Integrand size = 26, antiderivative size = 400 \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {b f x}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^2}{2 a^4 f}-\frac {2 f \cosh (c+d x)}{3 a d^2}-\frac {b^2 f \cosh (c+d x)}{a^3 d^2}-\frac {f \cosh ^3(c+d x)}{9 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {b \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {2 (e+f x) \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x) \sinh (c+d x)}{a^3 d}+\frac {b f \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b (e+f x) \sinh ^2(c+d x)}{2 a^2 d} \]
-1/4*b*f*x/a^2/d+1/2*b*(a^2+b^2)*(f*x+e)^2/a^4/f-2/3*f*cosh(d*x+c)/a/d^2-b ^2*f*cosh(d*x+c)/a^3/d^2-1/9*f*cosh(d*x+c)^3/a/d^2-b*(a^2+b^2)*(f*x+e)*ln( 1+a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d-b*(a^2+b^2)*(f*x+e)*ln(1+a*exp(d *x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d-b*(a^2+b^2)*f*polylog(2,-a*exp(d*x+c)/(b- (a^2+b^2)^(1/2)))/a^4/d^2-b*(a^2+b^2)*f*polylog(2,-a*exp(d*x+c)/(b+(a^2+b^ 2)^(1/2)))/a^4/d^2+2/3*(f*x+e)*sinh(d*x+c)/a/d+b^2*(f*x+e)*sinh(d*x+c)/a^3 /d+1/4*b*f*cosh(d*x+c)*sinh(d*x+c)/a^2/d^2+1/3*(f*x+e)*cosh(d*x+c)^2*sinh( d*x+c)/a/d-1/2*b*(f*x+e)*sinh(d*x+c)^2/a^2/d
Time = 1.18 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.74 \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {36 a^2 b c^2 f+36 b^3 c^2 f-36 a^2 b d^2 f x^2-36 b^3 d^2 f x^2+54 a^3 f \cosh (c+d x)+72 a b^2 f \cosh (c+d x)+18 a^2 b d f x \cosh (2 (c+d x))+2 a^3 f \cosh (3 (c+d x))+72 a^2 b c f \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+72 b^3 c f \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+72 a^2 b d f x \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+72 b^3 d f x \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+72 a^2 b c f \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+72 b^3 c f \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+72 a^2 b d f x \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+72 b^3 d f x \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )-72 a^2 b c f \log \left (a-2 b e^{c+d x}-a e^{2 (c+d x)}\right )-72 b^3 c f \log \left (a-2 b e^{c+d x}-a e^{2 (c+d x)}\right )+72 a^2 b d e \log (b+a \sinh (c+d x))+72 b^3 d e \log (b+a \sinh (c+d x))+72 b \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )+72 b \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )-72 a^3 d e \sinh (c+d x)-72 a b^2 d e \sinh (c+d x)-54 a^3 d f x \sinh (c+d x)-72 a b^2 d f x \sinh (c+d x)+36 a^2 b d e \sinh ^2(c+d x)-24 a^3 d e \sinh ^3(c+d x)-9 a^2 b f \sinh (2 (c+d x))-6 a^3 d f x \sinh (3 (c+d x))}{72 a^4 d^2} \]
-1/72*(36*a^2*b*c^2*f + 36*b^3*c^2*f - 36*a^2*b*d^2*f*x^2 - 36*b^3*d^2*f*x ^2 + 54*a^3*f*Cosh[c + d*x] + 72*a*b^2*f*Cosh[c + d*x] + 18*a^2*b*d*f*x*Co sh[2*(c + d*x)] + 2*a^3*f*Cosh[3*(c + d*x)] + 72*a^2*b*c*f*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 72*b^3*c*f*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 72*a^2*b*d*f*x*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 72*b^3*d*f*x*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 72 *a^2*b*c*f*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 72*b^3*c*f*Log [1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 72*a^2*b*d*f*x*Log[1 + (a*E^ (c + d*x))/(b + Sqrt[a^2 + b^2])] + 72*b^3*d*f*x*Log[1 + (a*E^(c + d*x))/( b + Sqrt[a^2 + b^2])] - 72*a^2*b*c*f*Log[a - 2*b*E^(c + d*x) - a*E^(2*(c + d*x))] - 72*b^3*c*f*Log[a - 2*b*E^(c + d*x) - a*E^(2*(c + d*x))] + 72*a^2 *b*d*e*Log[b + a*Sinh[c + d*x]] + 72*b^3*d*e*Log[b + a*Sinh[c + d*x]] + 72 *b*(a^2 + b^2)*f*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] + 72*b *(a^2 + b^2)*f*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 72*a ^3*d*e*Sinh[c + d*x] - 72*a*b^2*d*e*Sinh[c + d*x] - 54*a^3*d*f*x*Sinh[c + d*x] - 72*a*b^2*d*f*x*Sinh[c + d*x] + 36*a^2*b*d*e*Sinh[c + d*x]^2 - 24*a^ 3*d*e*Sinh[c + d*x]^3 - 9*a^2*b*f*Sinh[2*(c + d*x)] - 6*a^3*d*f*x*Sinh[3*( c + d*x)])/(a^4*d^2)
Time = 2.30 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.90, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6128, 6113, 3042, 3791, 3042, 3777, 26, 3042, 26, 3118, 6099, 3042, 3777, 26, 3042, 26, 3118, 5969, 3042, 25, 3115, 24, 6095, 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.
| \(\displaystyle \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 6128 |
| \(\displaystyle \int \frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{a \sinh (c+d x)+b}dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x) \cosh ^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{a}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {\frac {2}{3} \int (e+f x) \cosh (c+d x)dx-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2}{3} \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \int (e+f x) \cosh (c+d x)dx}{a^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{a^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{a^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{a^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{a^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{a^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{a}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{a^2}\right )}{a}\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}-\frac {f \int \sinh ^2(c+d x)dx}{2 d}}{a}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{a^2}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}-\frac {f \int -\sin (i c+i d x)^2dx}{2 d}}{a}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{a^2}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \int \sin (i c+i d x)^2dx}{2 d}}{a}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{a^2}\right )}{a}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}+\frac {\frac {f \left (\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{a^2}\right )}{a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{a^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \left (\int \frac {e^{c+d x} (e+f x)}{e^{c+d x} a+b-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)}{e^{c+d x} a+b+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^2}{2 a f}\right )}{a^2}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{a^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \left (-\frac {f \int \log \left (\frac {e^{c+d x} a}{b-\sqrt {a^2+b^2}}+1\right )dx}{a d}-\frac {f \int \log \left (\frac {e^{c+d x} a}{b+\sqrt {a^2+b^2}}+1\right )dx}{a d}+\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}+\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {(e+f x)^2}{2 a f}\right )}{a^2}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{a^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \left (-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} a}{b-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{a d^2}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} a}{b+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{a d^2}+\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}+\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {(e+f x)^2}{2 a f}\right )}{a^2}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{a^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )-\frac {f \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}+\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {(e+f x)^2}{2 a f}\right )}{a^2}-\frac {b \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{a^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\right )}{a}\) |
(-1/9*(f*Cosh[c + d*x]^3)/d^2 + ((e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/ (3*d) + (2*(-((f*Cosh[c + d*x])/d^2) + ((e + f*x)*Sinh[c + d*x])/d))/3)/a - (b*(((a^2 + b^2)*(-1/2*(e + f*x)^2/(a*f) + ((e + f*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a*d) + ((e + f*x)*Log[1 + (a*E^(c + d*x))/( b + Sqrt[a^2 + b^2])])/(a*d) + (f*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a ^2 + b^2]))])/(a*d^2) + (f*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^ 2]))])/(a*d^2)))/a^2 - (b*(-((f*Cosh[c + d*x])/d^2) + ((e + f*x)*Sinh[c + d*x])/d))/a^2 + (((e + f*x)*Sinh[c + d*x]^2)/(2*d) + (f*(x/2 - (Cosh[c + d *x]*Sinh[c + d*x])/(2*d)))/(2*d))/a))/a
3.1.28.3.1 Defintions of rubi rules used
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[(((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[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[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)* (x_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sinh[a + b*x]^(n + 1)/(b*(n + 1 ))), x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 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]
Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Int[(e + f*x)^m*Sinh[c + d*x]*(F [c + d*x]^n/(b + a*Sinh[c + d*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && H yperbolicQ[F] && IntegersQ[m, n]
Leaf count of result is larger than twice the leaf count of optimal. \(1101\) vs. \(2(372)=744\).
Time = 17.62 (sec) , antiderivative size = 1102, normalized size of antiderivative = 2.76
-2/d^2*b^3/a^4*c*f*ln(exp(d*x+c))+1/d^2*b^3/a^4*c*f*ln(exp(2*d*x+2*c)*a+2* exp(d*x+c)*b-a)+1/d^2*b^3/a^4*f*c^2-1/d*b^3/a^4*e*ln(exp(2*d*x+2*c)*a+2*ex p(d*x+c)*b-a)+2/d*b^3/a^4*e*ln(exp(d*x+c))-1/d^2*b^3/a^4*f*dilog((-a*exp(d *x+c)+(a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))-1/d^2*b^3/a^4*f*dilog((a*ex p(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))+1/2/a^2*b*f*x^2-1/a^2*b*e *x-1/72*(3*d*f*x+3*d*e+f)/a/d^2*exp(-3*d*x-3*c)-1/16*b*(2*d*f*x+2*d*e-f)/a ^2/d^2*exp(2*d*x+2*c)-1/8*(3*a^2+4*b^2)*(d*f*x+d*e+f)/a^3/d^2*exp(-d*x-c)- 1/16*b*(2*d*f*x+2*d*e+f)/a^2/d^2*exp(-2*d*x-2*c)+2/d*b^3/a^4*f*c*x-1/d*b^3 /a^4*f*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))*x-1/d*b^ 3/a^4*f*ln((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))*x-1/d^2*b ^3/a^4*f*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))*c-1/d^ 2*b^3/a^4*f*ln((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))*c+1/7 2*(3*d*f*x+3*d*e-f)/a/d^2*exp(3*d*x+3*c)+1/2*b^3/a^4*f*x^2-b^3/a^4*e*x+1/8 *(3*a^2*d*f*x+4*b^2*d*f*x+3*a^2*d*e+4*b^2*d*e-3*a^2*f-4*b^2*f)/a^3/d^2*exp (d*x+c)+2/d/a^2*b*f*c*x-1/d/a^2*b*f*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/( -b+(a^2+b^2)^(1/2)))*x-1/d/a^2*b*f*ln((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+ (a^2+b^2)^(1/2)))*x-1/d^2/a^2*b*f*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b +(a^2+b^2)^(1/2)))*c-1/d^2/a^2*b*f*ln((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+ (a^2+b^2)^(1/2)))*c+1/d^2/a^2*b*c*f*ln(exp(2*d*x+2*c)*a+2*exp(d*x+c)*b-a)- 2/d^2/a^2*b*c*f*ln(exp(d*x+c))+1/d^2/a^2*b*f*c^2-1/d^2/a^2*b*f*dilog((-...
Leaf count of result is larger than twice the leaf count of optimal. 2465 vs. \(2 (370) = 740\).
Time = 0.31 (sec) , antiderivative size = 2465, normalized size of antiderivative = 6.16 \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \]
1/144*(2*(3*a^3*d*f*x + 3*a^3*d*e - a^3*f)*cosh(d*x + c)^6 + 2*(3*a^3*d*f* x + 3*a^3*d*e - a^3*f)*sinh(d*x + c)^6 - 6*a^3*d*f*x - 9*(2*a^2*b*d*f*x + 2*a^2*b*d*e - a^2*b*f)*cosh(d*x + c)^5 - 3*(6*a^2*b*d*f*x + 6*a^2*b*d*e - 3*a^2*b*f - 4*(3*a^3*d*f*x + 3*a^3*d*e - a^3*f)*cosh(d*x + c))*sinh(d*x + c)^5 - 6*a^3*d*e + 18*((3*a^3 + 4*a*b^2)*d*f*x + (3*a^3 + 4*a*b^2)*d*e - ( 3*a^3 + 4*a*b^2)*f)*cosh(d*x + c)^4 + 3*(6*(3*a^3 + 4*a*b^2)*d*f*x + 6*(3* a^3 + 4*a*b^2)*d*e + 10*(3*a^3*d*f*x + 3*a^3*d*e - a^3*f)*cosh(d*x + c)^2 - 6*(3*a^3 + 4*a*b^2)*f - 15*(2*a^2*b*d*f*x + 2*a^2*b*d*e - a^2*b*f)*cosh( d*x + c))*sinh(d*x + c)^4 - 2*a^3*f + 72*((a^2*b + b^3)*d^2*f*x^2 + 2*(a^2 *b + b^3)*d^2*e*x + 4*(a^2*b + b^3)*c*d*e - 2*(a^2*b + b^3)*c^2*f)*cosh(d* x + c)^3 + 2*(36*(a^2*b + b^3)*d^2*f*x^2 + 72*(a^2*b + b^3)*d^2*e*x + 144* (a^2*b + b^3)*c*d*e - 72*(a^2*b + b^3)*c^2*f + 20*(3*a^3*d*f*x + 3*a^3*d*e - a^3*f)*cosh(d*x + c)^3 - 45*(2*a^2*b*d*f*x + 2*a^2*b*d*e - a^2*b*f)*cos h(d*x + c)^2 + 36*((3*a^3 + 4*a*b^2)*d*f*x + (3*a^3 + 4*a*b^2)*d*e - (3*a^ 3 + 4*a*b^2)*f)*cosh(d*x + c))*sinh(d*x + c)^3 - 18*((3*a^3 + 4*a*b^2)*d*f *x + (3*a^3 + 4*a*b^2)*d*e + (3*a^3 + 4*a*b^2)*f)*cosh(d*x + c)^2 + 6*(5*( 3*a^3*d*f*x + 3*a^3*d*e - a^3*f)*cosh(d*x + c)^4 - 3*(3*a^3 + 4*a*b^2)*d*f *x - 15*(2*a^2*b*d*f*x + 2*a^2*b*d*e - a^2*b*f)*cosh(d*x + c)^3 - 3*(3*a^3 + 4*a*b^2)*d*e + 18*((3*a^3 + 4*a*b^2)*d*f*x + (3*a^3 + 4*a*b^2)*d*e - (3 *a^3 + 4*a*b^2)*f)*cosh(d*x + c)^2 - 3*(3*a^3 + 4*a*b^2)*f + 36*((a^2*b...
\[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cosh ^{3}{\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{3}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]
-1/24*e*((3*a*b*e^(-d*x - c) - a^2 - 3*(3*a^2 + 4*b^2)*e^(-2*d*x - 2*c))*e ^(3*d*x + 3*c)/(a^3*d) + 24*(a^2*b + b^3)*(d*x + c)/(a^4*d) + (3*a*b*e^(-2 *d*x - 2*c) + a^2*e^(-3*d*x - 3*c) + 3*(3*a^2 + 4*b^2)*e^(-d*x - c))/(a^3* d) + 24*(a^2*b + b^3)*log(-2*b*e^(-d*x - c) + a*e^(-2*d*x - 2*c) - a)/(a^4 *d)) - 1/144*f*((72*(a^2*b*d^2*e^(3*c) + b^3*d^2*e^(3*c))*x^2 - 2*(3*a^3*d *x*e^(6*c) - a^3*e^(6*c))*e^(3*d*x) + 9*(2*a^2*b*d*x*e^(5*c) - a^2*b*e^(5* c))*e^(2*d*x) + 18*(3*a^3*e^(4*c) + 4*a*b^2*e^(4*c) - (3*a^3*d*e^(4*c) + 4 *a*b^2*d*e^(4*c))*x)*e^(d*x) + 18*(3*a^3*e^(2*c) + 4*a*b^2*e^(2*c) + (3*a^ 3*d*e^(2*c) + 4*a*b^2*d*e^(2*c))*x)*e^(-d*x) + 9*(2*a^2*b*d*x*e^c + a^2*b* e^c)*e^(-2*d*x) + 2*(3*a^3*d*x + a^3)*e^(-3*d*x))*e^(-3*c)/(a^4*d^2) - 18* integrate(16*((a^2*b^2*e^c + b^4*e^c)*x*e^(d*x) - (a^3*b + a*b^3)*x)/(a^5* e^(2*d*x + 2*c) + 2*a^4*b*e^(d*x + c) - a^5), x))
\[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{3}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]