Integrand size = 32, antiderivative size = 1218 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:
3*a^2*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2) /d^2+3*a^2*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2 +b^2)/d^2-6*a^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b /(a^2+b^2)/d^3-6*a^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2 )))/b/(a^2+b^2)/d^3-6*I*a^3*f^2*(f*x+e)*polylog(3,I*exp(d*x+c))/b^2/(a^2+b ^2)/d^3-3*I*a^3*f*(f*x+e)^2*polylog(2,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^2+(f* x+e)^3*ln(1+exp(2*d*x+2*c))/b/d+a^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^ 2)^(1/2)))/b/(a^2+b^2)/d+a^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2 )))/b/(a^2+b^2)/d+6*a^2*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b /(a^2+b^2)/d^4+6*a^2*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a ^2+b^2)/d^4-a^2*(f*x+e)^3*ln(1+exp(2*d*x+2*c))/b/(a^2+b^2)/d+3/4*f^3*polyl og(4,-exp(2*d*x+2*c))/b/d^4+6*I*a*f^3*polylog(4,-I*exp(d*x+c))/b^2/d^4-2*a *(f*x+e)^3*arctan(exp(d*x+c))/b^2/d-3/2*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2 *c))/b/d^3+3/2*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/b/d^2-1/4*(f*x+e)^4/ b/f+2*a^3*(f*x+e)^3*arctan(exp(d*x+c))/b^2/(a^2+b^2)/d-3/4*a^2*f^3*polylog (4,-exp(2*d*x+2*c))/b/(a^2+b^2)/d^4-6*I*a*f^3*polylog(4,I*exp(d*x+c))/b^2/ d^4+3/2*a^2*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/b/(a^2+b^2)/d^3-3/2*a^2 *f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/b/(a^2+b^2)/d^2-6*I*a*f^2*(f*x+e)* polylog(3,-I*exp(d*x+c))/b^2/d^3-3*I*a*f*(f*x+e)^2*polylog(2,I*exp(d*x+c)) /b^2/d^2-6*I*a^3*f^3*polylog(4,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^4+6*I*a^3...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3251\) vs. \(2(1218)=2436\).
Time = 10.61 (sec) , antiderivative size = 3251, normalized size of antiderivative = 2.67 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^3*Sinh[c + d*x]*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x]
Output:
-1/4*(8*b*d^4*e^3*E^(2*c)*x + 12*b*d^4*e^2*E^(2*c)*f*x^2 + 8*b*d^4*e*E^(2* c)*f^2*x^3 + 2*b*d^4*E^(2*c)*f^3*x^4 + 8*a*d^3*e^3*ArcTan[E^(c + d*x)] + 8 *a*d^3*e^3*E^(2*c)*ArcTan[E^(c + d*x)] + (12*I)*a*d^3*e^2*f*x*Log[1 - I*E^ (c + d*x)] + (12*I)*a*d^3*e^2*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (12*I)* a*d^3*e*f^2*x^2*Log[1 - I*E^(c + d*x)] + (12*I)*a*d^3*e*E^(2*c)*f^2*x^2*Lo g[1 - I*E^(c + d*x)] + (4*I)*a*d^3*f^3*x^3*Log[1 - I*E^(c + d*x)] + (4*I)* a*d^3*E^(2*c)*f^3*x^3*Log[1 - I*E^(c + d*x)] - (12*I)*a*d^3*e^2*f*x*Log[1 + I*E^(c + d*x)] - (12*I)*a*d^3*e^2*E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] - ( 12*I)*a*d^3*e*f^2*x^2*Log[1 + I*E^(c + d*x)] - (12*I)*a*d^3*e*E^(2*c)*f^2* x^2*Log[1 + I*E^(c + d*x)] - (4*I)*a*d^3*f^3*x^3*Log[1 + I*E^(c + d*x)] - (4*I)*a*d^3*E^(2*c)*f^3*x^3*Log[1 + I*E^(c + d*x)] - 4*b*d^3*e^3*Log[1 + E ^(2*(c + d*x))] - 4*b*d^3*e^3*E^(2*c)*Log[1 + E^(2*(c + d*x))] - 12*b*d^3* e^2*f*x*Log[1 + E^(2*(c + d*x))] - 12*b*d^3*e^2*E^(2*c)*f*x*Log[1 + E^(2*( c + d*x))] - 12*b*d^3*e*f^2*x^2*Log[1 + E^(2*(c + d*x))] - 12*b*d^3*e*E^(2 *c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] - 4*b*d^3*f^3*x^3*Log[1 + E^(2*(c + d *x))] - 4*b*d^3*E^(2*c)*f^3*x^3*Log[1 + E^(2*(c + d*x))] - (12*I)*a*d^2*(1 + E^(2*c))*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)] + (12*I)*a*d^2*(1 + E^(2*c))*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)] - 6*b*d^2*e^2*f*PolyLog[ 2, -E^(2*(c + d*x))] - 6*b*d^2*e^2*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))] - 12*b*d^2*e*f^2*x*PolyLog[2, -E^(2*(c + d*x))] - 12*b*d^2*e*E^(2*c)*f^...
Time = 5.67 (sec) , antiderivative size = 1036, normalized size of antiderivative = 0.85, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.594, Rules used = {6115, 3042, 26, 4201, 2620, 3011, 6101, 3042, 4668, 3011, 6107, 6095, 2620, 3011, 7163, 2720, 7143, 7293, 2009}
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.
\(\displaystyle \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) |
\(\Big \downarrow \) 6115 |
\(\displaystyle \frac {\int (e+f x)^3 \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int -i (e+f x)^3 \tan (i c+i d x)dx}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \int (e+f x)^3 \tan (i c+i d x)dx}{b}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \int \frac {e^{2 (c+d x)} (e+f x)^3}{1+e^{2 (c+d x)}}dx-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \int (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )dx}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 6101 |
\(\displaystyle -\frac {a \left (\frac {\int (e+f x)^3 \text {sech}(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {3 i f \int (e+f x)^2 \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {3 i f \int (e+f x)^2 \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 6107 |
\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {b^2 \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 6095 |
\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {b^2 \left (\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {b^2 \left (-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {b^2 \left (-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {b^2 \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 )}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,i e^{c+d x}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d}-\frac {f \int \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )dx}{2 d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{2 d}-\frac {3 f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d}-\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}-\frac {a \left (\frac {\frac {2 \arctan \left (e^{c+d x}\right ) (e+f x)^3}{d}+\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,i e^{c+d x}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (\frac {\left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{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 {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}\right ) b^2}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \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 )}{a^2+b^2}\right )}{b}+\frac {\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {\int \left (a (e+f x)^3 \text {sech}(c+d x)-b (e+f x)^3 \tanh (c+d x)\right )dx}{a^2+b^2}+\frac {b^2 \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 )}{a^2+b^2}\right )}{b}+\frac {\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \left (\frac {\frac {2 \arctan \left (e^{c+d x}\right ) (e+f x)^3}{d}+\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (\frac {\left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{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 {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}\right ) b^2}{a^2+b^2}+\frac {\frac {b (e+f x)^4}{4 f}+\frac {2 a \arctan \left (e^{c+d x}\right ) (e+f x)^3}{d}-\frac {b \log \left (1+e^{2 (c+d x)}\right ) (e+f x)^3}{d}-\frac {3 i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) (e+f x)^2}{d^2}+\frac {3 i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) (e+f x)^2}{d^2}-\frac {3 b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) (e+f x)^2}{2 d^2}+\frac {6 i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) (e+f x)}{d^3}-\frac {6 i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) (e+f x)}{d^3}+\frac {3 b f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) (e+f x)}{2 d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{d^4}-\frac {3 b f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 d^4}}{a^2+b^2}\right )}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{2 d}-\frac {3 f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\) |
Input:
Int[((e + f*x)^3*Sinh[c + d*x]*Tanh[c + d*x])/(a + b*Sinh[c + d*x]),x]
Output:
-((a*(((2*(e + f*x)^3*ArcTan[E^(c + d*x)])/d + ((3*I)*f*(-(((e + f*x)^2*Po lyLog[2, (-I)*E^(c + d*x)])/d) + (2*f*(((e + f*x)*PolyLog[3, (-I)*E^(c + d *x)])/d - (f*PolyLog[4, (-I)*E^(c + d*x)])/d^2))/d))/d - ((3*I)*f*(-(((e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/d) + (2*f*(((e + f*x)*PolyLog[3, I*E^(c + d*x)])/d - (f*PolyLog[4, I*E^(c + d*x)])/d^2))/d))/d)/b - (a*((b^2*(-1/ 4*(e + f*x)^4/(b*f) + ((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[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 + Sqrt[a^2 + b^2]))])/d^2))/d))/(b*d)))/(a^2 + b^2) + ((b*(e + f*x)^4)/(4*f) + (2*a*(e + f*x)^3*ArcTan[E^(c + d*x)])/d - (b*(e + f*x)^3*Log[1 + E^(2*(c + d*x))] )/d - ((3*I)*a*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/d^2 + ((3*I)*a* f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/d^2 - (3*b*f*(e + f*x)^2*PolyLog[ 2, -E^(2*(c + d*x))])/(2*d^2) + ((6*I)*a*f^2*(e + f*x)*PolyLog[3, (-I)*E^( c + d*x)])/d^3 - ((6*I)*a*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/d^3 + ( 3*b*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*d^3) - ((6*I)*a*f^3*...
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_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[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[(((e_.) + (f_.)*(x_))^(m_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/b Int[(e + f*x)^m*Sech[ c + d*x]*Tanh[c + d*x]^(n - 1), x], x] - Simp[a/b Int[(e + f*x)^m*Sech[c + d*x]*(Tanh[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b , c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b^2/(a^2 + b^2) Int[(e + f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Simp[1/(a^2 + b^2) Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; F reeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0 ]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/b Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x], x] - S imp[a/b Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*(Tanh[c + d*x]^n/(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]
\[\int \frac {\left (f x +e \right )^{3} \sinh \left (d x +c \right ) \tanh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^3*sinh(d*x+c)*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*sinh(d*x+c)*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)
Time = 0.16 (sec) , antiderivative size = 1962, normalized size of antiderivative = 1.61 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*sinh(d*x+c)*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm ="fricas")
Output:
-1/4*((a^2 + b^2)*d^4*f^3*x^4 + 4*(a^2 + b^2)*d^4*e*f^2*x^3 + 6*(a^2 + b^2 )*d^4*e^2*f*x^2 + 4*(a^2 + b^2)*d^4*e^3*x - 24*a^2*f^3*polylog(4, (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) - 24*a^2*f^3*polylog(4, (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) - 12*(a^ 2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x + a^2*d^2*e^2*f)*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^2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x + a^2*d^2*e^2*f) *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*(I*a*b*d^2*f^3*x^2 - b^2*d^2* f^3*x^2 + 2*I*a*b*d^2*e*f^2*x - 2*b^2*d^2*e*f^2*x + I*a*b*d^2*e^2*f - b^2* d^2*e^2*f)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + 12*(-I*a*b*d^2*f^3*x ^2 - b^2*d^2*f^3*x^2 - 2*I*a*b*d^2*e*f^2*x - 2*b^2*d^2*e*f^2*x - I*a*b*d^2 *e^2*f - b^2*d^2*e^2*f)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) - 4*(a^2 *d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*log(2*b*co sh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 4*(a^ 2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*log(2*b*c osh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 4*(a ^2*d^3*f^3*x^3 + 3*a^2*d^3*e*f^2*x^2 + 3*a^2*d^3*e^2*f*x + 3*a^2*c*d^2*e^2 *f - 3*a^2*c^2*d*e*f^2 + a^2*c^3*f^3)*log(-(a*cosh(d*x + c) + a*sinh(d*...
\[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \sinh {\left (c + d x \right )} \tanh {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**3*sinh(d*x+c)*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)**3*sinh(c + d*x)*tanh(c + d*x)/(a + b*sinh(c + d*x)), x )
\[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sinh \left (d x + c\right ) \tanh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*sinh(d*x+c)*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm ="maxima")
Output:
e^3*(a^2*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^2*b + b^3)*d) + 2*a*arctan(e^(-d*x - c))/((a^2 + b^2)*d) + b*log(e^(-2*d*x - 2*c) + 1)/ ((a^2 + b^2)*d) + (d*x + c)/(b*d)) + 1/4*(f^3*x^4 + 4*e*f^2*x^3 + 6*e^2*f* x^2)/b - integrate(2*(a^2*b*f^3*x^3 + 3*a^2*b*e*f^2*x^2 + 3*a^2*b*e^2*f*x - (a^3*f^3*x^3*e^c + 3*a^3*e*f^2*x^2*e^c + 3*a^3*e^2*f*x*e^c)*e^(d*x))/(a^ 2*b^2 + b^4 - (a^2*b^2*e^(2*c) + b^4*e^(2*c))*e^(2*d*x) - 2*(a^3*b*e^c + a *b^3*e^c)*e^(d*x)), x) - integrate(2*(b*f^3*x^3 + 3*b*e*f^2*x^2 + 3*b*e^2* f*x + (a*f^3*x^3*e^c + 3*a*e*f^2*x^2*e^c + 3*a*e^2*f*x*e^c)*e^(d*x))/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)), x)
Timed out. \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^3*sinh(d*x+c)*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm ="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {sinh}\left (c+d\,x\right )\,\mathrm {tanh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((sinh(c + d*x)*tanh(c + d*x)*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)
Output:
int((sinh(c + d*x)*tanh(c + d*x)*(e + f*x)^3)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
int((f*x+e)^3*sinh(d*x+c)*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
( - 2*atan(e**(c + d*x))*a*b*e**3 + e**(4*c)*int((e**(4*d*x)*x**3)/(e**(4* c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*a**2*b*d*f* *3 + e**(4*c)*int((e**(4*d*x)*x**3)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d* x)*a + 2*e**(c + d*x)*a - b),x)*b**3*d*f**3 + 3*e**(4*c)*int((e**(4*d*x)*x **2)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x) *a**2*b*d*e*f**2 + 3*e**(4*c)*int((e**(4*d*x)*x**2)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*b**3*d*e*f**2 + 3*e**(4*c) *int((e**(4*d*x)*x)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*a**2*b*d*e**2*f + 3*e**(4*c)*int((e**(4*d*x)*x)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*b**3*d*e**2*f - 2*e**(2*c)*int((e**(2*d*x)*x**3)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x )*a + 2*e**(c + d*x)*a - b),x)*a**2*b*d*f**3 - 2*e**(2*c)*int((e**(2*d*x)* x**3)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x )*b**3*d*f**3 - 6*e**(2*c)*int((e**(2*d*x)*x**2)/(e**(4*c + 4*d*x)*b + 2*e **(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*a**2*b*d*e*f**2 - 6*e**(2*c)* int((e**(2*d*x)*x**2)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*b**3*d*e*f**2 - 6*e**(2*c)*int((e**(2*d*x)*x)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*a**2*b*d*e**2* f - 6*e**(2*c)*int((e**(2*d*x)*x)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x) *a + 2*e**(c + d*x)*a - b),x)*b**3*d*e**2*f + int(x**3/(e**(4*c + 4*d*x...