Integrand size = 34, antiderivative size = 795 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b (e+f x)^2}{\left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right )}{a d^2}+\frac {4 b^2 f (e+f x) \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (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 )^{3/2} d}+\frac {b^3 (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 )^{3/2} d}+\frac {2 b f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {2 b^3 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 )^{3/2} d^2}+\frac {2 b^3 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 )^{3/2} d^2}+\frac {b f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {2 b^3 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 )^{3/2} d^3}-\frac {2 b^3 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 )^{3/2} d^3}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d} \]
-b*(f*x+e)^2/(a^2+b^2)/d-4*f*(f*x+e)*arctan(exp(d*x+c))/a/d^2+4*b^2*f*(f*x +e)*arctan(exp(d*x+c))/a/(a^2+b^2)/d^2-2*(f*x+e)^2*arctanh(exp(d*x+c))/a/d +2*b*f*(f*x+e)*ln(1+exp(2*d*x+2*c))/(a^2+b^2)/d^2-b^3*(f*x+e)^2*ln(1+b*exp (d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)^(3/2)/d+b^3*(f*x+e)^2*ln(1+b*exp( d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^(3/2)/d-2*f*(f*x+e)*polylog(2,-exp (d*x+c))/a/d^2+2*I*b^2*f^2*polylog(2,I*exp(d*x+c))/a/(a^2+b^2)/d^3-2*I*f^2 *polylog(2,I*exp(d*x+c))/a/d^3-2*I*b^2*f^2*polylog(2,-I*exp(d*x+c))/a/(a^2 +b^2)/d^3+2*I*f^2*polylog(2,-I*exp(d*x+c))/a/d^3+2*f*(f*x+e)*polylog(2,exp (d*x+c))/a/d^2+b*f^2*polylog(2,-exp(2*d*x+2*c))/(a^2+b^2)/d^3-2*b^3*f*(f*x +e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)^(3/2)/d^2+2*b ^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^(3/2 )/d^2+2*f^2*polylog(3,-exp(d*x+c))/a/d^3-2*f^2*polylog(3,exp(d*x+c))/a/d^3 +2*b^3*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)^(3/2)/ d^3-2*b^3*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^(3/ 2)/d^3+(f*x+e)^2*sech(d*x+c)/a/d-b^2*(f*x+e)^2*sech(d*x+c)/a/(a^2+b^2)/d-b *(f*x+e)^2*tanh(d*x+c)/(a^2+b^2)/d
Time = 8.30 (sec) , antiderivative size = 928, normalized size of antiderivative = 1.17 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=4 \left (-\frac {f \text {csch}(c+d x) \left (4 b d^2 e e^{2 c} x-4 b d^2 e \left (1+e^{2 c}\right ) x+2 b d^2 e^{2 c} f x^2-2 b d^2 \left (1+e^{2 c}\right ) f x^2+4 a d e \left (1+e^{2 c}\right ) \arctan \left (e^{c+d x}\right )+2 b d e \left (1+e^{2 c}\right ) \left (2 d x-\log \left (1+e^{2 (c+d x)}\right )\right )+2 i a \left (1+e^{2 c}\right ) f \left (d x \left (\log \left (1-i e^{c+d x}\right )-\log \left (1+i e^{c+d x}\right )\right )-\operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+\operatorname {PolyLog}\left (2,i e^{c+d x}\right )\right )+b \left (1+e^{2 c}\right ) f \left (2 d x \left (d x-\log \left (1+e^{2 (c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )\right )\right ) (a+b \sinh (c+d x))}{4 \left (a^2+b^2\right ) d^3 \left (1+e^{2 c}\right ) (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \left ((e+f x)^2 \log \left (1-e^{c+d x}\right )-(e+f x)^2 \log \left (1+e^{c+d x}\right )-\frac {2 f \left (d (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )-f \operatorname {PolyLog}\left (3,-e^{c+d x}\right )\right )}{d^2}+\frac {2 f \left (d (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )-f \operatorname {PolyLog}\left (3,e^{c+d x}\right )\right )}{d^2}\right ) (a+b \sinh (c+d x))}{4 a d (b+a \text {csch}(c+d x))}-\frac {b^3 \text {csch}(c+d x) \left (-2 d^2 e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right ) (a+b \sinh (c+d x))}{4 a \left (a^2+b^2\right )^{3/2} d^3 (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \text {sech}(c) \text {sech}(c+d x) \left (a e^2 \cosh (c)+2 a e f x \cosh (c)+a f^2 x^2 \cosh (c)-b e^2 \sinh (d x)-2 b e f x \sinh (d x)-b f^2 x^2 \sinh (d x)\right ) (a+b \sinh (c+d x))}{4 \left (a^2+b^2\right ) d (b+a \text {csch}(c+d x))}\right ) \]
4*(-1/4*(f*Csch[c + d*x]*(4*b*d^2*e*E^(2*c)*x - 4*b*d^2*e*(1 + E^(2*c))*x + 2*b*d^2*E^(2*c)*f*x^2 - 2*b*d^2*(1 + E^(2*c))*f*x^2 + 4*a*d*e*(1 + E^(2* c))*ArcTan[E^(c + d*x)] + 2*b*d*e*(1 + E^(2*c))*(2*d*x - Log[1 + E^(2*(c + d*x))]) + (2*I)*a*(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)] ) + b*(1 + E^(2*c))*f*(2*d*x*(d*x - Log[1 + E^(2*(c + d*x))]) - PolyLog[2, -E^(2*(c + d*x))]))*(a + b*Sinh[c + d*x]))/((a^2 + b^2)*d^3*(1 + E^(2*c)) *(b + a*Csch[c + d*x])) + (Csch[c + d*x]*((e + f*x)^2*Log[1 - E^(c + d*x)] - (e + f*x)^2*Log[1 + E^(c + d*x)] - (2*f*(d*(e + f*x)*PolyLog[2, -E^(c + d*x)] - f*PolyLog[3, -E^(c + d*x)]))/d^2 + (2*f*(d*(e + f*x)*PolyLog[2, E ^(c + d*x)] - f*PolyLog[3, E^(c + d*x)]))/d^2)*(a + b*Sinh[c + d*x]))/(4*a *d*(b + a*Csch[c + d*x])) - (b^3*Csch[c + d*x]*(-2*d^2*e^2*ArcTanh[(a + b* E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a - S qrt[a^2 + b^2])] + d^2*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2 ])] - 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - d^2*f^2 *x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*f*(e + f*x)*Poly Log[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*d*f*(e + f*x)*PolyLog[2 , -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2*f^2*PolyLog[3, (b*E^(c + d *x))/(-a + Sqrt[a^2 + b^2])] + 2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqr t[a^2 + b^2]))])*(a + b*Sinh[c + d*x]))/(4*a*(a^2 + b^2)^(3/2)*d^3*(b +...
Time = 3.74 (sec) , antiderivative size = 695, normalized size of antiderivative = 0.87, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6123, 5985, 25, 6107, 3042, 3803, 25, 2694, 27, 2620, 3011, 2720, 7143, 7292, 27, 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.
| \(\displaystyle \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6123 |
| \(\displaystyle \frac {\int (e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5985 |
| \(\displaystyle \frac {-2 f \int -\left ((e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle \frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {b^2 \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}\right )}{a}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle \frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {2 b^2 \int -\frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \int \frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}\right )}{a}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle \frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 f \int \frac {(e+f x) (\text {arctanh}(\cosh (c+d x))-\text {sech}(c+d x))}{d}dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 f \int (e+f x) (\text {arctanh}(\cosh (c+d x))-\text {sech}(c+d x))dx}{d}-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\frac {2 f \int ((e+f x) \text {arctanh}(\cosh (c+d x))-(e+f x) \text {sech}(c+d x))dx}{d}-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int \left (a (e+f x)^2 \text {sech}^2(c+d x)-b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 f \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{f}+\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 f}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}+\frac {f \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{d^2}-\frac {f \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}+\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}+\frac {-\frac {a f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{d^3}-\frac {2 a f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{d^2}+\frac {a (e+f x)^2 \tanh (c+d x)}{d}+\frac {a (e+f x)^2}{d}-\frac {4 b f (e+f x) \arctan \left (e^{c+d x}\right )}{d^2}+\frac {2 i b f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^3}-\frac {2 i b f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^3}+\frac {b (e+f x)^2 \text {sech}(c+d x)}{d}}{a^2+b^2}\right )}{a}\) |
(-(((e + f*x)^2*ArcTanh[Cosh[c + d*x]])/d) + (2*f*((-2*(e + f*x)*ArcTan[E^ (c + d*x)])/d - ((e + f*x)^2*ArcTanh[E^(c + d*x)])/f + ((e + f*x)^2*ArcTan h[Cosh[c + d*x]])/(2*f) - ((e + f*x)*PolyLog[2, -E^(c + d*x)])/d + (I*f*Po lyLog[2, (-I)*E^(c + d*x)])/d^2 - (I*f*PolyLog[2, I*E^(c + d*x)])/d^2 + (( e + f*x)*PolyLog[2, E^(c + d*x)])/d + (f*PolyLog[3, -E^(c + d*x)])/d^2 - ( f*PolyLog[3, E^(c + d*x)])/d^2))/d + ((e + f*x)^2*Sech[c + d*x])/d)/a - (b *((-2*b^2*(-1/2*(b*(((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)))/Sqrt[a^2 + b^2] + (b*(((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*Sqrt[a^2 + b^2])))/(a^2 + b^2) + ((a* (e + f*x)^2)/d - (4*b*f*(e + f*x)*ArcTan[E^(c + d*x)])/d^2 - (2*a*f*(e + f *x)*Log[1 + E^(2*(c + d*x))])/d^2 + ((2*I)*b*f^2*PolyLog[2, (-I)*E^(c + d* x)])/d^3 - ((2*I)*b*f^2*PolyLog[2, I*E^(c + d*x)])/d^3 - (a*f^2*PolyLog[2, -E^(2*(c + d*x))])/d^3 + (b*(e + f*x)^2*Sech[c + d*x])/d + (a*(e + f*x)^2 *Tanh[c + d*x])/d)/(a^2 + b^2)))/a
3.5.41.3.1 Defintions of rubi rules used
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[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> With[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Simp[(c + d*x)^m u, x] - Simp[d*m Int[(c + d*x)^(m - 1)*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n , p]
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]
\[\int \frac {\left (f x +e \right )^{2} \operatorname {csch}\left (d x +c \right ) \operatorname {sech}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 5562 vs. \(2 (729) = 1458\).
Time = 0.40 (sec) , antiderivative size = 5562, normalized size of antiderivative = 7.00 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right ) \operatorname {sech}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-2*b*e*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)) - 4*a*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) - 4*b*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) - (b^3*log((b*e^(- d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(( a^3 + a*b^2)*sqrt(a^2 + b^2)*d) - 2*(a*e^(-d*x - c) - b)/((a^2 + b^2 + (a^ 2 + b^2)*e^(-2*d*x - 2*c))*d) + log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a*d))*e^2 - 4*a*e*f*arctan(e^(d*x + c))/((a^2 + b^2)*d^2) + 2*( b*f^2*x^2 + 2*b*e*f*x + (a*f^2*x^2*e^c + 2*a*e*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)) - 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*d ilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*f^2/(a*d^3) + (d^2*x^2*lo g(-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) - integrate(-2*(b^3*f^2*x^2*e^c + 2*b^3*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)
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]