 2*(d*e*f - f^2)*b^3*e^(4*c) - (4*a^2*b*d^2*f^2*e^(4*c) + b^3*d^2*f^2*e^(4*c))*x^2 - 2*(4*(d^2*e*f - d*f^2)*a^2*b*e^(4*c) + (d^2*e*f - d*f^2)*b^3*e^(4*c))*x)*e ^(d*x) - 54*(8*(d*e*f + f^2)*a^2*b*e^(2*c) + 2*(d*e*f + f^2)*b^3*e^(2*c) + (4*a^2*b*d^2*f^2*e^(2*c) + b^3*d^2*f^2*e^(2*c))*x^2 + 2*(4*(d^2*e*f + d*f ^2)*a^2*b*e^(2*c) + (d^2*e*f + d*f^2)*b^3*e^(2*c))*x)*e^(-d*x) - 27*(2*a*b ^2*d^2*f^2*x^2*e^c + 2*(2*d^2*e*f + d*f^2)*a*b^2*x*e^c + (2*d*e*f + f^2)*a *b^2*e^c)*e^(-2*d*x) - 2*(9*b^3*d^2*f^2*x^2 + 6*(3*d^2*e*f + d*f^2)*b^3*x + 2*(3*d*e*f + f^2)*b^3)*e^(-3*d*x))*e^(-3*c)/(b^4*d^3) + integrate(2*((a^ 4*f^2*e^c + a^2*b^2*f^2*e^c)*x^2 + 2*(a^4*e*f*e^c + a^2*b^2*e*f*e^c)*x)*e^ (d*x)/(b^5*e^(2*d*x + 2*c) + 2*a*b^4*e^(d*x + c) - b^5), x)
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]