Integrand size = 26, antiderivative size = 327 \[ \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {e x}{2 a}+\frac {b^2 e x}{a^3}+\frac {f x^2}{4 a}+\frac {b^2 f x^2}{2 a^3}-\frac {b (e+f x) \cosh (c+d x)}{a^2 d}-\frac {f \cosh ^2(c+d x)}{4 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {b \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b f \sinh (c+d x)}{a^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d} \]
1/2*e*x/a+b^2*e*x/a^3+1/4*f*x^2/a+1/2*b^2*f*x^2/a^3-b*(f*x+e)*cosh(d*x+c)/ a^2/d-1/4*f*cosh(d*x+c)^2/a/d^2+b*f*sinh(d*x+c)/a^2/d^2+1/2*(f*x+e)*cosh(d *x+c)*sinh(d*x+c)/a/d-b*(f*x+e)*ln(1+a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))*(a^ 2+b^2)^(1/2)/a^3/d+b*(f*x+e)*ln(1+a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))*(a^2+b ^2)^(1/2)/a^3/d-b*f*polylog(2,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))*(a^2+b^2) ^(1/2)/a^3/d^2+b*f*polylog(2,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))*(a^2+b^2)^ (1/2)/a^3/d^2
Time = 1.97 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.87 \[ \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {\text {csch}(c+d x) (b+a \sinh (c+d x)) \left (2 a^2 e \left (\frac {c}{d}+x-\frac {2 b \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2} d}\right )+a^2 f \left (x^2-\frac {2 b \left (d x \left (\log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )-\log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )-\operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^2}\right )+\frac {2 e \left (\left (a^2+4 b^2\right ) (c+d x)-\frac {2 b \left (3 a^2+4 b^2\right ) \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-4 a b \cosh (c+d x)+a^2 \sinh (2 (c+d x))\right )}{d}+\frac {f \left (\left (a^2+4 b^2\right ) (-c+d x) (c+d x)-8 a b d x \cosh (c+d x)-a^2 \cosh (2 (c+d x))-\frac {2 b \left (3 a^2+4 b^2\right ) \left (2 c \text {arctanh}\left (\frac {b+a e^{c+d x}}{\sqrt {a^2+b^2}}\right )+(c+d x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )-(c+d x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+\operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )-\operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}+8 a b \sinh (c+d x)+2 a^2 d x \sinh (2 (c+d x))\right )}{d^2}\right )}{8 a^3 (a+b \text {csch}(c+d x))} \]
(Csch[c + d*x]*(b + a*Sinh[c + d*x])*(2*a^2*e*(c/d + x - (2*b*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/(Sqrt[-a^2 - b^2]*d)) + a^2*f*(x^2 - (2*b*(d*x*(Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] - Log[1 + (a* E^(c + d*x))/(b + Sqrt[a^2 + b^2])]) + PolyLog[2, (a*E^(c + d*x))/(-b + Sq rt[a^2 + b^2])] - PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/( Sqrt[a^2 + b^2]*d^2)) + (2*e*((a^2 + 4*b^2)*(c + d*x) - (2*b*(3*a^2 + 4*b^ 2)*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 4*a*b*Cosh[c + d*x] + a^2*Sinh[2*(c + d*x)]))/d + (f*((a^2 + 4*b^2)*(-c + d*x)*(c + d*x) - 8*a*b*d*x*Cosh[c + d*x] - a^2*Cosh[2*(c + d*x)] - (2*b*(3 *a^2 + 4*b^2)*(2*c*ArcTanh[(b + a*E^(c + d*x))/Sqrt[a^2 + b^2]] + (c + d*x )*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] - (c + d*x)*Log[1 + (a*E^ (c + d*x))/(b + Sqrt[a^2 + b^2])] + PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[ a^2 + b^2])] - PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/Sqrt [a^2 + b^2] + 8*a*b*Sinh[c + d*x] + 2*a^2*d*x*Sinh[2*(c + d*x)]))/d^2))/(8 *a^3*(a + b*Csch[c + d*x]))
Result contains complex when optimal does not.
Time = 1.59 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.99, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.731, Rules used = {6128, 6113, 3042, 3791, 17, 6099, 17, 3042, 26, 3777, 3042, 3117, 3803, 25, 2694, 27, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 6128 |
| \(\displaystyle \int \frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{a \sinh (c+d x)+b}dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x) \cosh ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{a}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {\frac {1}{2} \int (e+f x)dx-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \int (e+f x)dx}{a^2}+\frac {\int (e+f x) \sinh (c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{b+a \sinh (c+d x)}dx}{a^2}+\frac {\int (e+f x) \sinh (c+d x)dx}{a}-\frac {b (e+f x)^2}{2 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{b-i a \sin (i c+i d x)}dx}{a^2}+\frac {\int -i (e+f x) \sin (i c+i d x)dx}{a}-\frac {b (e+f x)^2}{2 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \int (e+f x) \sin (i c+i d x)dx}{a}-\frac {b (e+f x)^2}{2 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{a}-\frac {b (e+f x)^2}{2 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{a}-\frac {b (e+f x)^2}{2 a^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {b (e+f x)^2}{2 a^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {2 \left (a^2+b^2\right ) \int -\frac {e^{c+d x} (e+f x)}{-e^{2 (c+d x)} a+a-2 b e^{c+d x}}dx}{a^2}-\frac {b (e+f x)^2}{2 a^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)}{-e^{2 (c+d x)} a+a-2 b e^{c+d x}}dx}{a^2}-\frac {b (e+f x)^2}{2 a^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \int -\frac {e^{c+d x} (e+f x)}{2 \left (e^{c+d x} a+b-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {a \int -\frac {e^{c+d x} (e+f x)}{2 \left (e^{c+d x} a+b+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^2}{2 a^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \int \frac {e^{c+d x} (e+f x)}{e^{c+d x} a+b+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {a \int \frac {e^{c+d x} (e+f x)}{e^{c+d x} a+b-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^2}{2 a^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \left (\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {f \int \log \left (\frac {e^{c+d x} a}{b+\sqrt {a^2+b^2}}+1\right )dx}{a d}\right )}{2 \sqrt {a^2+b^2}}-\frac {a \left (\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {f \int \log \left (\frac {e^{c+d x} a}{b-\sqrt {a^2+b^2}}+1\right )dx}{a d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^2}{2 a^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \left (\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} a}{b+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{a d^2}\right )}{2 \sqrt {a^2+b^2}}-\frac {a \left (\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} a}{b-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{a d^2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^2}{2 a^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}-\frac {a \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^2}{2 a^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{a}\right )}{a}\) |
((e + f*x)^2/(4*f) - (f*Cosh[c + d*x]^2)/(4*d^2) + ((e + f*x)*Cosh[c + d*x ]*Sinh[c + d*x])/(2*d))/a - (b*(-1/2*(b*(e + f*x)^2)/(a^2*f) - (2*(a^2 + b ^2)*(-1/2*(a*(((e + f*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/( a*d) + (f*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a*d^2)))/ Sqrt[a^2 + b^2] + (a*(((e + f*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b ^2])])/(a*d) + (f*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a *d^2)))/(2*Sqrt[a^2 + b^2])))/a^2 - (I*((I*(e + f*x)*Cosh[c + d*x])/d - (I *f*Sinh[c + d*x])/d^2))/a))/a
3.1.24.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
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[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
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[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. )*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-a/b^2 Int[(e + f*x)^m*Cos h[c + d*x]^(n - 2), x], x] + (Simp[1/b Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)*Sinh[c + d*x], x], x] + Simp[(a^2 + b^2)/b^2 Int[(e + f*x)^m*(Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/b Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x] - S imp[a/b Int[(e + f*x)^m*Cosh[c + d*x]^p*(Sinh[c + d*x]^(n - 1)/(a + b*Sin h[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[ n, 0] && IGtQ[p, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Int[(e + f*x)^m*Sinh[c + d*x]*(F [c + d*x]^n/(b + a*Sinh[c + d*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && H yperbolicQ[F] && IntegersQ[m, n]
Leaf count of result is larger than twice the leaf count of optimal. \(1011\) vs. \(2(297)=594\).
Time = 4.51 (sec) , antiderivative size = 1012, normalized size of antiderivative = 3.09
1/4*f*x^2/a+1/2*e*x/a+1/2*b^2*f*x^2/a^3+b^2*e*x/a^3+1/16*(2*d*f*x+2*d*e-f) /a/d^2*exp(2*d*x+2*c)-1/2*b*(d*f*x+d*e-f)/a^2/d^2*exp(d*x+c)-1/2*b*(d*f*x+ d*e+f)/a^2/d^2*exp(-d*x-c)-1/16*(2*d*f*x+2*d*e+f)/a/d^2*exp(-2*d*x-2*c)+2/ d/a*b*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*exp(d*x+c)+2*b)/(a^2+b^2)^(1/2))+ 2/d/a^3*b^3*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*exp(d*x+c)+2*b)/(a^2+b^2)^( 1/2))-1/d/a*b*f/(a^2+b^2)^(1/2)*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b+( a^2+b^2)^(1/2)))*x+1/d/a*b*f/(a^2+b^2)^(1/2)*ln((a*exp(d*x+c)+(a^2+b^2)^(1 /2)+b)/(b+(a^2+b^2)^(1/2)))*x-1/d^2/a*b*f/(a^2+b^2)^(1/2)*ln((-a*exp(d*x+c )+(a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))*c+1/d^2/a*b*f/(a^2+b^2)^(1/2)*l n((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))*c-1/d^2/a*b*f/(a^2 +b^2)^(1/2)*dilog((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))+ 1/d^2/a*b*f/(a^2+b^2)^(1/2)*dilog((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2 +b^2)^(1/2)))-1/d/a^3*b^3*f/(a^2+b^2)^(1/2)*ln((-a*exp(d*x+c)+(a^2+b^2)^(1 /2)-b)/(-b+(a^2+b^2)^(1/2)))*x+1/d/a^3*b^3*f/(a^2+b^2)^(1/2)*ln((a*exp(d*x +c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))*x-1/d^2/a^3*b^3*f/(a^2+b^2)^(1 /2)*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))*c+1/d^2/a^3 *b^3*f/(a^2+b^2)^(1/2)*ln((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1 /2)))*c-1/d^2/a^3*b^3*f/(a^2+b^2)^(1/2)*dilog((-a*exp(d*x+c)+(a^2+b^2)^(1/ 2)-b)/(-b+(a^2+b^2)^(1/2)))+1/d^2/a^3*b^3*f/(a^2+b^2)^(1/2)*dilog((a*exp(d *x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))-2/d^2/a*b*f*c/(a^2+b^2)^(...
Leaf count of result is larger than twice the leaf count of optimal. 1284 vs. \(2 (295) = 590\).
Time = 0.30 (sec) , antiderivative size = 1284, normalized size of antiderivative = 3.93 \[ \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \]
-1/16*(2*a^2*d*f*x - (2*a^2*d*f*x + 2*a^2*d*e - a^2*f)*cosh(d*x + c)^4 - ( 2*a^2*d*f*x + 2*a^2*d*e - a^2*f)*sinh(d*x + c)^4 + 2*a^2*d*e + 8*(a*b*d*f* x + a*b*d*e - a*b*f)*cosh(d*x + c)^3 + 4*(2*a*b*d*f*x + 2*a*b*d*e - 2*a*b* f - (2*a^2*d*f*x + 2*a^2*d*e - a^2*f)*cosh(d*x + c))*sinh(d*x + c)^3 + a^2 *f - 4*((a^2 + 2*b^2)*d^2*f*x^2 + 2*(a^2 + 2*b^2)*d^2*e*x)*cosh(d*x + c)^2 - 2*(2*(a^2 + 2*b^2)*d^2*f*x^2 + 4*(a^2 + 2*b^2)*d^2*e*x + 3*(2*a^2*d*f*x + 2*a^2*d*e - a^2*f)*cosh(d*x + c)^2 - 12*(a*b*d*f*x + a*b*d*e - a*b*f)*c osh(d*x + c))*sinh(d*x + c)^2 + 16*(a*b*f*cosh(d*x + c)^2 + 2*a*b*f*cosh(d *x + c)*sinh(d*x + c) + a*b*f*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*dilog ((b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))* sqrt((a^2 + b^2)/a^2) - a)/a + 1) - 16*(a*b*f*cosh(d*x + c)^2 + 2*a*b*f*co sh(d*x + c)*sinh(d*x + c) + a*b*f*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*d ilog((b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) - 16*((a*b*d*e - a*b*c*f)*cosh(d*x + c)^2 + 2*(a*b*d*e - a*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d*e - a*b *c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*log(2*a*cosh(d*x + c) + 2*a*s inh(d*x + c) + 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 16*((a*b*d*e - a*b*c*f)* cosh(d*x + c)^2 + 2*(a*b*d*e - a*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b *d*e - a*b*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*log(2*a*cosh(d*x + c) + 2*a*sinh(d*x + c) - 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 16*((a*b*d*...
\[ \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cosh ^{2}{\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{2}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]
-1/16*(32*(a^2*b*e^c + b^3*e^c)*integrate(x*e^(d*x)/(a^4*e^(2*d*x + 2*c) + 2*a^3*b*e^(d*x + c) - a^4), x) - (4*(a^2*d^2*e^(2*c) + 2*b^2*d^2*e^(2*c)) *x^2 + (2*a^2*d*x*e^(4*c) - a^2*e^(4*c))*e^(2*d*x) - 8*(a*b*d*x*e^(3*c) - a*b*e^(3*c))*e^(d*x) - 8*(a*b*d*x*e^c + a*b*e^c)*e^(-d*x) - (2*a^2*d*x + a ^2)*e^(-2*d*x))*e^(-2*c)/(a^3*d^2))*f - 1/8*e*((4*b*e^(-d*x - c) - a)*e^(2 *d*x + 2*c)/(a^2*d) - 4*(a^2 + 2*b^2)*(d*x + c)/(a^3*d) + (4*b*e^(-d*x - c ) + a*e^(-2*d*x - 2*c))/(a^2*d) + 8*(a^2*b + b^3)*log((a*e^(-d*x - c) - b - sqrt(a^2 + b^2))/(a*e^(-d*x - c) - b + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2 )*a^3*d))
\[ \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{2}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]