Integrand size = 36, antiderivative size = 755 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 f^2 x}{4 b^3 d^2}+\frac {a^4 (e+f x)^3}{3 b^5 f}+\frac {a^2 (e+f x)^3}{6 b^3 f}-\frac {(e+f x)^3}{24 b f}-\frac {2 a^3 f^2 \cosh (c+d x)}{b^4 d^3}-\frac {4 a f^2 \cosh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^2 \cosh (c+d x)}{b^4 d}-\frac {a^2 f (e+f x) \cosh ^2(c+d x)}{2 b^3 d^2}-\frac {2 a f^2 \cosh ^3(c+d x)}{27 b^2 d^3}-\frac {a (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 b d^2}-\frac {a^3 \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^3 \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {2 a^3 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^3 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^3 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {2 a^3 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^3}+\frac {2 a^3 f (e+f x) \sinh (c+d x)}{b^4 d^2}+\frac {4 a f (e+f x) \sinh (c+d x)}{9 b^2 d^2}+\frac {a^2 f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {2 a f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^2}+\frac {f^2 \sinh (4 c+4 d x)}{256 b d^3}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 b d} \] Output:
1/4*a^2*f^2*x/b^3/d^2+1/3*a^4*(f*x+e)^3/b^5/f+1/6*a^2*(f*x+e)^3/b^3/f-1/24 *(f*x+e)^3/b/f-2*a^3*f^2*cosh(d*x+c)/b^4/d^3-4/9*a*f^2*cosh(d*x+c)/b^2/d^3 -a^3*(f*x+e)^2*cosh(d*x+c)/b^4/d-1/2*a^2*f*(f*x+e)*cosh(d*x+c)^2/b^3/d^2-2 /27*a*f^2*cosh(d*x+c)^3/b^2/d^3-1/3*a*(f*x+e)^2*cosh(d*x+c)^3/b^2/d-1/64*f *(f*x+e)*cosh(4*d*x+4*c)/b/d^2-a^3*(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^5/d+a^3*(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^5/d-2*a^3*(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^5/d^2+2*a^3*(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^5/d^2+2*a^3*(a^2+b^2)^(1/ 2)*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^5/d^3-2*a^3*(a^2+b^2 )^(1/2)*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^5/d^3+2*a^3*f*( f*x+e)*sinh(d*x+c)/b^4/d^2+4/9*a*f*(f*x+e)*sinh(d*x+c)/b^2/d^2+1/4*a^2*f^2 *cosh(d*x+c)*sinh(d*x+c)/b^3/d^3+1/2*a^2*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c) /b^3/d+2/9*a*f*(f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/b^2/d^2+1/256*f^2*sinh(4* d*x+4*c)/b/d^3+1/32*(f*x+e)^2*sinh(4*d*x+4*c)/b/d
Leaf count is larger than twice the leaf count of optimal. \(3579\) vs. \(2(755)=1510\).
Time = 12.47 (sec) , antiderivative size = 3579, normalized size of antiderivative = 4.74 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x]^3)/(a + b*Sinh[c + d* x]),x]
Output:
-1/8*(e^2*(c/d + x - (2*a*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2 ]])/(Sqrt[-a^2 - b^2]*d)))/b - (e*f*(x^2 - (2*a*(d*x*(Log[1 + (b*E^(c + d* x))/(a - Sqrt[a^2 + b^2])] - Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]) ]) + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - PolyLog[2, -((b* E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^2)))/(8*b) - (f^ 2*(x^3 - (3*a*(d^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - d^ 2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (b *E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*d*x*PolyLog[2, -((b*E^(c + d*x)) /(a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b ^2])] + 2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3)))/(24*b) - (f^2*(2*(4*a^2 + b^2)*x^3 - (6*a*(4*a^2 + 3*b^2)*( d^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - d^2*x^2*Log[1 + ( b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (b*E^(c + d*x))/( -a + Sqrt[a^2 + b^2])] - 2*d*x*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*PolyL og[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) - (24*a*b*Cosh[d*x]*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c]))/d^3 + (3*b^2*Co sh[2*d*x]*(-2*d*x*Cosh[2*c] + (1 + 2*d^2*x^2)*Sinh[2*c]))/d^3 - (24*a*b*(- 2*d*x*Cosh[c] + (2 + d^2*x^2)*Sinh[c])*Sinh[d*x])/d^3 + (3*b^2*((1 + 2*d^2 *x^2)*Cosh[2*c] - 2*d*x*Sinh[2*c])*Sinh[2*d*x])/d^3))/(96*b^3) - (e^2*(...
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.
| \(\displaystyle \int \frac {(e+f x)^2 \sinh ^3(c+d x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {\int \left (\frac {1}{8} (e+f x)^2 \cosh (4 c+4 d x)-\frac {1}{8} (e+f x)^2\right )dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\int (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 5970 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \int (e+f x) \cosh ^3(c+d x)dx}{3 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \int (e+f x) \cosh (c+d x)dx-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\int (e+f x)^2 \cosh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f^2 \int \cosh ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^2dx-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f^2 \int \cosh ^2(c+d x)dx}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {f^2 \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f^2 \left (\frac {\int 1dx}{2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2dx}{b^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}+\frac {\int -i (e+f x)^2 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \int (e+f x)^2 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {f^2 \sinh (4 c+4 d x)}{256 d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^3}{24 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 d}-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {-\frac {f (e+f x) \cosh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{b}\right )}{b}\right )}{b}\) |
Input:
Int[((e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 6459 vs. \(2 (693) = 1386\).
Time = 0.20 (sec) , antiderivative size = 6459, normalized size of antiderivative = 8.55 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algor ithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**2*cosh(d*x+c)**2*sinh(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algor ithm="maxima")
Output:
-1/192*e^2*(192*sqrt(a^2 + b^2)*a^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b ^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(b^5*d) + (8*a*b^2*e^(-d*x - c) - 24*a^2*b*e^(-2*d*x - 2*c) - 3*b^3 + 24*(4*a^3 + a*b^2)*e^(-3*d*x - 3* c))*e^(4*d*x + 4*c)/(b^4*d) - 24*(8*a^4 + 4*a^2*b^2 - b^4)*(d*x + c)/(b^5* d) + (24*a^2*b*e^(-2*d*x - 2*c) + 8*a*b^2*e^(-3*d*x - 3*c) + 3*b^3*e^(-4*d *x - 4*c) + 24*(4*a^3 + a*b^2)*e^(-d*x - c))/(b^4*d)) + 1/13824*(576*(8*a^ 4*d^3*f^2*e^(4*c) + 4*a^2*b^2*d^3*f^2*e^(4*c) - b^4*d^3*f^2*e^(4*c))*x^3 + 1728*(8*a^4*d^3*e*f*e^(4*c) + 4*a^2*b^2*d^3*e*f*e^(4*c) - b^4*d^3*e*f*e^( 4*c))*x^2 + 27*(8*b^4*d^2*f^2*x^2*e^(8*c) + 4*(4*d^2*e*f - d*f^2)*b^4*x*e^ (8*c) - (4*d*e*f - f^2)*b^4*e^(8*c))*e^(4*d*x) - 64*(9*a*b^3*d^2*f^2*x^2*e ^(7*c) + 6*(3*d^2*e*f - d*f^2)*a*b^3*x*e^(7*c) - 2*(3*d*e*f - f^2)*a*b^3*e ^(7*c))*e^(3*d*x) + 864*(2*a^2*b^2*d^2*f^2*x^2*e^(6*c) + 2*(2*d^2*e*f - d* f^2)*a^2*b^2*x*e^(6*c) - (2*d*e*f - f^2)*a^2*b^2*e^(6*c))*e^(2*d*x) + 1728 *(8*(d*e*f - f^2)*a^3*b*e^(5*c) + 2*(d*e*f - f^2)*a*b^3*e^(5*c) - (4*a^3*b *d^2*f^2*e^(5*c) + a*b^3*d^2*f^2*e^(5*c))*x^2 - 2*(4*(d^2*e*f - d*f^2)*a^3 *b*e^(5*c) + (d^2*e*f - d*f^2)*a*b^3*e^(5*c))*x)*e^(d*x) - 1728*(8*(d*e*f + f^2)*a^3*b*e^(3*c) + 2*(d*e*f + f^2)*a*b^3*e^(3*c) + (4*a^3*b*d^2*f^2*e^ (3*c) + a*b^3*d^2*f^2*e^(3*c))*x^2 + 2*(4*(d^2*e*f + d*f^2)*a^3*b*e^(3*c) + (d^2*e*f + d*f^2)*a*b^3*e^(3*c))*x)*e^(-d*x) - 864*(2*a^2*b^2*d^2*f^2*x^ 2*e^(2*c) + 2*(2*d^2*e*f + d*f^2)*a^2*b^2*x*e^(2*c) + (2*d*e*f + f^2)*a...
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)^3/(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)^3/(b*sinh(d*x + c) + a ), x)
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((cosh(c + d*x)^2*sinh(c + d*x)^3*(e + f*x)^2)/(a + b*sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)^2*sinh(c + d*x)^3*(e + f*x)^2)/(a + b*sinh(c + d*x)), x )
\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x \] Input:
int((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)