Integrand size = 32, antiderivative size = 448 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a (e+f x)^4}{4 b^2 f}-\frac {6 f^3 \cosh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {a (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {3 a f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {3 a f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}+\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^3}-\frac {6 a f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^4}-\frac {6 a f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d} \]
1/4*a*(f*x+e)^4/b^2/f-6*f^3*cosh(d*x+c)/b/d^4-3*f*(f*x+e)^2*cosh(d*x+c)/b/ d^2-a*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/d-a*(f*x+e)^3*l n(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/d-3*a*f*(f*x+e)^2*polylog(2,-b*e xp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/d^2-3*a*f*(f*x+e)^2*polylog(2,-b*exp(d* x+c)/(a+(a^2+b^2)^(1/2)))/b^2/d^2+6*a*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/ (a-(a^2+b^2)^(1/2)))/b^2/d^3+6*a*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a ^2+b^2)^(1/2)))/b^2/d^3-6*a*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2) ))/b^2/d^4-6*a*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/d^4+6* f^2*(f*x+e)*sinh(d*x+c)/b/d^3+(f*x+e)^3*sinh(d*x+c)/b/d
Leaf count is larger than twice the leaf count of optimal. \(2809\) vs. \(2(448)=896\).
Time = 9.73 (sec) , antiderivative size = 2809, normalized size of antiderivative = 6.27 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]
((a*(4*e^3*E^(2*c)*x + 6*e^2*E^(2*c)*f*x^2 + 4*e*E^(2*c)*f^2*x^3 + E^(2*c) *f^3*x^4 + (4*a*Sqrt[a^2 + b^2]*e^3*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) + (4*a*Sqrt[-a^2 - b^2]*e^3*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) - (2*e^3*E^(2*c) *Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))])/d + (2*e^3*Log[2*a*E^(c + d *x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/ (a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e^2*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 + (6*e*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*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 + (2*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)] )])/d - (2*E^(2*c)*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a ^2 + b^2)*E^(2*c)])])/d - (6*e^2*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 + (6*e*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*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 + (2*f^3*x^3*Lo g[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (2*E^(2* c)*f^3*x^3*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)^2*PolyLog[2, -((b*E^(2*c + d*x))/(a*...
Result contains complex when optimal does not.
Time = 2.28 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.01, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.594, Rules used = {6113, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 6095, 2620, 3011, 7163, 2720, 7143}
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.
\(\displaystyle \int \frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx\) |
\(\Big \downarrow \) 6113 |
\(\displaystyle \frac {\int (e+f x)^3 \cosh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 i f \int -i (e+f x)^2 \sinh (c+d x)dx}{d}}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 f \int (e+f x)^2 \sinh (c+d x)dx}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 f \int -i (e+f x)^2 \sin (i c+i d x)dx}{d}}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \int (e+f x)^2 \sin (i c+i d x)dx}{d}}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 6095 |
\(\displaystyle -\frac {a \left (\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^4}{4 b f}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {a \left (-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {a \left (-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle -\frac {a \left (-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {a \left (-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {a \left (-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
-((a*(-1/4*(e + f*x)^4/(b*f) + ((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - S qrt[a^2 + b^2])])/(b*d) + ((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a ^2 + b^2])])/(b*d) - (3*f*(-(((e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/d) + (2*f*(((e + f*x)*PolyLog[3, -((b*E^(c + d*x))/( a - Sqrt[a^2 + b^2]))])/d - (f*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/d^2))/d))/(b*d) - (3*f*(-(((e + f*x)^2*PolyLog[2, -((b*E^(c + d *x))/(a + Sqrt[a^2 + b^2]))])/d) + (2*f*(((e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/d - (f*PolyLog[4, -((b*E^(c + d*x))/(a + S qrt[a^2 + b^2]))])/d^2))/d))/(b*d)))/b) + (((e + f*x)^3*Sinh[c + d*x])/d + ((3*I)*f*((I*(e + f*x)^2*Cosh[c + d*x])/d - ((2*I)*f*(-((f*Cosh[c + d*x]) /d^2) + ((e + f*x)*Sinh[c + d*x])/d))/d))/d)/b
3.4.33.3.1 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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
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[(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_)]^(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[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1976 vs. \(2 (420) = 840\).
Time = 0.29 (sec) , antiderivative size = 1976, normalized size of antiderivative = 4.41 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
-1/4*(2*b*d^3*f^3*x^3 + 2*b*d^3*e^3 + 6*b*d^2*e^2*f + 12*b*d*e*f^2 + 12*b* f^3 + 6*(b*d^3*e*f^2 + b*d^2*f^3)*x^2 - 2*(b*d^3*f^3*x^3 + b*d^3*e^3 - 3*b *d^2*e^2*f + 6*b*d*e*f^2 - 6*b*f^3 + 3*(b*d^3*e*f^2 - b*d^2*f^3)*x^2 + 3*( b*d^3*e^2*f - 2*b*d^2*e*f^2 + 2*b*d*f^3)*x)*cosh(d*x + c)^2 - 2*(b*d^3*f^3 *x^3 + b*d^3*e^3 - 3*b*d^2*e^2*f + 6*b*d*e*f^2 - 6*b*f^3 + 3*(b*d^3*e*f^2 - b*d^2*f^3)*x^2 + 3*(b*d^3*e^2*f - 2*b*d^2*e*f^2 + 2*b*d*f^3)*x)*sinh(d*x + c)^2 + 6*(b*d^3*e^2*f + 2*b*d^2*e*f^2 + 2*b*d*f^3)*x - (a*d^4*f^3*x^4 + 4*a*d^4*e*f^2*x^3 + 6*a*d^4*e^2*f*x^2 + 4*a*d^4*e^3*x + 8*a*c*d^3*e^3 - 1 2*a*c^2*d^2*e^2*f + 8*a*c^3*d*e*f^2 - 2*a*c^4*f^3)*cosh(d*x + c) + 12*((a* d^2*f^3*x^2 + 2*a*d^2*e*f^2*x + a*d^2*e^2*f)*cosh(d*x + c) + (a*d^2*f^3*x^ 2 + 2*a*d^2*e*f^2*x + a*d^2*e^2*f)*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^ 2) - b)/b + 1) + 12*((a*d^2*f^3*x^2 + 2*a*d^2*e*f^2*x + a*d^2*e^2*f)*cosh( d*x + c) + (a*d^2*f^3*x^2 + 2*a*d^2*e*f^2*x + a*d^2*e^2*f)*sinh(d*x + c))* dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 4*((a*d^3*e^3 - 3*a*c*d^2*e^2*f + 3*a*c^2*d*e*f^2 - a*c^3*f^3)*cosh(d*x + c) + (a*d^3*e^3 - 3*a*c*d^2*e^2*f + 3*a*c^2*d*e*f^2 - a*c^3*f^3)*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b *sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 4*((a*d^3*e^3 - 3*a*c* d^2*e^2*f + 3*a*c^2*d*e*f^2 - a*c^3*f^3)*cosh(d*x + c) + (a*d^3*e^3 - 3...
Timed out. \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-1/2*e^3*(2*(d*x + c)*a/(b^2*d) - e^(d*x + c)/(b*d) + e^(-d*x - c)/(b*d) + 2*a*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^2*d)) - 1/4*(a*d^4 *f^3*x^4*e^c + 4*a*d^4*e*f^2*x^3*e^c + 6*a*d^4*e^2*f*x^2*e^c - 2*(b*d^3*f^ 3*x^3*e^(2*c) + 3*(d^3*e*f^2 - d^2*f^3)*b*x^2*e^(2*c) + 3*(d^3*e^2*f - 2*d ^2*e*f^2 + 2*d*f^3)*b*x*e^(2*c) - 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*b*e^(2 *c))*e^(d*x) + 2*(b*d^3*f^3*x^3 + 3*(d^3*e*f^2 + d^2*f^3)*b*x^2 + 3*(d^3*e ^2*f + 2*d^2*e*f^2 + 2*d*f^3)*b*x + 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*b)*e ^(-d*x))*e^(-c)/(b^2*d^4) + integrate(-2*(a*b*f^3*x^3 + 3*a*b*e*f^2*x^2 + 3*a*b*e^2*f*x - (a^2*f^3*x^3*e^c + 3*a^2*e*f^2*x^2*e^c + 3*a^2*e^2*f*x*e^c )*e^(d*x))/(b^3*e^(2*d*x + 2*c) + 2*a*b^2*e^(d*x + c) - b^3), x)
\[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]