Integrand size = 36, antiderivative size = 649 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a f^2 x}{4 b^2 d^2}-\frac {a^3 (e+f x)^3}{3 b^4 f}-\frac {a (e+f x)^3}{6 b^2 f}+\frac {2 a^2 f^2 \cosh (c+d x)}{b^3 d^3}+\frac {4 f^2 \cosh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \cosh (c+d x)}{b^3 d}+\frac {a f (e+f x) \cosh ^2(c+d x)}{2 b^2 d^2}+\frac {2 f^2 \cosh ^3(c+d x)}{27 b d^3}+\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^2 \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {2 a^2 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a^2 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}-\frac {2 a^2 f (e+f x) \sinh (c+d x)}{b^3 d^2}-\frac {4 f (e+f x) \sinh (c+d x)}{9 b d^2}-\frac {a f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {2 f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b d^2} \] Output:
-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+a^2*(a^2+b^2)^(1/2)*(f*x+e)^2*ln(1+b*exp(d *x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d-a^2*(a^2+b^2)^(1/2)*(f*x+e)^2*ln(1+b*exp( d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d+2*a^2*(a^2+b^2)^(1/2)*f*(f*x+e)*polylog( 2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d^2-2*a^2*(a^2+b^2)^(1/2)*f*(f*x+ e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d^2-2*a^2*(a^2+b^2)^(1 /2)*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d^3+2*a^2*(a^2+b^ 2)^(1/2)*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d^3-2*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
Time = 2.68 (sec) , antiderivative size = 966, normalized size of antiderivative = 1.49 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {216 a^3 d^3 e^2 x+108 a b^2 d^3 e^2 x+216 a^3 d^3 e f x^2+108 a b^2 d^3 e f x^2+72 a^3 d^3 f^2 x^3+36 a b^2 d^3 f^2 x^3+432 a^2 \sqrt {a^2+b^2} d^2 e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-216 a^2 b d^2 e^2 \cosh (c+d x)-54 b^3 d^2 e^2 \cosh (c+d x)-432 a^2 b f^2 \cosh (c+d x)-108 b^3 f^2 \cosh (c+d x)-432 a^2 b d^2 e f x \cosh (c+d x)-108 b^3 d^2 e f x \cosh (c+d x)-216 a^2 b d^2 f^2 x^2 \cosh (c+d x)-54 b^3 d^2 f^2 x^2 \cosh (c+d x)-54 a b^2 d e f \cosh (2 (c+d x))-54 a b^2 d f^2 x \cosh (2 (c+d x))-18 b^3 d^2 e^2 \cosh (3 (c+d x))-4 b^3 f^2 \cosh (3 (c+d x))-36 b^3 d^2 e f x \cosh (3 (c+d x))-18 b^3 d^2 f^2 x^2 \cosh (3 (c+d x))-432 a^2 \sqrt {a^2+b^2} d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-216 a^2 \sqrt {a^2+b^2} d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+432 a^2 \sqrt {a^2+b^2} d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+216 a^2 \sqrt {a^2+b^2} d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-432 a^2 \sqrt {a^2+b^2} d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+432 a^2 \sqrt {a^2+b^2} d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+432 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-432 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+432 a^2 b d e f \sinh (c+d x)+108 b^3 d e f \sinh (c+d x)+432 a^2 b d f^2 x \sinh (c+d x)+108 b^3 d f^2 x \sinh (c+d x)+54 a b^2 d^2 e^2 \sinh (2 (c+d x))+27 a b^2 f^2 \sinh (2 (c+d x))+108 a b^2 d^2 e f x \sinh (2 (c+d x))+54 a b^2 d^2 f^2 x^2 \sinh (2 (c+d x))+12 b^3 d e f \sinh (3 (c+d x))+12 b^3 d f^2 x \sinh (3 (c+d x))}{216 b^4 d^3} \] Input:
Integrate[((e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x]^2)/(a + b*Sinh[c + d* x]),x]
Output:
-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 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{b}\right )}{b}\) |
Input:
Int[((e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 4311 vs. \(2 (595) = 1190\).
Time = 0.18 (sec) , antiderivative size = 4311, normalized size of antiderivative = 6.64 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algor ithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**2*cosh(d*x+c)**2*sinh(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algor ithm="maxima")
Output:
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)
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algor ithm="giac")
Output:
integrate((f*x + e)^2*cosh(d*x + c)^2*sinh(d*x + c)^2/(b*sinh(d*x + c) + a ), x)
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((cosh(c + d*x)^2*sinh(c + d*x)^2*(e + f*x)^2)/(a + b*sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)^2*sinh(c + d*x)^2*(e + f*x)^2)/(a + b*sinh(c + d*x)), x )
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
(864*e**(3*c + 3*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt (a**2 + b**2))*a**2*b**3*d**2*e**2*i + 18*e**(6*c + 6*d*x)*b**6*d**2*e**2 + 36*e**(6*c + 6*d*x)*b**6*d**2*e*f*x + 18*e**(6*c + 6*d*x)*b**6*d**2*f**2 *x**2 - 12*e**(6*c + 6*d*x)*b**6*d*e*f - 12*e**(6*c + 6*d*x)*b**6*d*f**2*x + 4*e**(6*c + 6*d*x)*b**6*f**2 - 54*e**(5*c + 5*d*x)*a*b**5*d**2*e**2 - 1 08*e**(5*c + 5*d*x)*a*b**5*d**2*e*f*x - 54*e**(5*c + 5*d*x)*a*b**5*d**2*f* *2*x**2 + 54*e**(5*c + 5*d*x)*a*b**5*d*e*f + 54*e**(5*c + 5*d*x)*a*b**5*d* f**2*x - 27*e**(5*c + 5*d*x)*a*b**5*f**2 + 216*e**(4*c + 4*d*x)*a**2*b**4* d**2*e**2 + 432*e**(4*c + 4*d*x)*a**2*b**4*d**2*e*f*x + 216*e**(4*c + 4*d* x)*a**2*b**4*d**2*f**2*x**2 - 432*e**(4*c + 4*d*x)*a**2*b**4*d*e*f - 432*e **(4*c + 4*d*x)*a**2*b**4*d*f**2*x + 432*e**(4*c + 4*d*x)*a**2*b**4*f**2 + 54*e**(4*c + 4*d*x)*b**6*d**2*e**2 + 108*e**(4*c + 4*d*x)*b**6*d**2*e*f*x + 54*e**(4*c + 4*d*x)*b**6*d**2*f**2*x**2 - 108*e**(4*c + 4*d*x)*b**6*d*e *f - 108*e**(4*c + 4*d*x)*b**6*d*f**2*x + 108*e**(4*c + 4*d*x)*b**6*f**2 + 3456*e**(3*c + 3*d*x)*int(x**2/(e**(5*c + 5*d*x)*b + 2*e**(4*c + 4*d*x)*a - e**(3*c + 3*d*x)*b),x)*a**6*b*d**3*f**2 + 4320*e**(3*c + 3*d*x)*int(x** 2/(e**(5*c + 5*d*x)*b + 2*e**(4*c + 4*d*x)*a - e**(3*c + 3*d*x)*b),x)*a**4 *b**3*d**3*f**2 + 864*e**(3*c + 3*d*x)*int(x**2/(e**(5*c + 5*d*x)*b + 2*e* *(4*c + 4*d*x)*a - e**(3*c + 3*d*x)*b),x)*a**2*b**5*d**3*f**2 + 6912*e**(3 *c + 3*d*x)*int(x/(e**(5*c + 5*d*x)*b + 2*e**(4*c + 4*d*x)*a - e**(3*c ...