Integrand size = 28, antiderivative size = 1118 \[ \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:
-(f*x+e)^3*sech(d*x+c)/b/d-3*a^3*f^2*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/b^ 2/(a^2+b^2)/d^3-3*a^3*f*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/b^2/(a^2+b^2)/d^2-6 *a^2*f*(f*x+e)^2*arctan(exp(d*x+c))/b/(a^2+b^2)/d^2-6*I*a^2*f^3*polylog(3, -I*exp(d*x+c))/b/(a^2+b^2)/d^4-6*I*a^2*f^2*(f*x+e)*polylog(2,I*exp(d*x+c)) /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)))/(a^2+b^2)^(3/2)/d^3-6*a^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^ 2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^3-3*a^2*f*(f*x+e)^2*polylog(2,-b*exp(d*x+ c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^2+3*a^2*f*(f*x+e)^2*polylog(2,-b *exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^2+3/2*a^3*f^3*polylog(3 ,-exp(2*d*x+2*c))/b^2/(a^2+b^2)/d^4+3*a*f^2*(f*x+e)*polylog(2,-exp(2*d*x+2 *c))/b^2/d^3+3*a*f*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/b^2/d^2-6*I*f^2*(f*x+e)* polylog(2,-I*exp(d*x+c))/b/d^3-a^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2 )^(1/2)))/(a^2+b^2)^(3/2)/d+a^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^( 1/2)))/(a^2+b^2)^(3/2)/d-6*a^2*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1 /2)))/(a^2+b^2)^(3/2)/d^4+6*a^2*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^( 1/2)))/(a^2+b^2)^(3/2)/d^4-a*(f*x+e)^3/b^2/d+a^3*(f*x+e)^3*tanh(d*x+c)/b^2 /(a^2+b^2)/d+a^2*(f*x+e)^3*sech(d*x+c)/b/(a^2+b^2)/d+6*I*f^3*polylog(3,-I* exp(d*x+c))/b/d^4-a*(f*x+e)^3*tanh(d*x+c)/b^2/d+a^3*(f*x+e)^3/b^2/(a^2+b^2 )/d-3/2*a*f^3*polylog(3,-exp(2*d*x+2*c))/b^2/d^4+6*f*(f*x+e)^2*arctan(exp( d*x+c))/b/d^2-6*I*f^3*polylog(3,I*exp(d*x+c))/b/d^4+6*I*f^2*(f*x+e)*pol...
Time = 7.53 (sec) , antiderivative size = 1071, normalized size of antiderivative = 0.96 \[ \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^3*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
((f*(-12*a*d^3*e^2*E^(2*c)*x + 12*a*d^3*e^2*(1 + E^(2*c))*x + 12*a*d^3*e*f *x^2 + 4*a*d^3*f^2*x^3 + 12*b*d^2*e^2*(1 + E^(2*c))*ArcTan[E^(c + d*x)] - 6*a*d^2*e^2*(1 + E^(2*c))*(2*d*x - Log[1 + E^(2*(c + d*x))]) + (12*I)*b*d* e*(1 + E^(2*c))*f*(d*x*(Log[1 - I*E^(c + d*x)] - Log[1 + I*E^(c + d*x)]) - PolyLog[2, (-I)*E^(c + d*x)] + PolyLog[2, I*E^(c + d*x)]) - 6*a*d*e*(1 + E^(2*c))*f*(2*d*x*(d*x - Log[1 + E^(2*(c + d*x))]) - PolyLog[2, -E^(2*(c + d*x))]) + (6*I)*b*(1 + E^(2*c))*f^2*(d^2*x^2*Log[1 - I*E^(c + d*x)] - d^2 *x^2*Log[1 + I*E^(c + d*x)] - 2*d*x*PolyLog[2, (-I)*E^(c + d*x)] + 2*d*x*P olyLog[2, I*E^(c + d*x)] + 2*PolyLog[3, (-I)*E^(c + d*x)] - 2*PolyLog[3, I *E^(c + d*x)]) - a*(1 + E^(2*c))*f^2*(2*d^2*x^2*(2*d*x - 3*Log[1 + E^(2*(c + d*x))]) - 6*d*x*PolyLog[2, -E^(2*(c + d*x))] + 3*PolyLog[3, -E^(2*(c + d*x))])))/((a^2 + b^2)*(1 + E^(2*c))) + (2*a^2*(-2*d^3*e^3*ArcTanh[(a + b* E^(c + d*x))/Sqrt[a^2 + b^2]] + 3*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 3*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 3 *d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 3*d^3*e*f^2* x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - d^3*f^3*x^3*Log[1 + ( b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 3*d^2*f*(e + f*x)^2*PolyLog[2, (b* E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 3*d^2*f*(e + f*x)^2*PolyLog[2, -((b *E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 6*d*e*f^2*PolyLog[3, (b*E^(c + ...
Time = 6.91 (sec) , antiderivative size = 949, normalized size of antiderivative = 0.85, number of steps used = 31, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6101, 5974, 3042, 4668, 3011, 2720, 6117, 3042, 4672, 26, 3042, 26, 4201, 2620, 3011, 2720, 6107, 3042, 3803, 25, 2694, 27, 2620, 3011, 7143, 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.
\(\displaystyle \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
\(\Big \downarrow \) 6101 |
\(\displaystyle \frac {\int (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
\(\Big \downarrow \) 5974 |
\(\displaystyle \frac {\frac {3 f \int (e+f x)^2 \text {sech}(c+d x)dx}{d}-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{b}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (-\frac {2 i f \int (e+f x) \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {2 i f \int (e+f x) \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 6117 |
\(\displaystyle -\frac {a \left (\frac {\int (e+f x)^3 \text {sech}^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{b}\right )}{b}\) |
\(\Big \downarrow \) 4672 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {3 i f \int -i (e+f x)^2 \tanh (c+d x)dx}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a \left (\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {3 f \int (e+f x)^2 \tanh (c+d x)dx}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {3 f \int -i (e+f x)^2 \tan (i c+i d x)dx}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \int (e+f x)^2 \tan (i c+i d x)dx}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}}dx-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int (e+f x) \log \left (1+e^{2 (c+d x)}\right )dx}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 6107 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {b^2 \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 3803 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {2 b^2 \int -\frac {e^{c+d x} (e+f x)^3}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \int \frac {e^{c+d x} (e+f x)^3}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 2694 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int -\frac {e^{c+d x} (e+f x)^3}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)^3}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\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}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\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}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\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 )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\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 )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int \left (a (e+f x)^3 \text {sech}^2(c+d x)-b (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\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 )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\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 )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {3 f \left (\frac {2 \arctan \left (e^{c+d x}\right ) (e+f x)^2}{d}+\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {\tanh (c+d x) (e+f x)^3}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{2 d}-\frac {f \left (\frac {f \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}-\frac {a \left (\frac {-\frac {6 i b \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) f^3}{d^4}+\frac {6 i b \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) f^3}{d^4}+\frac {3 a \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) f^3}{2 d^4}+\frac {6 i b (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) f^2}{d^3}-\frac {6 i b (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) f^2}{d^3}-\frac {3 a (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) f^2}{d^3}-\frac {6 b (e+f x)^2 \arctan \left (e^{c+d x}\right ) f}{d^2}-\frac {3 a (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) f}{d^2}+\frac {a (e+f x)^3}{d}+\frac {b (e+f x)^3 \text {sech}(c+d x)}{d}+\frac {a (e+f x)^3 \tanh (c+d x)}{d}}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\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 )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\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 )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\) |
Input:
Int[((e + f*x)^3*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
((3*f*((2*(e + f*x)^2*ArcTan[E^(c + d*x)])/d + ((2*I)*f*(-(((e + f*x)*Poly Log[2, (-I)*E^(c + d*x)])/d) + (f*PolyLog[3, (-I)*E^(c + d*x)])/d^2))/d - ((2*I)*f*(-(((e + f*x)*PolyLog[2, I*E^(c + d*x)])/d) + (f*PolyLog[3, I*E^( c + d*x)])/d^2))/d))/d - ((e + f*x)^3*Sech[c + d*x])/d)/b - (a*((((3*I)*f* (((-1/3*I)*(e + f*x)^3)/f + (2*I)*(((e + f*x)^2*Log[1 + E^(2*(c + d*x))])/ (2*d) - (f*(-1/2*((e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/d + (f*PolyLog[3 , -E^(2*(c + d*x))])/(4*d^2)))/d)))/d + ((e + f*x)^3*Tanh[c + d*x])/d)/b - (a*((-2*b^2*(-1/2*(b*(((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 - S qrt[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)))/Sqrt[a^2 + b^2] + (b*(((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)))/(2*Sqrt[a^2 + b^2])))/( a^2 + b^2) + ((a*(e + f*x)^3)/d - (6*b*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/ d^2 - (3*a*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/d^2 + ((6*I)*b*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/d^3 - ((6*I)*b*f^2*(e + f*x)*PolyLog[2 , I*E^(c + d*x)])/d^3 - (3*a*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))]...
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[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_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( -I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[((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[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Sech[a + b*x]^n/(b*n)) , x] + Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 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_.)*Sech[(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*Sech[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1), x], x] - Simp[a/b Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*(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] && 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]
\[\int \frac {\left (f x +e \right )^{3} \tanh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^3*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 6494 vs. \(2 (1024) = 2048\).
Time = 0.25 (sec) , antiderivative size = 6494, normalized size of antiderivative = 5.81 \[ \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \tanh ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**3*tanh(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)**3*tanh(c + d*x)**2/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \tanh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
-3*a*e^2*f*(2*(d*x + c)/((a^2 + b^2)*d^2) - log(e^(2*d*x + 2*c) + 1)/((a^2 + b^2)*d^2)) + 6*b*f^3*integrate(x^2*e^(d*x + c)/(a^2*d*e^(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 6*a*f^3*integrate(x^2/(a^2*d *e^(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) + 12*b*e*f^2 *integrate(x*e^(d*x + c)/(a^2*d*e^(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 12*a*e*f^2*integrate(x/(a^2*d*e^(2*d*x + 2*c) + b^2*d *e^(2*d*x + 2*c) + a^2*d + b^2*d), x) + e^3*(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)))/((a^2 + b^2)^(3/ 2)*d) - 2*(b*e^(-d*x - c) + a)/((a^2 + b^2 + (a^2 + b^2)*e^(-2*d*x - 2*c)) *d)) + 6*b*e^2*f*arctan(e^(d*x + c))/((a^2 + b^2)*d^2) + 2*(a*f^3*x^3 + 3* a*e*f^2*x^2 + 3*a*e^2*f*x - (b*f^3*x^3*e^c + 3*b*e*f^2*x^2*e^c + 3*b*e^2*f *x*e^c)*e^(d*x))/(a^2*d + b^2*d + (a^2*d*e^(2*c) + b^2*d*e^(2*c))*e^(2*d*x )) + integrate(-2*(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)/(a^2*b + b^3 - (a^2*b*e^(2*c) + b^3*e^(2*c))*e^(2*d*x) - 2*(a^ 3*e^c + a*b^2*e^c)*e^(d*x)), x)
Timed out. \[ \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^3*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((tanh(c + d*x)^2*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)
Output:
int((tanh(c + d*x)^2*(e + f*x)^3)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^3*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
(12*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**4*b*d**2*e**2*f + 24*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**4*b*d*e*f**2 + 24*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**4*b*f**3 + 24*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**2*b**3*d* *2*e**2*f + 48*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**2*b**3*d*e*f**2 + 48 *e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**2*b**3*f**3 + 12*e**(2*c + 2*d*x)* atan(e**(c + d*x))*b**5*d**2*e**2*f + 24*e**(2*c + 2*d*x)*atan(e**(c + d*x ))*b**5*d*e*f**2 + 24*e**(2*c + 2*d*x)*atan(e**(c + d*x))*b**5*f**3 + 12*a tan(e**(c + d*x))*a**4*b*d**2*e**2*f + 24*atan(e**(c + d*x))*a**4*b*d*e*f* *2 + 24*atan(e**(c + d*x))*a**4*b*f**3 + 24*atan(e**(c + d*x))*a**2*b**3*d **2*e**2*f + 48*atan(e**(c + d*x))*a**2*b**3*d*e*f**2 + 48*atan(e**(c + d* x))*a**2*b**3*f**3 + 12*atan(e**(c + d*x))*b**5*d**2*e**2*f + 24*atan(e**( c + d*x))*b**5*d*e*f**2 + 24*atan(e**(c + d*x))*b**5*f**3 + 4*e**(2*c + 2* d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a* *2*b**2*d**3*e**3*i + 4*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sq rt(a**2 + b**2))*a**2*b**2*d**3*e**3*i + 16*e**(5*c + 2*d*x)*int((e**(3*d* x)*x**3)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a + e**(4*c + 4*d*x)*b + 4*e**(3*c + 3*d*x)*a - e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**6 *d**4*f**3 + 32*e**(5*c + 2*d*x)*int((e**(3*d*x)*x**3)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a + e**(4*c + 4*d*x)*b + 4*e**(3*c + 3*d*x)*a - e**( 2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**4*b**2*d**4*f**3 + 16*e**(...