Integrand size = 32, antiderivative size = 734 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 b (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a d^2}-\frac {2 i b f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 i b f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {b^2 f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a d^3} \]
-2*b*(f*x+e)^2*arctan(exp(d*x+c))/(a^2+b^2)/d-2*(f*x+e)^2*arctanh(exp(2*d* x+2*c))/a/d+b^2*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/a/(a^2+b^2)/d-b^2*(f*x+e)^2 *ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d-b^2*(f*x+e)^2*ln(1+b *exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d-2*I*b*f*(f*x+e)*polylog(2,I *exp(d*x+c))/(a^2+b^2)/d^2+2*I*b*f^2*polylog(3,I*exp(d*x+c))/(a^2+b^2)/d^3 +b^2*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/a/(a^2+b^2)/d^2-f*(f*x+e)*polylo g(2,-exp(2*d*x+2*c))/a/d^2+f*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a/d^2-2*b^2 *f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^2-2* b^2*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^2 +2*I*b*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/(a^2+b^2)/d^2-2*I*b*f^2*polylog( 3,-I*exp(d*x+c))/(a^2+b^2)/d^3-1/2*b^2*f^2*polylog(3,-exp(2*d*x+2*c))/a/(a ^2+b^2)/d^3+1/2*f^2*polylog(3,-exp(2*d*x+2*c))/a/d^3-1/2*f^2*polylog(3,exp (2*d*x+2*c))/a/d^3+2*b^2*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/ a/(a^2+b^2)/d^3+2*b^2*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/( a^2+b^2)/d^3
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3730\) vs. \(2(734)=1468\).
Time = 13.22 (sec) , antiderivative size = 3730, normalized size of antiderivative = 5.08 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]
2*((a*E^c*((e + f*x)^3/(3*E^c*f) + ((1 + E^(-c))*(e + f*x)^2*Log[1 + E^(-c - d*x)])/d - (2*(1 + E^c)*f*(d*(e + f*x)*PolyLog[2, -E^(-c - d*x)] + f*Po lyLog[3, -E^(-c - d*x)]))/(d^3*E^c)))/(2*(a^2 + b^2)*(1 + E^c)) + (d^2*(d* x*((-3*I)*b*e*f*x + a*((-3*I)*e^2*E^c + 3*e*f*x + f^2*x^2)) + 3*(1 + I*E^c )*f*x*(2*a*e - (2*I)*b*e + a*f*x)*Log[1 - I*E^(-c - d*x)] + 3*a*e^2*(1 + I *E^c)*Log[I - E^(c + d*x)]) - (6*I)*d*(-I + E^c)*f*((-I)*b*e + a*(e + f*x) )*PolyLog[2, I*E^(-c - d*x)] - (6*I)*a*(-I + E^c)*f^2*PolyLog[3, I*E^(-c - d*x)])/(6*(a - I*b)*((-I)*a + b)*d^3*(-I + E^c)) - (b^2*E^(2*c)*((2*(e + f*x)^3)/(E^(2*c)*f) - (3*(1 - E^(-2*c))*(e + f*x)^2*Log[1 - E^(-c - d*x)]) /d - (3*(1 - E^(-2*c))*(e + f*x)^2*Log[1 + E^(-c - d*x)])/d + (6*(-1 + E^( 2*c))*f*(d*(e + f*x)*PolyLog[2, -E^(-c - d*x)] + f*PolyLog[3, -E^(-c - d*x )]))/(d^3*E^(2*c)) + (6*(-1 + E^(2*c))*f*(d*(e + f*x)*PolyLog[2, E^(-c - d *x)] + f*PolyLog[3, E^(-c - d*x)]))/(d^3*E^(2*c))))/(6*a*(a^2 + b^2)*(-1 + E^(2*c))) - ((I/2)*b*((-2*I)*d^2*e^2*ArcTan[E^(c + d*x)] + d^2*f^2*x^2*Lo g[1 - I*E^(c + d*x)] - d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] - 2*d*f^2*x*Poly Log[2, (-I)*E^(c + d*x)] + 2*d*f^2*x*PolyLog[2, I*E^(c + d*x)] + 2*f^2*Pol yLog[3, (-I)*E^(c + d*x)] - 2*f^2*PolyLog[3, I*E^(c + d*x)]))/((a^2 + b^2) *d^3) - ((-I)*b*d^3*e*E^(2*c)*f*x^2 + 2*a*d^2*e^2*ArcTan[1 - (1 + I)*E^(c + d*x)] + (2*I)*a*d^2*e^2*E^(2*c)*ArcTan[1 - (1 + I)*E^(c + d*x)] + (2*I)* a*d^2*e*f*x*Log[1 - E^(c + d*x)] - 2*a*d^2*e*E^(2*c)*f*x*Log[1 - E^(c +...
Time = 3.13 (sec) , antiderivative size = 651, normalized size of antiderivative = 0.89, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {6123, 5984, 3042, 26, 4670, 3011, 2720, 6107, 6095, 2620, 3011, 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.
| \(\displaystyle \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6123 |
| \(\displaystyle \frac {\int (e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle \frac {2 \int (e+f x)^2 \text {csch}(2 c+2 d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {2 \int i (e+f x)^2 \csc (2 i c+2 i d x)dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {2 i \int (e+f x)^2 \csc (2 i c+2 i d x)dx}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {2 i \left (\frac {i f \int (e+f x) \log \left (1-e^{2 c+2 d x}\right )dx}{d}-\frac {i f \int (e+f x) \log \left (1+e^{2 c+2 d x}\right )dx}{d}+\frac {i (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {2 i \left (-\frac {i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i f \left (\frac {f \int \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {2 i \left (-\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle -\frac {b \left (\frac {b^2 \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}+\frac {2 i \left (-\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle -\frac {b \left (\frac {b^2 \left (\int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^3}{3 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}+\frac {2 i \left (-\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {b \left (\frac {b^2 \left (-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^3}{3 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}+\frac {2 i \left (-\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \left (\frac {b^2 \left (-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^3}{3 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}+\frac {2 i \left (-\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {b \left (\frac {b^2 \left (-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^3}{3 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}+\frac {2 i \left (-\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^3}{3 b f}\right )}{a^2+b^2}\right )}{a}+\frac {2 i \left (\frac {i (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}-\frac {i f \left (\frac {f \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i f \left (\frac {f \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {b \left (\frac {\int \left (a (e+f x)^2 \text {sech}(c+d x)-b (e+f x)^2 \tanh (c+d x)\right )dx}{a^2+b^2}+\frac {b^2 \left (-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^3}{3 b f}\right )}{a^2+b^2}\right )}{a}+\frac {2 i \left (\frac {i (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}-\frac {i f \left (\frac {f \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i f \left (\frac {f \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 i \left (\frac {i (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}-\frac {i f \left (\frac {f \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i f \left (\frac {f \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{d}\right )}{a}-\frac {b \left (\frac {b^2 \left (-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^3}{3 b f}\right )}{a^2+b^2}+\frac {\frac {2 a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d^3}-\frac {2 i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d^3}-\frac {2 i a f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {2 i a f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}+\frac {b f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d^3}-\frac {b f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{d^2}-\frac {b (e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{d}+\frac {b (e+f x)^3}{3 f}}{a^2+b^2}\right )}{a}\) |
-((b*((b^2*(-1/3*(e + f*x)^3/(b*f) + ((e + f*x)^2*Log[1 + (b*E^(c + d*x))/ (a - Sqrt[a^2 + b^2])])/(b*d) + ((e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) - (2*f*(-(((e + f*x)*PolyLog[2, -((b*E^(c + d*x)) /(a - Sqrt[a^2 + b^2]))])/d) + (f*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a ^2 + b^2]))])/d^2))/(b*d) - (2*f*(-(((e + f*x)*PolyLog[2, -((b*E^(c + d*x) )/(a + Sqrt[a^2 + b^2]))])/d) + (f*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[ a^2 + b^2]))])/d^2))/(b*d)))/(a^2 + b^2) + ((b*(e + f*x)^3)/(3*f) + (2*a*( e + f*x)^2*ArcTan[E^(c + d*x)])/d - (b*(e + f*x)^2*Log[1 + E^(2*(c + d*x)) ])/d - ((2*I)*a*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/d^2 + ((2*I)*a*f *(e + f*x)*PolyLog[2, I*E^(c + d*x)])/d^2 - (b*f*(e + f*x)*PolyLog[2, -E^( 2*(c + d*x))])/d^2 + ((2*I)*a*f^2*PolyLog[3, (-I)*E^(c + d*x)])/d^3 - ((2* I)*a*f^2*PolyLog[3, I*E^(c + d*x)])/d^3 + (b*f^2*PolyLog[3, -E^(2*(c + d*x ))])/(2*d^3))/(a^2 + b^2)))/a) + ((2*I)*((I*(e + f*x)^2*ArcTanh[E^(2*c + 2 *d*x)])/d - (I*f*(-1/2*((e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/d + (f*Pol yLog[3, -E^(2*c + 2*d*x)])/(4*d^2)))/d + (I*f*(-1/2*((e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/d + (f*PolyLog[3, E^(2*c + 2*d*x)])/(4*d^2)))/d))/a
3.5.36.3.1 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[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csch[2*a + 2*b*x ]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
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_.)*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[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/a Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Simp[b/ a Int[(e + f*x)^m*Sech[c + d*x]^p*(Csch[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]
\[\int \frac {\left (f x +e \right )^{2} \operatorname {csch}\left (d x +c \right ) \operatorname {sech}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1535 vs. \(2 (677) = 1354\).
Time = 0.31 (sec) , antiderivative size = 1535, normalized size of antiderivative = 2.09 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
(2*b^2*f^2*polylog(3, (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) + 2*b^2*f^2*polylog(3, (a*c osh(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) - 2*(a^2 + b^2)*f^2*polylog(3, cosh(d*x + c) + sinh(d *x + c)) - 2*(a^2 + b^2)*f^2*polylog(3, -cosh(d*x + c) - sinh(d*x + c)) - 2*(b^2*d*f^2*x + b^2*d*e*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) - 2*(b^ 2*d*f^2*x + b^2*d*e*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) + 2*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*dilog(cosh(d*x + c) + sinh(d*x + c)) - 2 *(a^2*d*f^2*x + I*a*b*d*f^2*x + a^2*d*e*f + I*a*b*d*e*f)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) - 2*(a^2*d*f^2*x - I*a*b*d*f^2*x + a^2*d*e*f - I*a *b*d*e*f)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) + 2*((a^2 + b^2)*d*f^2 *x + (a^2 + b^2)*d*e*f)*dilog(-cosh(d*x + c) - sinh(d*x + c)) - (b^2*d^2*e ^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c ) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - (b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2* c^2*f^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/ b^2) + 2*a) - (b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2 *f^2)*log(-(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) - (b^2*d^2*f^2*x^2 + 2*b^2*d^2*...
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right ) \operatorname {sech}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-e^2*(b^2*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^3 + a*b^2)*d ) - 2*b*arctan(e^(-d*x - c))/((a^2 + b^2)*d) + a*log(e^(-2*d*x - 2*c) + 1) /((a^2 + b^2)*d) - log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a* d)) + 2*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*e*f/(a*d^2) + 2*( d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))*e*f/(a*d^2) + (d^2*x^2*log (e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)) )*f^2/(a*d^3) + (d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*f^2/(a*d^3) - 2/3*(d^3*f^2*x^3 + 3*d^3*e*f*x^ 2)/(a*d^3) + integrate(2*(b^3*f^2*x^2 + 2*b^3*e*f*x - (a*b^2*f^2*x^2*e^c + 2*a*b^2*e*f*x*e^c)*e^(d*x))/(a^3*b + a*b^3 - (a^3*b*e^(2*c) + a*b^3*e^(2* c))*e^(2*d*x) - 2*(a^4*e^c + a^2*b^2*e^c)*e^(d*x)), x) - integrate(-2*(a*f ^2*x^2 + 2*a*e*f*x - (b*f^2*x^2*e^c + 2*b*e*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)
\[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right ) \operatorname {sech}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]