Integrand size = 26, antiderivative size = 502 \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 i f (e+f x)^2}{2 a d^2}-\frac {6 i f^2 (e+f x) \arctan \left (e^{i (c+d x)}\right )}{a d^3}-\frac {i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{a d}+\frac {3 f^2 (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{a d^3}+\frac {3 i f^3 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{a d^4}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{2 a d^2}-\frac {3 i f^3 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^4}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{2 a d^2}-\frac {3 i f^3 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f^3 \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right )}{a d^4}+\frac {3 i f^3 \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right )}{a d^4}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \sec ^2(c+d x)}{2 a d}+\frac {3 f (e+f x)^2 \tan (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \sec (c+d x) \tan (c+d x)}{2 a d} \]
-3*I*f^3*polylog(4,-I*exp(I*(d*x+c)))/a/d^4+3*I*f^3*polylog(4,I*exp(I*(d*x +c)))/a/d^4-3/2*I*f*(f*x+e)^2/a/d^2+3*f^2*(f*x+e)*ln(1+exp(2*I*(d*x+c)))/a /d^3-3*I*f^3*polylog(2,I*exp(I*(d*x+c)))/a/d^4-6*I*f^2*(f*x+e)*arctan(exp( I*(d*x+c)))/a/d^3-I*(f*x+e)^3*arctan(exp(I*(d*x+c)))/a/d-3/2*I*f*(f*x+e)^2 *polylog(2,I*exp(I*(d*x+c)))/a/d^2+3/2*I*f*(f*x+e)^2*polylog(2,-I*exp(I*(d *x+c)))/a/d^2-3*f^2*(f*x+e)*polylog(3,-I*exp(I*(d*x+c)))/a/d^3+3*f^2*(f*x+ e)*polylog(3,I*exp(I*(d*x+c)))/a/d^3+3*I*f^3*polylog(2,-I*exp(I*(d*x+c)))/ a/d^4-3/2*I*f^3*polylog(2,-exp(2*I*(d*x+c)))/a/d^4-3/2*f*(f*x+e)^2*sec(d*x +c)/a/d^2-1/2*(f*x+e)^3*sec(d*x+c)^2/a/d+3/2*f*(f*x+e)^2*tan(d*x+c)/a/d^2+ 1/2*(f*x+e)^3*sec(d*x+c)*tan(d*x+c)/a/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1025\) vs. \(2(502)=1004\).
Time = 8.46 (sec) , antiderivative size = 1025, normalized size of antiderivative = 2.04 \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )}{8 a \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )}-\frac {(\cos (c)+i \sin (c)) \left (\frac {(e+f x)^3 \log (1-i \cos (c+d x)-\sin (c+d x)) (1-i \cos (c)-\sin (c))}{d}+\frac {(e+f x)^4 (\cos (c)-i \sin (c))}{4 f}+\frac {3 f \left (d^2 (e+f x)^2 \operatorname {PolyLog}(2,i \cos (c+d x)+\sin (c+d x))-2 i d f (e+f x) \operatorname {PolyLog}(3,i \cos (c+d x)+\sin (c+d x))-2 f^2 \operatorname {PolyLog}(4,i \cos (c+d x)+\sin (c+d x))\right ) (\cos (c)+i (-1+\sin (c))) (i \cos (c)+\sin (c))}{d^4}\right )}{2 a (\cos (c)+i (-1+\sin (c)))}-\frac {(\cos (c)+i \sin (c)) \left (\frac {\left (12 f^2+d^2 (e+f x)^2\right )^2 (\cos (c)-i \sin (c))}{4 d^2 f}+\frac {3 f \left (d^2 e^2+4 f^2\right ) \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i \sin (c)) (1-i \cos (c)+\sin (c))}{d^2}+6 e f^2 x \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i \sin (c)) (1-i \cos (c)+\sin (c))+3 f^3 x^2 \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (1+\sin (c)))-\frac {6 f^3 \operatorname {PolyLog}(4,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (1+\sin (c)))}{d^2}-\frac {3 f \left (d^2 e^2+4 f^2\right ) x \log (1+i \cos (c+d x)+\sin (c+d x)) (\cos (c)-i \sin (c)) (\cos (c)+i (1+\sin (c)))}{d}-3 d e f^2 x^2 \log (1+i \cos (c+d x)+\sin (c+d x)) (\cos (c)-i \sin (c)) (\cos (c)+i (1+\sin (c)))-d f^3 x^3 \log (1+i \cos (c+d x)+\sin (c+d x)) (\cos (c)-i \sin (c)) (\cos (c)+i (1+\sin (c)))-\frac {e \left (d^2 e^2+12 f^2\right ) \log (\cos (c+d x)+i (1+\sin (c+d x))) (\cos (c)-i \sin (c)) (\cos (c)+i (1+\sin (c)))}{d}-\frac {6 e f^2 \operatorname {PolyLog}(3,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i \sin (c)) (\cos (c)+i (1+\sin (c)))}{d}-\frac {6 f^3 x \operatorname {PolyLog}(3,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i \sin (c)) (\cos (c)+i (1+\sin (c)))}{d}+e \left (d^2 e^2+12 f^2\right ) x (i \cos (c)+\sin (c)) (\cos (c)+i (1+\sin (c)))\right )}{2 a d^2 (\cos (c)+i (1+\sin (c)))}-\frac {(e+f x)^3}{2 a d \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {3 \left (e^2 f \sin \left (\frac {d x}{2}\right )+2 e f^2 x \sin \left (\frac {d x}{2}\right )+f^3 x^2 \sin \left (\frac {d x}{2}\right )\right )}{a d^2 \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]
(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))/(8*a*(Cos[c/2] - Sin[c/2]) *(Cos[c/2] + Sin[c/2])) - ((Cos[c] + I*Sin[c])*(((e + f*x)^3*Log[1 - I*Cos [c + d*x] - Sin[c + d*x]]*(1 - I*Cos[c] - Sin[c]))/d + ((e + f*x)^4*(Cos[c ] - I*Sin[c]))/(4*f) + (3*f*(d^2*(e + f*x)^2*PolyLog[2, I*Cos[c + d*x] + S in[c + d*x]] - (2*I)*d*f*(e + f*x)*PolyLog[3, I*Cos[c + d*x] + Sin[c + d*x ]] - 2*f^2*PolyLog[4, I*Cos[c + d*x] + Sin[c + d*x]])*(Cos[c] + I*(-1 + Si n[c]))*(I*Cos[c] + Sin[c]))/d^4))/(2*a*(Cos[c] + I*(-1 + Sin[c]))) - ((Cos [c] + I*Sin[c])*(((12*f^2 + d^2*(e + f*x)^2)^2*(Cos[c] - I*Sin[c]))/(4*d^2 *f) + (3*f*(d^2*e^2 + 4*f^2)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]* (Cos[c] - I*Sin[c])*(1 - I*Cos[c] + Sin[c]))/d^2 + 6*e*f^2*x*PolyLog[2, (- I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(1 - I*Cos[c] + Sin[c] ) + 3*f^3*x^2*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])) - (6*f^3*PolyLog[4, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])))/d^2 - (3*f*(d^2*e^2 + 4*f^2)*x*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])))/d - 3*d*e*f^2 *x^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])) - d*f^3*x^3*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])) - (e*(d^2*e^2 + 12*f^2)*Log[Cos[c + d*x] + I*(1 + Sin[c + d*x])]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c]) ))/d - (6*e*f^2*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - ...
Time = 2.89 (sec) , antiderivative size = 496, normalized size of antiderivative = 0.99, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {5042, 3042, 4674, 3042, 4669, 2715, 2838, 3011, 4909, 3042, 4672, 25, 3042, 4202, 2620, 2715, 2838, 7163, 2720, 7143}
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.
\(\displaystyle \int \frac {(e+f x)^3 \sec (c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 5042 |
\(\displaystyle \frac {\int (e+f x)^3 \sec ^3(c+d x)dx}{a}-\frac {\int (e+f x)^3 \sec ^2(c+d x) \tan (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{a}-\frac {\int (e+f x)^3 \sec ^2(c+d x) \tan (c+d x)dx}{a}\) |
\(\Big \downarrow \) 4674 |
\(\displaystyle \frac {\frac {3 f^2 \int (e+f x) \sec (c+d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \sec (c+d x)dx-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}-\frac {\int (e+f x)^3 \sec ^2(c+d x) \tan (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {3 f^2 \int (e+f x) \csc \left (c+d x+\frac {\pi }{2}\right )dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}-\frac {\int (e+f x)^3 \sec ^2(c+d x) \tan (c+d x)dx}{a}\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle -\frac {\int (e+f x)^3 \sec ^2(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3 f^2 \left (-\frac {f \int \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {f \int \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (-\frac {3 f \int (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {\int (e+f x)^3 \sec ^2(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3 f^2 \left (\frac {i f \int e^{-i (c+d x)} \log \left (1-i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i f \int e^{-i (c+d x)} \log \left (1+i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (-\frac {3 f \int (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {\int (e+f x)^3 \sec ^2(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {1}{2} \left (-\frac {3 f \int (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {\int (e+f x)^3 \sec ^2(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}\) |
\(\Big \downarrow \) 4909 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^2(c+d x)}{2 d}-\frac {3 f \int (e+f x)^2 \sec ^2(c+d x)dx}{2 d}}{a}+\frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^2(c+d x)}{2 d}-\frac {3 f \int (e+f x)^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2dx}{2 d}}{a}+\frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}\) |
\(\Big \downarrow \) 4672 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^2(c+d x)}{2 d}-\frac {3 f \left (\frac {2 f \int -((e+f x) \tan (c+d x))dx}{d}+\frac {(e+f x)^2 \tan (c+d x)}{d}\right )}{2 d}}{a}+\frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \int (e+f x) \tan (c+d x)dx}{d}\right )}{2 d}}{a}+\frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \int (e+f x) \tan (c+d x)dx}{d}\right )}{2 d}}{a}+\frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \int \frac {e^{2 i (c+d x)} (e+f x)}{1+e^{2 i (c+d x)}}dx\right )}{d}\right )}{2 d}}{a}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {i f \int \log \left (1+e^{2 i (c+d x)}\right )dx}{2 d}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )}{2 d}}{a}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {f \int e^{-2 i (c+d x)} \log \left (1+e^{2 i (c+d x)}\right )de^{2 i (c+d x)}}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )}{2 d}}{a}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )}{2 d}}{a}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {i f \int \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )dx}{d}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {i f \int \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )dx}{d}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )}{2 d}}{a}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )}{2 d}}{a}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}+\frac {1}{2} \left (-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right )}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right )}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}\right )-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )}{2 d}}{a}\) |
((3*f^2*(((-2*I)*(e + f*x)*ArcTan[E^(I*(c + d*x))])/d + (I*f*PolyLog[2, (- I)*E^(I*(c + d*x))])/d^2 - (I*f*PolyLog[2, I*E^(I*(c + d*x))])/d^2))/d^2 + (((-2*I)*(e + f*x)^3*ArcTan[E^(I*(c + d*x))])/d + (3*f*((I*(e + f*x)^2*Po lyLog[2, (-I)*E^(I*(c + d*x))])/d - ((2*I)*f*(((-I)*(e + f*x)*PolyLog[3, ( -I)*E^(I*(c + d*x))])/d + (f*PolyLog[4, (-I)*E^(I*(c + d*x))])/d^2))/d))/d - (3*f*((I*(e + f*x)^2*PolyLog[2, I*E^(I*(c + d*x))])/d - ((2*I)*f*(((-I) *(e + f*x)*PolyLog[3, I*E^(I*(c + d*x))])/d + (f*PolyLog[4, I*E^(I*(c + d* x))])/d^2))/d))/d)/2 - (3*f*(e + f*x)^2*Sec[c + d*x])/(2*d^2) + ((e + f*x) ^3*Sec[c + d*x]*Tan[c + d*x])/(2*d))/a - (((e + f*x)^3*Sec[c + d*x]^2)/(2* d) - (3*f*((-2*f*(((I/2)*(e + f*x)^2)/f - (2*I)*(((-1/2*I)*(e + f*x)*Log[1 + E^((2*I)*(c + d*x))])/d - (f*PolyLog[2, -E^((2*I)*(c + d*x))])/(4*d^2)) ))/d + ((e + f*x)^2*Tan[c + d*x])/d))/(2*d))/a
3.3.69.3.1 Defintions of rubi rules used
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[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[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[(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[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, 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[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ (n - 1)) Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c , d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b _.)*(x_)]^(p_.), x_Symbol] :> Simp[(c + d*x)^m*(Sec[a + b*x]^n/(b*n)), x] - Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; FreeQ[{ a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sec[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_. )*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a Int[(e + f*x)^m*Sec[c + d*x]^(n + 2), x], x] - Simp[1/b Int[(e + f*x)^m*Sec[c + d*x]^(n + 1)*Tan [c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a ^2 - b^2, 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1195 vs. \(2 (444 ) = 888\).
Time = 0.67 (sec) , antiderivative size = 1196, normalized size of antiderivative = 2.38
-I*(d*exp(I*(d*x+c))*f^3*x^3+3*d*exp(I*(d*x+c))*e*f^2*x^2+3*d*exp(I*(d*x+c ))*e^2*f*x+d*exp(I*(d*x+c))*e^3+3*f^3*x^2-3*I*f^3*x^2*exp(I*(d*x+c))+6*e*f ^2*x-6*I*e*f^2*x*exp(I*(d*x+c))+3*e^2*f-3*I*e^2*f*exp(I*(d*x+c)))/d^2/(exp (I*(d*x+c))+I)^2/a+3*I/a/d^2*e^2*f*c*arctan(exp(I*(d*x+c)))-3*I/a/d^3*e*f^ 2*c^2*arctan(exp(I*(d*x+c)))-3*I/a/d^2*e*f^2*polylog(2,I*exp(I*(d*x+c)))*x +3*I/a/d^2*e*f^2*polylog(2,-I*exp(I*(d*x+c)))*x+I/a/d^4*f^3*c^3*arctan(exp (I*(d*x+c)))+3/2/a/d*e^2*f*ln(1-I*exp(I*(d*x+c)))*x+3/2/a/d^2*e^2*f*ln(1-I *exp(I*(d*x+c)))*c-3/2/a/d*e^2*f*ln(1+I*exp(I*(d*x+c)))*x-3/2/a/d^2*e^2*f* ln(1+I*exp(I*(d*x+c)))*c+3/2/a/d*e*f^2*ln(1-I*exp(I*(d*x+c)))*x^2-3/2/a/d* e*f^2*ln(1+I*exp(I*(d*x+c)))*x^2-3/2/a/d^3*c^2*e*f^2*ln(1-I*exp(I*(d*x+c)) )+3/a/d^3*e*f^2*polylog(3,I*exp(I*(d*x+c)))-3/a/d^3*e*f^2*polylog(3,-I*exp (I*(d*x+c)))+6/a/d^4*f^3*c*ln(exp(I*(d*x+c)))-3/a/d^4*f^3*c*ln(1+exp(2*I*( d*x+c)))+1/2/a/d^4*c^3*f^3*ln(1-I*exp(I*(d*x+c)))-1/2/a/d^4*c^3*f^3*ln(1+I *exp(I*(d*x+c)))+6/a/d^3*f^3*ln(1-I*exp(I*(d*x+c)))*x+6/a/d^4*f^3*ln(1-I*e xp(I*(d*x+c)))*c-6/a/d^3*e*f^2*ln(exp(I*(d*x+c)))+3/a/d^3*e*f^2*ln(1+exp(2 *I*(d*x+c)))+1/2/a/d*f^3*ln(1-I*exp(I*(d*x+c)))*x^3+3/a/d^3*f^3*polylog(3, I*exp(I*(d*x+c)))*x-1/2/a/d*f^3*ln(1+I*exp(I*(d*x+c)))*x^3-3/a/d^3*f^3*pol ylog(3,-I*exp(I*(d*x+c)))*x-I/a/d*e^3*arctan(exp(I*(d*x+c)))-3*I/a/d^2*f^3 *x^2-3*I/a/d^4*f^3*c^2-6*I/a/d^4*f^3*polylog(2,I*exp(I*(d*x+c)))+3/2/a/d^3 *c^2*e*f^2*ln(1+I*exp(I*(d*x+c)))-6*I/a/d^3*f^3*c*x+6*I/a/d^4*f^3*c*arc...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1884 vs. \(2 (421) = 842\).
Time = 0.37 (sec) , antiderivative size = 1884, normalized size of antiderivative = 3.75 \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
-1/4*(2*d^3*f^3*x^3 + 6*d^3*e*f^2*x^2 + 6*d^3*e^2*f*x + 2*d^3*e^3 + 6*(d^2 *f^3*x^2 + 2*d^2*e*f^2*x + d^2*e^2*f)*cos(d*x + c) + 3*(I*d^2*f^3*x^2 + 2* I*d^2*e*f^2*x + I*d^2*e^2*f + (I*d^2*f^3*x^2 + 2*I*d^2*e*f^2*x + I*d^2*e^2 *f)*sin(d*x + c))*dilog(I*cos(d*x + c) + sin(d*x + c)) + 3*(I*d^2*f^3*x^2 + 2*I*d^2*e*f^2*x + I*d^2*e^2*f + 4*I*f^3 + (I*d^2*f^3*x^2 + 2*I*d^2*e*f^2 *x + I*d^2*e^2*f + 4*I*f^3)*sin(d*x + c))*dilog(I*cos(d*x + c) - sin(d*x + c)) + 3*(-I*d^2*f^3*x^2 - 2*I*d^2*e*f^2*x - I*d^2*e^2*f + (-I*d^2*f^3*x^2 - 2*I*d^2*e*f^2*x - I*d^2*e^2*f)*sin(d*x + c))*dilog(-I*cos(d*x + c) + si n(d*x + c)) + 3*(-I*d^2*f^3*x^2 - 2*I*d^2*e*f^2*x - I*d^2*e^2*f - 4*I*f^3 + (-I*d^2*f^3*x^2 - 2*I*d^2*e*f^2*x - I*d^2*e^2*f - 4*I*f^3)*sin(d*x + c)) *dilog(-I*cos(d*x + c) - sin(d*x + c)) - (d^3*e^3 - 3*c*d^2*e^2*f + 3*(c^2 + 4)*d*e*f^2 - (c^3 + 12*c)*f^3 + (d^3*e^3 - 3*c*d^2*e^2*f + 3*(c^2 + 4)* d*e*f^2 - (c^3 + 12*c)*f^3)*sin(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c ) + I) + (d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3 + (d^3*e^3 - 3 *c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*sin(d*x + c))*log(cos(d*x + c) - I *sin(d*x + c) + I) - (d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*c*d^2*e^2*f - 3*c^ 2*d*e*f^2 + (c^3 + 12*c)*f^3 + 3*(d^3*e^2*f + 4*d*f^3)*x + (d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + (c^3 + 12*c)*f^3 + 3*(d^ 3*e^2*f + 4*d*f^3)*x)*sin(d*x + c))*log(I*cos(d*x + c) + sin(d*x + c) + 1) + (d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + 3*c*d^2*e^2*f - 3*c...
\[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{3} \sec {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{3} x^{3} \sec {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e f^{2} x^{2} \sec {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e^{2} f x \sec {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
(Integral(e**3*sec(c + d*x)/(sin(c + d*x) + 1), x) + Integral(f**3*x**3*se c(c + d*x)/(sin(c + d*x) + 1), x) + Integral(3*e*f**2*x**2*sec(c + d*x)/(s in(c + d*x) + 1), x) + Integral(3*e**2*f*x*sec(c + d*x)/(sin(c + d*x) + 1) , x))/a
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3854 vs. \(2 (421) = 842\).
Time = 1.04 (sec) , antiderivative size = 3854, normalized size of antiderivative = 7.68 \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
1/4*(3*c*e^2*f*(2/(a*d*sin(d*x + c) + a*d) - log(sin(d*x + c) + 1)/(a*d) + log(sin(d*x + c) - 1)/(a*d)) + e^3*(log(sin(d*x + c) + 1)/a - log(sin(d*x + c) - 1)/a - 2/(a*sin(d*x + c) + a)) - 4*(12*d^2*e^2*f - 24*c*d*e*f^2 + 12*c^2*f^3 + 2*(3*(c^2 + 4)*d*e*f^2 - (c^3 + 12*c)*f^3 - (3*(c^2 + 4)*d*e* f^2 - (c^3 + 12*c)*f^3)*cos(2*d*x + 2*c) + 2*(3*(-I*c^2 - 4*I)*d*e*f^2 + ( I*c^3 + 12*I*c)*f^3)*cos(d*x + c) + (3*(-I*c^2 - 4*I)*d*e*f^2 + (I*c^3 + 1 2*I*c)*f^3)*sin(2*d*x + 2*c) + 2*(3*(c^2 + 4)*d*e*f^2 - (c^3 + 12*c)*f^3)* sin(d*x + c))*arctan2(sin(d*x + c) + 1, cos(d*x + c)) - 2*(3*c^2*d*e*f^2 - c^3*f^3 - (3*c^2*d*e*f^2 - c^3*f^3)*cos(2*d*x + 2*c) - 2*(3*I*c^2*d*e*f^2 - I*c^3*f^3)*cos(d*x + c) - (3*I*c^2*d*e*f^2 - I*c^3*f^3)*sin(2*d*x + 2*c ) + 2*(3*c^2*d*e*f^2 - c^3*f^3)*sin(d*x + c))*arctan2(sin(d*x + c) - 1, co s(d*x + c)) - 2*((d*x + c)^3*f^3 + 3*(d*e*f^2 - c*f^3)*(d*x + c)^2 + 3*(d^ 2*e^2*f - 2*c*d*e*f^2 + (c^2 + 4)*f^3)*(d*x + c) - ((d*x + c)^3*f^3 + 3*(d *e*f^2 - c*f^3)*(d*x + c)^2 + 3*(d^2*e^2*f - 2*c*d*e*f^2 + (c^2 + 4)*f^3)* (d*x + c))*cos(2*d*x + 2*c) - 2*(I*(d*x + c)^3*f^3 + 3*(I*d*e*f^2 - I*c*f^ 3)*(d*x + c)^2 + 3*(I*d^2*e^2*f - 2*I*c*d*e*f^2 + (I*c^2 + 4*I)*f^3)*(d*x + c))*cos(d*x + c) - (I*(d*x + c)^3*f^3 + 3*(I*d*e*f^2 - I*c*f^3)*(d*x + c )^2 + 3*(I*d^2*e^2*f - 2*I*c*d*e*f^2 + (I*c^2 + 4*I)*f^3)*(d*x + c))*sin(2 *d*x + 2*c) + 2*((d*x + c)^3*f^3 + 3*(d*e*f^2 - c*f^3)*(d*x + c)^2 + 3*(d^ 2*e^2*f - 2*c*d*e*f^2 + (c^2 + 4)*f^3)*(d*x + c))*sin(d*x + c))*arctan2...
\[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sec \left (d x + c\right )}{a \sin \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Hanged} \]