Integrand size = 34, antiderivative size = 474 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^4 e x}{b^5}+\frac {a^2 e x}{2 b^3}+\frac {a^4 f x^2}{2 b^5}+\frac {a^2 f x^2}{4 b^3}-\frac {(e+f x)^2}{16 b f}-\frac {a^3 (e+f x) \cosh (c+d x)}{b^4 d}-\frac {a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac {a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac {f \cosh (4 c+4 d x)}{128 b d^2}-\frac {a^3 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^3 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {a^3 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {a^3 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {a^3 f \sinh (c+d x)}{b^4 d^2}+\frac {a f \sinh (c+d x)}{3 b^2 d^2}+\frac {a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 b d} \]
a^4*e*x/b^5+1/2*a^2*e*x/b^3+1/2*a^4*f*x^2/b^5+1/4*a^2*f*x^2/b^3-1/16*(f*x+ e)^2/b/f-a^3*(f*x+e)*cosh(d*x+c)/b^4/d-1/4*a^2*f*cosh(d*x+c)^2/b^3/d^2-1/3 *a*(f*x+e)*cosh(d*x+c)^3/b^2/d-1/128*f*cosh(4*d*x+4*c)/b/d^2+a^3*f*sinh(d* x+c)/b^4/d^2+1/3*a*f*sinh(d*x+c)/b^2/d^2+1/2*a^2*(f*x+e)*cosh(d*x+c)*sinh( d*x+c)/b^3/d+1/9*a*f*sinh(d*x+c)^3/b^2/d^2+1/32*(f*x+e)*sinh(4*d*x+4*c)/b/ d-a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d +a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d- a^3*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d^2 +a^3*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d^ 2
Leaf count is larger than twice the leaf count of optimal. \(1627\) vs. \(2(474)=948\).
Time = 4.22 (sec) , antiderivative size = 1627, normalized size of antiderivative = 3.43 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \]
(1152*a^4*Sqrt[-(a^2 + b^2)^2]*c*d*e + 576*a^2*b^2*Sqrt[-(a^2 + b^2)^2]*c* d*e - 144*b^4*Sqrt[-(a^2 + b^2)^2]*c*d*e - 576*a^4*Sqrt[-(a^2 + b^2)^2]*c^ 2*f - 288*a^2*b^2*Sqrt[-(a^2 + b^2)^2]*c^2*f + 1152*a^4*Sqrt[-(a^2 + b^2)^ 2]*d^2*e*x + 576*a^2*b^2*Sqrt[-(a^2 + b^2)^2]*d^2*e*x - 144*b^4*Sqrt[-(a^2 + b^2)^2]*d^2*e*x + 576*a^4*Sqrt[-(a^2 + b^2)^2]*d^2*f*x^2 + 288*a^2*b^2* Sqrt[-(a^2 + b^2)^2]*d^2*f*x^2 - 72*b^4*Sqrt[-(a^2 + b^2)^2]*d^2*f*x^2 - 2 304*a^5*Sqrt[a^2 + b^2]*d*e*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b ^2]] - 2304*a^3*b^2*Sqrt[a^2 + b^2]*d*e*ArcTan[(b - a*Tanh[(c + d*x)/2])/S qrt[-a^2 - b^2]] - 2304*a^5*Sqrt[-a^2 - b^2]*c*f*ArcTanh[(a + b*E^(c + d*x ))/Sqrt[a^2 + b^2]] - 2304*a^3*b^2*Sqrt[-a^2 - b^2]*c*f*ArcTanh[(a + b*E^( c + d*x))/Sqrt[a^2 + b^2]] - 288*a*b^4*Sqrt[-a^2 - b^2]*c*f*ArcTanh[(a + b *E^(c + d*x))/Sqrt[a^2 + b^2]] - 1152*a^3*b*Sqrt[-(a^2 + b^2)^2]*d*e*Cosh[ c + d*x] - 288*a*b^3*Sqrt[-(a^2 + b^2)^2]*d*e*Cosh[c + d*x] - 1152*a^3*b*S qrt[-(a^2 + b^2)^2]*d*f*x*Cosh[c + d*x] - 288*a*b^3*Sqrt[-(a^2 + b^2)^2]*d *f*x*Cosh[c + d*x] - 144*a^2*b^2*Sqrt[-(a^2 + b^2)^2]*f*Cosh[2*(c + d*x)] - 96*a*b^3*Sqrt[-(a^2 + b^2)^2]*d*e*Cosh[3*(c + d*x)] - 96*a*b^3*Sqrt[-(a^ 2 + b^2)^2]*d*f*x*Cosh[3*(c + d*x)] - 9*b^4*Sqrt[-(a^2 + b^2)^2]*f*Cosh[4* (c + d*x)] - 1152*a^5*Sqrt[-a^2 - b^2]*c*f*Log[1 + (b*E^(c + d*x))/(a - Sq rt[a^2 + b^2])] - 1152*a^3*b^2*Sqrt[-a^2 - b^2]*c*f*Log[1 + (b*E^(c + d*x) )/(a - Sqrt[a^2 + b^2])] - 144*a*b^4*Sqrt[-a^2 - b^2]*c*f*Log[1 + (b*E^...
Result contains complex when optimal does not.
Time = 2.87 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.96, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, Rules used = {6113, 5971, 2009, 6113, 5970, 3042, 3113, 2009, 6113, 3042, 3791, 17, 6099, 17, 3042, 26, 3777, 3042, 3117, 3803, 25, 2694, 27, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {(e+f x) \sinh ^3(c+d x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {\int \left (\frac {1}{8} (-e-f x)+\frac {1}{8} (e+f x) \cosh (4 c+4 d x)\right )dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\int (e+f x) \cosh ^2(c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 5970 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {f \int \cosh ^3(c+d x)dx}{3 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{3 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \int \left (\sinh ^2(c+d x)+1\right )d(-i \sinh (c+d x))}{3 d^2}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x) \cosh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (-\frac {a \left (\frac {\frac {1}{2} \int (e+f x)dx-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)dx}{b^2}+\frac {\int (e+f x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}+\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x) \sinh (c+d x)dx}{b}-\frac {a (e+f x)^2}{2 b^2 f}\right )}{b}\right )}{b}+\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{b^2}+\frac {\int -i (e+f x) \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^2}{2 b^2 f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \int (e+f x) \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^2}{2 b^2 f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{b}-\frac {a (e+f x)^2}{2 b^2 f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b}-\frac {a (e+f x)^2}{2 b^2 f}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {2 \left (a^2+b^2\right ) \int -\frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {f \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^2}{16 f}}{b}-\frac {a \left (\frac {\frac {(e+f x) \cosh ^3(c+d x)}{3 d}-\frac {i f \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{3 d^2}}{b}-\frac {a \left (\frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\) |
-((a*((((e + f*x)*Cosh[c + d*x]^3)/(3*d) - ((I/3)*f*((-I)*Sinh[c + d*x] - (I/3)*Sinh[c + d*x]^3))/d^2)/b - (a*(((e + f*x)^2/(4*f) - (f*Cosh[c + d*x] ^2)/(4*d^2) + ((e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*d))/b - (a*(-1/2* (a*(e + f*x)^2)/(b^2*f) - (2*(a^2 + b^2)*(-1/2*(b*(((e + f*x)*Log[1 + (b*E ^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + (f*PolyLog[2, -((b*E^(c + d*x) )/(a - Sqrt[a^2 + b^2]))])/(b*d^2)))/Sqrt[a^2 + b^2] + (b*(((e + f*x)*Log[ 1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + (f*PolyLog[2, -((b*E^( c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^2)))/(2*Sqrt[a^2 + b^2])))/b^2 - ( I*((I*(e + f*x)*Cosh[c + d*x])/d - (I*f*Sinh[c + d*x])/d^2))/b))/b))/b))/b ) + (-1/16*(e + f*x)^2/f - (f*Cosh[4*c + 4*d*x])/(128*d^2) + ((e + f*x)*Si nh[4*c + 4*d*x])/(32*d))/b
3.4.98.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( -I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^m*(Cosh[a + b*x]^(n + 1)/(b*(n + 1 ))), x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. )*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-a/b^2 Int[(e + f*x)^m*Cos h[c + d*x]^(n - 2), x], x] + (Simp[1/b Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)*Sinh[c + d*x], x], x] + Simp[(a^2 + b^2)/b^2 Int[(e + f*x)^m*(Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/b Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x] - S imp[a/b Int[(e + f*x)^m*Cosh[c + d*x]^p*(Sinh[c + d*x]^(n - 1)/(a + b*Sin h[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[ n, 0] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1212\) vs. \(2(432)=864\).
Time = 76.74 (sec) , antiderivative size = 1213, normalized size of antiderivative = 2.56
-1/8*a*(4*a^2*d*f*x+b^2*d*f*x+4*a^2*d*e+b^2*d*e-4*a^2*f-b^2*f)/b^4/d^2*exp (d*x+c)+1/256*(4*d*f*x+4*d*e-f)/b/d^2*exp(4*d*x+4*c)-1/d*a^5/b^5*f/(a^2+b^ 2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/d* a^5/b^5*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2) ^(1/2)))*x-1/d^2*a^5/b^5*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/ 2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/d^2*a^5/b^5*f/(a^2+b^2)^(1/2)*ln((b*exp(d* x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/d*a^3/b^3*f/(a^2+b^2)^(1/ 2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/d*a^3/b^ 3*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2) ))*x-1/d^2*a^3/b^3*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/ (-a+(a^2+b^2)^(1/2)))*c+1/4*a^2*f*x^2/b^3+1/16*a^2*(2*d*f*x+2*d*e-f)/b^3/d ^2*exp(2*d*x+2*c)-1/16*a^2*(2*d*f*x+2*d*e+f)/b^3/d^2*exp(-2*d*x-2*c)+2/d*a ^3/b^3*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2)) -1/d^2*a^5/b^5*f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/( -a+(a^2+b^2)^(1/2)))+1/d^2*a^5/b^5*f/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+( a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/d^2*a^3/b^3*f/(a^2+b^2)^(1/2)*dil og((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/d^2*a^3/b^3*f /(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2) ))-1/16*f*x^2/b-1/256*(4*d*f*x+4*d*e+f)/b/d^2*exp(-4*d*x-4*c)+1/2*a^4*f*x^ 2/b^5-1/8*e*x/b+1/2*a^2*e*x/b^3+1/d^2*a^3/b^3*f/(a^2+b^2)^(1/2)*ln((b*e...
Leaf count of result is larger than twice the leaf count of optimal. 3228 vs. \(2 (430) = 860\).
Time = 0.33 (sec) , antiderivative size = 3228, normalized size of antiderivative = 6.81 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
1/2304*(9*(4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*cosh(d*x + c)^8 + 9*(4*b^4*d*f *x + 4*b^4*d*e - b^4*f)*sinh(d*x + c)^8 - 32*(3*a*b^3*d*f*x + 3*a*b^3*d*e - a*b^3*f)*cosh(d*x + c)^7 - 8*(12*a*b^3*d*f*x + 12*a*b^3*d*e - 4*a*b^3*f - 9*(4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*cosh(d*x + c))*sinh(d*x + c)^7 - 36* b^4*d*f*x + 144*(2*a^2*b^2*d*f*x + 2*a^2*b^2*d*e - a^2*b^2*f)*cosh(d*x + c )^6 + 4*(72*a^2*b^2*d*f*x + 72*a^2*b^2*d*e - 36*a^2*b^2*f + 63*(4*b^4*d*f* x + 4*b^4*d*e - b^4*f)*cosh(d*x + c)^2 - 56*(3*a*b^3*d*f*x + 3*a*b^3*d*e - a*b^3*f)*cosh(d*x + c))*sinh(d*x + c)^6 - 36*b^4*d*e - 288*((4*a^3*b + a* b^3)*d*f*x + (4*a^3*b + a*b^3)*d*e - (4*a^3*b + a*b^3)*f)*cosh(d*x + c)^5 - 24*(12*(4*a^3*b + a*b^3)*d*f*x - 21*(4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*co sh(d*x + c)^3 + 12*(4*a^3*b + a*b^3)*d*e + 28*(3*a*b^3*d*f*x + 3*a*b^3*d*e - a*b^3*f)*cosh(d*x + c)^2 - 12*(4*a^3*b + a*b^3)*f - 36*(2*a^2*b^2*d*f*x + 2*a^2*b^2*d*e - a^2*b^2*f)*cosh(d*x + c))*sinh(d*x + c)^5 - 9*b^4*f + 1 44*((8*a^4 + 4*a^2*b^2 - b^4)*d^2*f*x^2 + 2*(8*a^4 + 4*a^2*b^2 - b^4)*d^2* e*x)*cosh(d*x + c)^4 + 2*(72*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*f*x^2 + 144*(8* a^4 + 4*a^2*b^2 - b^4)*d^2*e*x + 315*(4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*cos h(d*x + c)^4 - 560*(3*a*b^3*d*f*x + 3*a*b^3*d*e - a*b^3*f)*cosh(d*x + c)^3 + 1080*(2*a^2*b^2*d*f*x + 2*a^2*b^2*d*e - a^2*b^2*f)*cosh(d*x + c)^2 - 72 0*((4*a^3*b + a*b^3)*d*f*x + (4*a^3*b + a*b^3)*d*e - (4*a^3*b + a*b^3)*f)* cosh(d*x + c))*sinh(d*x + c)^4 - 288*((4*a^3*b + a*b^3)*d*f*x + (4*a^3*...
Timed out. \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-1/2304*(4608*(a^5*e^c + a^3*b^2*e^c)*integrate(x*e^(d*x)/(b^6*e^(2*d*x + 2*c) + 2*a*b^5*e^(d*x + c) - b^6), x) - (144*(8*a^4*d^2*e^(4*c) + 4*a^2*b^ 2*d^2*e^(4*c) - b^4*d^2*e^(4*c))*x^2 + 9*(4*b^4*d*x*e^(8*c) - b^4*e^(8*c)) *e^(4*d*x) - 32*(3*a*b^3*d*x*e^(7*c) - a*b^3*e^(7*c))*e^(3*d*x) + 144*(2*a ^2*b^2*d*x*e^(6*c) - a^2*b^2*e^(6*c))*e^(2*d*x) + 288*(4*a^3*b*e^(5*c) + a *b^3*e^(5*c) - (4*a^3*b*d*e^(5*c) + a*b^3*d*e^(5*c))*x)*e^(d*x) - 288*(4*a ^3*b*e^(3*c) + a*b^3*e^(3*c) + (4*a^3*b*d*e^(3*c) + a*b^3*d*e^(3*c))*x)*e^ (-d*x) - 144*(2*a^2*b^2*d*x*e^(2*c) + a^2*b^2*e^(2*c))*e^(-2*d*x) - 32*(3* a*b^3*d*x*e^c + a*b^3*e^c)*e^(-3*d*x) - 9*(4*b^4*d*x + b^4)*e^(-4*d*x))*e^ (-4*c)/(b^5*d^2))*f - 1/192*e*(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))
\[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]