Integrand size = 36, antiderivative size = 883 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {3 a f (e+f x)^2}{8 b^2 d^2}-\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {a (e+f x)^4}{8 b^2 f}+\frac {6 a^2 f^2 (e+f x) \cosh (c+d x)}{b^3 d^3}+\frac {4 f^2 (e+f x) \cosh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \cosh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh ^2(c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^2 \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {3 a^2 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {3 a^2 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {6 a^2 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {6 a^2 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {6 a^2 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^4}-\frac {6 a^2 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^4}-\frac {6 a^2 f^3 \sinh (c+d x)}{b^3 d^4}-\frac {14 f^3 \sinh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \sinh (c+d x)}{b^3 d^2}-\frac {2 f (e+f x)^2 \sinh (c+d x)}{3 b d^2}-\frac {3 a f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d^2}-\frac {2 f^3 \sinh ^3(c+d x)}{27 b d^4} \] Output:
-3*a^2*(a^2+b^2)^(1/2)*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1 /2)))/b^4/d^2+3*a^2*(a^2+b^2)^(1/2)*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a -(a^2+b^2)^(1/2)))/b^4/d^2+4/3*f^2*(f*x+e)*cosh(d*x+c)/b/d^3-2/3*f*(f*x+e) ^2*sinh(d*x+c)/b/d^2-1/8*a*(f*x+e)^4/b^2/f-14/9*f^3*sinh(d*x+c)/b/d^4+6*a^ 2*(a^2+b^2)^(1/2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2))) /b^4/d^3-6*a^2*(a^2+b^2)^(1/2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2 +b^2)^(1/2)))/b^4/d^3-6*a^2*f^3*sinh(d*x+c)/b^3/d^4+3/8*a*f^3*cosh(d*x+c)^ 2/b^2/d^4+2/9*f^2*(f*x+e)*cosh(d*x+c)^3/b/d^3-3/8*a*f*(f*x+e)^2/b^2/d^2+a^ 2*(f*x+e)^3*cosh(d*x+c)/b^3/d-1/4*a^3*(f*x+e)^4/b^4/f-2/27*f^3*sinh(d*x+c) ^3/b/d^4+1/3*(f*x+e)^3*cosh(d*x+c)^3/b/d-3/4*a*f^2*(f*x+e)*cosh(d*x+c)*sin h(d*x+c)/b^2/d^3-3*a^2*f*(f*x+e)^2*sinh(d*x+c)/b^3/d^2+6*a^2*f^2*(f*x+e)*c osh(d*x+c)/b^3/d^3+3/4*a*f*(f*x+e)^2*cosh(d*x+c)^2/b^2/d^2-1/2*a*(f*x+e)^3 *cosh(d*x+c)*sinh(d*x+c)/b^2/d-1/3*f*(f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/b /d^2-a^2*(a^2+b^2)^(1/2)*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/ b^4/d+a^2*(a^2+b^2)^(1/2)*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2))) /b^4/d-6*a^2*(a^2+b^2)^(1/2)*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2 )))/b^4/d^4+6*a^2*(a^2+b^2)^(1/2)*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2) ^(1/2)))/b^4/d^4
Time = 4.14 (sec) , antiderivative size = 1667, normalized size of antiderivative = 1.89 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^3*Cosh[c + d*x]^2*Sinh[c + d*x]^2)/(a + b*Sinh[c + d* x]),x]
Output:
-1/432*(432*a^3*d^4*e^3*x + 216*a*b^2*d^4*e^3*x + 648*a^3*d^4*e^2*f*x^2 + 324*a*b^2*d^4*e^2*f*x^2 + 432*a^3*d^4*e*f^2*x^3 + 216*a*b^2*d^4*e*f^2*x^3 + 108*a^3*d^4*f^3*x^4 + 54*a*b^2*d^4*f^3*x^4 + 864*a^2*Sqrt[a^2 + b^2]*d^3 *e^3*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 432*a^2*b*d^3*e^3*Cosh [c + d*x] - 108*b^3*d^3*e^3*Cosh[c + d*x] - 2592*a^2*b*d*e*f^2*Cosh[c + d* x] - 648*b^3*d*e*f^2*Cosh[c + d*x] - 1296*a^2*b*d^3*e^2*f*x*Cosh[c + d*x] - 324*b^3*d^3*e^2*f*x*Cosh[c + d*x] - 2592*a^2*b*d*f^3*x*Cosh[c + d*x] - 6 48*b^3*d*f^3*x*Cosh[c + d*x] - 1296*a^2*b*d^3*e*f^2*x^2*Cosh[c + d*x] - 32 4*b^3*d^3*e*f^2*x^2*Cosh[c + d*x] - 432*a^2*b*d^3*f^3*x^3*Cosh[c + d*x] - 108*b^3*d^3*f^3*x^3*Cosh[c + d*x] - 162*a*b^2*d^2*e^2*f*Cosh[2*(c + d*x)] - 81*a*b^2*f^3*Cosh[2*(c + d*x)] - 324*a*b^2*d^2*e*f^2*x*Cosh[2*(c + d*x)] - 162*a*b^2*d^2*f^3*x^2*Cosh[2*(c + d*x)] - 36*b^3*d^3*e^3*Cosh[3*(c + d* x)] - 24*b^3*d*e*f^2*Cosh[3*(c + d*x)] - 108*b^3*d^3*e^2*f*x*Cosh[3*(c + d *x)] - 24*b^3*d*f^3*x*Cosh[3*(c + d*x)] - 108*b^3*d^3*e*f^2*x^2*Cosh[3*(c + d*x)] - 36*b^3*d^3*f^3*x^3*Cosh[3*(c + d*x)] - 1296*a^2*Sqrt[a^2 + b^2]* d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 1296*a^2*Sqrt [a^2 + b^2]*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 432*a^2*Sqrt[a^2 + b^2]*d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 1296*a^2*Sqrt[a^2 + b^2]*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/( a + Sqrt[a^2 + b^2])] + 1296*a^2*Sqrt[a^2 + b^2]*d^3*e*f^2*x^2*Log[1 + ...
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.
\(\displaystyle \int \frac {(e+f x)^3 \sinh ^2(c+d x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
\(\Big \downarrow \) 6113 |
\(\displaystyle \frac {\int (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
\(\Big \downarrow \) 5970 |
\(\displaystyle \frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \int (e+f x)^2 \cosh ^3(c+d x)dx}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{d}}{b}\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle \frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 f^2 \int \cosh ^3(c+d x)dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \cosh (c+d x)dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 f^2 \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \int \left (\sinh ^2(c+d x)+1\right )d(-i \sinh (c+d x))}{9 d^3}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 6113 |
\(\displaystyle -\frac {a \left (\frac {\int (e+f x)^3 \cosh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{b}\right )}{b}\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle -\frac {a \left (\frac {\frac {3 f^2 \int (e+f x) \cosh ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^3dx-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle -\frac {a \left (\frac {\frac {3 f^2 \int (e+f x) \cosh ^2(c+d x)dx}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {3 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}\right )}{b}\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle -\frac {a \left (\frac {\frac {3 f^2 \left (\frac {1}{2} \int (e+f x)dx-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle -\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 6099 |
\(\displaystyle -\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^3dx}{b^2}+\frac {\int (e+f x)^3 \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle -\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^3 \sinh (c+d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}+\frac {\int -i (e+f x)^3 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \int (e+f x)^3 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \cosh (c+d x)dx}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\right )}{b}\) |
Input:
Int[((e + f*x)^3*Cosh[c + d*x]^2*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 7042 vs. \(2 (813) = 1626\).
Time = 0.23 (sec) , antiderivative size = 7042, normalized size of antiderivative = 7.98 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algor ithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**3*cosh(d*x+c)**2*sinh(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algor ithm="maxima")
Output:
1/24*e^3*(24*sqrt(a^2 + b^2)*a^2*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2) )/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(b^4*d) - (3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 + b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) - 12*(2*a^ 3 + a*b^2)*(d*x + c)/(b^4*d) + (3*a*b*e^(-2*d*x - 2*c) + b^2*e^(-3*d*x - 3 *c) + 3*(4*a^2 + b^2)*e^(-d*x - c))/(b^3*d)) - 1/864*(108*(2*a^3*d^4*f^3*e ^(3*c) + a*b^2*d^4*f^3*e^(3*c))*x^4 + 432*(2*a^3*d^4*e*f^2*e^(3*c) + a*b^2 *d^4*e*f^2*e^(3*c))*x^3 + 648*(2*a^3*d^4*e^2*f*e^(3*c) + a*b^2*d^4*e^2*f*e ^(3*c))*x^2 - 4*(9*b^3*d^3*f^3*x^3*e^(6*c) + 9*(3*d^3*e*f^2 - d^2*f^3)*b^3 *x^2*e^(6*c) + 3*(9*d^3*e^2*f - 6*d^2*e*f^2 + 2*d*f^3)*b^3*x*e^(6*c) - (9* d^2*e^2*f - 6*d*e*f^2 + 2*f^3)*b^3*e^(6*c))*e^(3*d*x) + 27*(4*a*b^2*d^3*f^ 3*x^3*e^(5*c) + 6*(2*d^3*e*f^2 - d^2*f^3)*a*b^2*x^2*e^(5*c) + 6*(2*d^3*e^2 *f - 2*d^2*e*f^2 + d*f^3)*a*b^2*x*e^(5*c) - 3*(2*d^2*e^2*f - 2*d*e*f^2 + f ^3)*a*b^2*e^(5*c))*e^(2*d*x) + 108*(12*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a^2 *b*e^(4*c) + 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*b^3*e^(4*c) - (4*a^2*b*d^3* f^3*e^(4*c) + b^3*d^3*f^3*e^(4*c))*x^3 - 3*(4*(d^3*e*f^2 - d^2*f^3)*a^2*b* e^(4*c) + (d^3*e*f^2 - d^2*f^3)*b^3*e^(4*c))*x^2 - 3*(4*(d^3*e^2*f - 2*d^2 *e*f^2 + 2*d*f^3)*a^2*b*e^(4*c) + (d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*b^3* e^(4*c))*x)*e^(d*x) - 108*(12*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*a^2*b*e^(2*c ) + 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*b^3*e^(2*c) + (4*a^2*b*d^3*f^3*e^(2* c) + b^3*d^3*f^3*e^(2*c))*x^3 + 3*(4*(d^3*e*f^2 + d^2*f^3)*a^2*b*e^(2*c...
\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algor ithm="giac")
Output:
integrate((f*x + e)^3*cosh(d*x + c)^2*sinh(d*x + c)^2/(b*sinh(d*x + c) + a ), x)
Timed out. \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((cosh(c + d*x)^2*sinh(c + d*x)^2*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)^2*sinh(c + d*x)^2*(e + f*x)^3)/(a + b*sinh(c + d*x)), x )
\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x \] Input:
int((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)