Integrand size = 36, antiderivative size = 1021 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:
14/9*a^2*f^2*sinh(d*x+c)/b^3/d^3+2*a^3*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+ c)/(a+(a^2+b^2)^(1/2)))/b^6/d^3+2*a^3*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c )/(a-(a^2+b^2)^(1/2)))/b^6/d^3-a^3*(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/( a+(a^2+b^2)^(1/2)))/b^6/d-a^3*(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^ 2+b^2)^(1/2)))/b^6/d-2*a^3*(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+ (a^2+b^2)^(1/2)))/b^6/d^2-2*a^3*(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c )/(a-(a^2+b^2)^(1/2)))/b^6/d^2+2*a^4*f^2*sinh(d*x+c)/b^5/d^3-1/4*a^3*f^2*s inh(d*x+c)^2/b^4/d^3-1/2*a^3*(f*x+e)^2*sinh(d*x+c)^2/b^4/d+2/27*a^2*f^2*si nh(d*x+c)^3/b^3/d^3-1/200*f*(f*x+e)*cosh(5*d*x+5*c)/b/d^2-1/72*f*(f*x+e)*c osh(3*d*x+3*c)/b/d^2-3/32*a*f^2*cosh(d*x+c)^2/b^2/d^3-1/32*a*f^2*cosh(d*x+ c)^4/b^2/d^3-1/4*a*(f*x+e)^2*cosh(d*x+c)^4/b^2/d+1/3*a^3*(a^2+b^2)*(f*x+e) ^3/b^6/f+2/3*a^2*(f*x+e)^2*sinh(d*x+c)/b^3/d-2*a^4*f*(f*x+e)*cosh(d*x+c)/b ^5/d^2-2/9*a^2*f*(f*x+e)*cosh(d*x+c)^3/b^3/d^2+1/3*a^2*(f*x+e)^2*cosh(d*x+ c)^2*sinh(d*x+c)/b^3/d-4/3*a^2*f*(f*x+e)*cosh(d*x+c)/b^3/d^2+1/2*a^3*f*(f* x+e)*cosh(d*x+c)*sinh(d*x+c)/b^4/d^2+1/8*a*f*(f*x+e)*cosh(d*x+c)^3*sinh(d* x+c)/b^2/d^2+1/1000*f^2*sinh(5*d*x+5*c)/b/d^3+1/80*(f*x+e)^2*sinh(5*d*x+5* c)/b/d+1/216*f^2*sinh(3*d*x+3*c)/b/d^3+1/48*(f*x+e)^2*sinh(3*d*x+3*c)/b/d- 1/4*a^3*(f*x+e)^2/b^4/d-1/4*f^2*sinh(d*x+c)/b/d^3+3/32*a*(f*x+e)^2/b^2/d+a ^4*(f*x+e)^2*sinh(d*x+c)/b^5/d-1/8*(f*x+e)^2*sinh(d*x+c)/b/d+1/4*f*(f*x+e) *cosh(d*x+c)/b/d^2+3/16*a*f*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b^2/d^2
Time = 9.12 (sec) , antiderivative size = 1811, normalized size of antiderivative = 1.77 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^2*Cosh[c + d*x]^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d* x]),x]
Output:
((-8*a^3*(a^2 + b^2)*e^2*x*Coth[c])/b^6 - (8*a^3*(a^2 + b^2)*e*f*x^2*Coth[ c])/b^6 - (8*a^3*(a^2 + b^2)*f^2*x^3*Coth[c])/(3*b^6) + (8*a^3*(a^2 + b^2) *(6*e^2*E^(2*c)*x + 6*e*E^(2*c)*f*x^2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a^2 + b^2]*e^2*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^2 )^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTan[(a + b*E^(c + d*x)) /Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(3/2)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]*e^2* ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (6* a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (3*e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2* (c + d*x)))])/d - (3*e^2*E^(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d *x)))])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E ^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a ^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sq rt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x) )/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d *x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b *E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)* f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c -...
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.
| \(\displaystyle \int \frac {(e+f x)^2 \sinh ^3(c+d x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {\int \left (-\frac {1}{8} \cosh (c+d x) (e+f x)^2+\frac {1}{16} \cosh (3 c+3 d x) (e+f x)^2+\frac {1}{16} \cosh (5 c+5 d x) (e+f x)^2\right )dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\int (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 5970 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \int (e+f x) \cosh ^4(c+d x)dx}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^4dx}{2 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \int (e+f x) \cosh ^2(c+d x)dx-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (\frac {1}{2} \int (e+f x)dx-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\int (e+f x)^2 \cosh ^3(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 f^2 \int \cosh ^3(c+d x)dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \cosh (c+d x)dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 i f^2 \int \left (\sinh ^2(c+d x)+1\right )d(-i \sinh (c+d x))}{9 d^3}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2 \cosh (c+d x)dx}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
Input:
Int[((e + f*x)^2*Cosh[c + d*x]^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{3} \sinh \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^2*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*cosh(d*x+c)^3*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. 11318 vs. \(2 (949) = 1898\).
Time = 0.25 (sec) , antiderivative size = 11318, normalized size of antiderivative = 11.09 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algor ithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**2*cosh(d*x+c)**3*sinh(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algor ithm="maxima")
Output:
-1/960*e^2*((15*a*b^3*e^(-d*x - c) - 6*b^4 - 10*(4*a^2*b^2 + b^4)*e^(-2*d* x - 2*c) + 60*(2*a^3*b + a*b^3)*e^(-3*d*x - 3*c) - 60*(8*a^4 + 6*a^2*b^2 - b^4)*e^(-4*d*x - 4*c))*e^(5*d*x + 5*c)/(b^5*d) + 960*(a^5 + a^3*b^2)*(d*x + c)/(b^6*d) + (15*a*b^3*e^(-4*d*x - 4*c) + 6*b^4*e^(-5*d*x - 5*c) + 60*( 8*a^4 + 6*a^2*b^2 - b^4)*e^(-d*x - c) + 60*(2*a^3*b + a*b^3)*e^(-2*d*x - 2 *c) + 10*(4*a^2*b^2 + b^4)*e^(-3*d*x - 3*c))/(b^5*d) + 960*(a^5 + a^3*b^2) *log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^6*d)) - 1/1728000*(576 000*(a^5*d^3*f^2*e^(5*c) + a^3*b^2*d^3*f^2*e^(5*c))*x^3 + 1728000*(a^5*d^3 *e*f*e^(5*c) + a^3*b^2*d^3*e*f*e^(5*c))*x^2 - 432*(25*b^5*d^2*f^2*x^2*e^(1 0*c) + 10*(5*d^2*e*f - d*f^2)*b^5*x*e^(10*c) - 2*(5*d*e*f - f^2)*b^5*e^(10 *c))*e^(5*d*x) + 3375*(8*a*b^4*d^2*f^2*x^2*e^(9*c) + 4*(4*d^2*e*f - d*f^2) *a*b^4*x*e^(9*c) - (4*d*e*f - f^2)*a*b^4*e^(9*c))*e^(4*d*x) + 2000*(8*(3*d *e*f - f^2)*a^2*b^3*e^(8*c) + 2*(3*d*e*f - f^2)*b^5*e^(8*c) - 9*(4*a^2*b^3 *d^2*f^2*e^(8*c) + b^5*d^2*f^2*e^(8*c))*x^2 - 6*(4*(3*d^2*e*f - d*f^2)*a^2 *b^3*e^(8*c) + (3*d^2*e*f - d*f^2)*b^5*e^(8*c))*x)*e^(3*d*x) - 54000*(2*(2 *d*e*f - f^2)*a^3*b^2*e^(7*c) + (2*d*e*f - f^2)*a*b^4*e^(7*c) - 2*(2*a^3*b ^2*d^2*f^2*e^(7*c) + a*b^4*d^2*f^2*e^(7*c))*x^2 - 2*(2*(2*d^2*e*f - d*f^2) *a^3*b^2*e^(7*c) + (2*d^2*e*f - d*f^2)*a*b^4*e^(7*c))*x)*e^(2*d*x) + 10800 0*(16*(d*e*f - f^2)*a^4*b*e^(6*c) + 12*(d*e*f - f^2)*a^2*b^3*e^(6*c) - 2*( d*e*f - f^2)*b^5*e^(6*c) - (8*a^4*b*d^2*f^2*e^(6*c) + 6*a^2*b^3*d^2*f^2...
\[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)^3*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)^3*sinh(d*x + c)^3/(b*sinh(d*x + c) + a ), x)
Timed out. \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((cosh(c + d*x)^3*sinh(c + d*x)^3*(e + f*x)^2)/(a + b*sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)^3*sinh(c + d*x)^3*(e + f*x)^2)/(a + b*sinh(c + d*x)), x )
\[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{3} \sinh \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x \] Input:
int((f*x+e)^2*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)