Integrand size = 28, antiderivative size = 698 \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {i f (e+f x)^2}{2 a d^2}-\frac {5 i f^2 (e+f x) \arctan \left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {f^2 (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{a d^3}+\frac {5 i f^3 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{2 a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{8 a d^2}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{2 a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{8 a d^2}-\frac {i f^3 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{4 a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{4 a d^3}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right )}{4 a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right )}{4 a d^4}-\frac {f^3 \sec (c+d x)}{4 a d^4}-\frac {9 f (e+f x)^2 \sec (c+d x)}{8 a d^2}-\frac {f^2 (e+f x) \sec ^2(c+d x)}{4 a d^3}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 a d^2}-\frac {(e+f x)^3 \sec ^4(c+d x)}{4 a d}+\frac {f^3 \tan (c+d x)}{4 a d^4}+\frac {f (e+f x)^2 \tan (c+d x)}{2 a d^2}+\frac {f^2 (e+f x) \sec (c+d x) \tan (c+d x)}{4 a d^3}+\frac {3 (e+f x)^3 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {f (e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{4 a d^2}+\frac {(e+f x)^3 \sec ^3(c+d x) \tan (c+d x)}{4 a d} \]
-9/4*I*f^3*polylog(4,-I*exp(I*(d*x+c)))/a/d^4+9/4*I*f^3*polylog(4,I*exp(I* (d*x+c)))/a/d^4-1/2*I*f*(f*x+e)^2/a/d^2+f^2*(f*x+e)*ln(1+exp(2*I*(d*x+c))) /a/d^3-5/2*I*f^3*polylog(2,I*exp(I*(d*x+c)))/a/d^4-5*I*f^2*(f*x+e)*arctan( exp(I*(d*x+c)))/a/d^3-3/4*I*(f*x+e)^3*arctan(exp(I*(d*x+c)))/a/d-9/8*I*f*( f*x+e)^2*polylog(2,I*exp(I*(d*x+c)))/a/d^2+9/8*I*f*(f*x+e)^2*polylog(2,-I* exp(I*(d*x+c)))/a/d^2-9/4*f^2*(f*x+e)*polylog(3,-I*exp(I*(d*x+c)))/a/d^3+9 /4*f^2*(f*x+e)*polylog(3,I*exp(I*(d*x+c)))/a/d^3+5/2*I*f^3*polylog(2,-I*ex p(I*(d*x+c)))/a/d^4-1/2*I*f^3*polylog(2,-exp(2*I*(d*x+c)))/a/d^4-1/4*f^3*s ec(d*x+c)/a/d^4-9/8*f*(f*x+e)^2*sec(d*x+c)/a/d^2-1/4*f^2*(f*x+e)*sec(d*x+c )^2/a/d^3-1/4*f*(f*x+e)^2*sec(d*x+c)^3/a/d^2-1/4*(f*x+e)^3*sec(d*x+c)^4/a/ d+1/4*f^3*tan(d*x+c)/a/d^4+1/2*f*(f*x+e)^2*tan(d*x+c)/a/d^2+1/4*f^2*(f*x+e )*sec(d*x+c)*tan(d*x+c)/a/d^3+3/8*(f*x+e)^3*sec(d*x+c)*tan(d*x+c)/a/d+1/4* f*(f*x+e)^2*sec(d*x+c)^2*tan(d*x+c)/a/d^2+1/4*(f*x+e)^3*sec(d*x+c)^3*tan(d *x+c)/a/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2278\) vs. \(2(698)=1396\).
Time = 10.14 (sec) , antiderivative size = 2278, normalized size of antiderivative = 3.26 \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Result too large to show} \]
(-3*(6*d^4*e^2*f*x^2 + 8*d^2*f^3*x^2 + 4*d^4*e*f^2*x^3 + d^4*f^3*x^4 - (4* I)*d^4*e^3*x*Cos[c] - (16*I)*d^2*e*f^2*x*Cos[c] - (4*I)*d^3*e^3*Log[-Cos[c + d*x] - I*(-1 + Sin[c + d*x])] - (16*I)*d*e*f^2*Log[-Cos[c + d*x] - I*(- 1 + Sin[c + d*x])] - (12*I)*d^3*e^2*f*x*Log[1 - I*Cos[c + d*x] - Sin[c + d *x]] - (16*I)*d*f^3*x*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]] - (12*I)*d^3* e*f^2*x^2*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]] - (4*I)*d^3*f^3*x^3*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]] - 24*f^3*PolyLog[4, I*Cos[c + d*x] + Sin [c + d*x]] + 24*d*f^2*(e + f*x)*PolyLog[3, I*Cos[c + d*x] + Sin[c + d*x]]* (Cos[c] + I*(-1 + Sin[c])) + 4*f*(4*f^2 + 3*d^2*(e + f*x)^2)*PolyLog[2, I* Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] - Sin[c]) + 4*d^3*e^3*Log[-Cos[ c + d*x] - I*(-1 + Sin[c + d*x])]*(Cos[c] + I*Sin[c]) + 16*d*e*f^2*Log[-Co s[c + d*x] - I*(-1 + Sin[c + d*x])]*(Cos[c] + I*Sin[c]) + 12*d^3*e^2*f*x*L og[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*Sin[c]) + 16*d*f^3*x*Log [1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*Sin[c]) + 12*d^3*e*f^2*x^2 *Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*Sin[c]) + 4*d^3*f^3*x^ 3*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*Sin[c]) + 4*d^4*e^3*x *Sin[c] + 16*d^2*e*f^2*x*Sin[c] + 24*f^3*PolyLog[4, I*Cos[c + d*x] + Sin[c + d*x]]*((-I)*Cos[c] + Sin[c])))/(32*a*d^4*(Cos[c] + I*(-1 + Sin[c]))) - ((Cos[c] + I*Sin[c])*(((28*f^2 + 3*d^2*(e + f*x)^2)^2*(Cos[c] - I*Sin[c])) /(12*d^2*f) + (f*(9*d^2*e^2 + 28*f^2)*PolyLog[2, (-I)*Cos[c + d*x] - Si...
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.
\(\displaystyle \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 5042 |
\(\displaystyle \frac {\int (e+f x)^3 \sec ^5(c+d x)dx}{a}-\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^5dx}{a}-\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}\) |
\(\Big \downarrow \) 4674 |
\(\displaystyle \frac {\frac {f^2 \int (e+f x) \sec ^3(c+d x)dx}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \sec ^3(c+d x)dx-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {f^2 \int (e+f x) \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}\) |
\(\Big \downarrow \) 4673 |
\(\displaystyle \frac {\frac {f^2 \left (\frac {1}{2} \int (e+f x) \sec (c+d x)dx-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {f^2 \left (\frac {1}{2} \int (e+f x) \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {f^2 \left (\frac {1}{2} \left (-\frac {f \int \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {f \int \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {f^2 \left (\frac {1}{2} \left (\frac {i f \int e^{-i (c+d x)} \log \left (1-i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i f \int e^{-i (c+d x)} \log \left (1+i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 4674 |
\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \left (\frac {3 f^2 \int (e+f x) \sec (c+d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \sec (c+d x)dx-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \left (\frac {3 f^2 \int (e+f x) \csc \left (c+d x+\frac {\pi }{2}\right )dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \left (\frac {3 f^2 \left (-\frac {f \int \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {f \int \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (-\frac {3 f \int (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \left (\frac {3 f^2 \left (\frac {i f \int e^{-i (c+d x)} \log \left (1-i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i f \int e^{-i (c+d x)} \log \left (1+i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (-\frac {3 f \int (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (-\frac {3 f \int (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 4909 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \int (e+f x)^2 \sec ^4(c+d x)dx}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \int (e+f x)^2 \csc \left (c+d x+\frac {\pi }{2}\right )^4dx}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 4674 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {f^2 \int \sec ^2(c+d x)dx}{3 d^2}+\frac {2}{3} \int (e+f x)^2 \sec ^2(c+d x)dx-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {f^2 \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx}{3 d^2}+\frac {2}{3} \int (e+f x)^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (-\frac {f^2 \int 1d(-\tan (c+d x))}{3 d^3}+\frac {2}{3} \int (e+f x)^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \int (e+f x)^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 4672 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \left (\frac {2 f \int -((e+f x) \tan (c+d x))dx}{d}+\frac {(e+f x)^2 \tan (c+d x)}{d}\right )+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \int (e+f x) \tan (c+d x)dx}{d}\right )+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \int (e+f x) \tan (c+d x)dx}{d}\right )+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \int \frac {e^{2 i (c+d x)} (e+f x)}{1+e^{2 i (c+d x)}}dx\right )}{d}\right )+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {i f \int \log \left (1+e^{2 i (c+d x)}\right )dx}{2 d}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {f \int e^{-2 i (c+d x)} \log \left (1+e^{2 i (c+d x)}\right )de^{2 i (c+d x)}}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {f^2 \tan (c+d x)}{3 d^3}+\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {i f \int \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )dx}{d}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {i f \int \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )dx}{d}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {f^2 \tan (c+d x)}{3 d^3}+\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {f^2 \left (\frac {1}{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 )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \sec (c+d x) \tan (c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \left (\frac {\sec (c+d x) \tan (c+d x) (e+f x)^3}{2 d}-\frac {3 f \sec (c+d x) (e+f x)^2}{2 d^2}+\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 \arctan \left (e^{i (c+d x)}\right ) (e+f x)^3}{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 {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}\right )\right )}{a}-\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {\tan (c+d x) f^2}{3 d^3}-\frac {(e+f x) \sec ^2(c+d x) f}{3 d^2}+\frac {(e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {2}{3} \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 (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{4 d^2}\right )\right )}{d}\right )\right )}{4 d}}{a}\) |
3.3.81.3.1 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[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 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_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S imp[b^2*((n - 2)/(n - 1)) Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
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[((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. 2040 vs. \(2 (612 ) = 1224\).
Time = 1.18 (sec) , antiderivative size = 2041, normalized size of antiderivative = 2.92
9/8/a/d*e^2*f*ln(1-I*exp(I*(d*x+c)))*x+9/8/a/d^2*e^2*f*ln(1-I*exp(I*(d*x+c )))*c-9/8/a/d*e^2*f*ln(1+I*exp(I*(d*x+c)))*x-9/8/a/d^2*e^2*f*ln(1+I*exp(I* (d*x+c)))*c+9/8/a/d*e*f^2*ln(1-I*exp(I*(d*x+c)))*x^2-9/8/a/d*e*f^2*ln(1+I* exp(I*(d*x+c)))*x^2-9/8/a/d^3*c^2*e*f^2*ln(1-I*exp(I*(d*x+c)))-9/4*I/a/d^3 *e*f^2*c^2*arctan(exp(I*(d*x+c)))+9/4*I/a/d^2*e^2*f*c*arctan(exp(I*(d*x+c) ))-9/4*I/a/d^2*e*f^2*polylog(2,I*exp(I*(d*x+c)))*x+9/4*I/a/d^2*e*f^2*polyl og(2,-I*exp(I*(d*x+c)))*x+9/4/a/d^3*e*f^2*polylog(3,I*exp(I*(d*x+c)))-9/4/ a/d^3*e*f^2*polylog(3,-I*exp(I*(d*x+c)))+2/a/d^4*f^3*c*ln(exp(I*(d*x+c)))- 1/a/d^4*f^3*c*ln(1+exp(2*I*(d*x+c)))+3/8/a/d^4*c^3*f^3*ln(1-I*exp(I*(d*x+c )))-3/8/a/d^4*c^3*f^3*ln(1+I*exp(I*(d*x+c)))+7/2/a/d^3*f^3*ln(1-I*exp(I*(d *x+c)))*x+7/2/a/d^4*f^3*ln(1-I*exp(I*(d*x+c)))*c-2/a/d^3*e*f^2*ln(exp(I*(d *x+c)))+1/a/d^3*e*f^2*ln(1+exp(2*I*(d*x+c)))+3/8/a/d*f^3*ln(1-I*exp(I*(d*x +c)))*x^3+9/4/a/d^3*f^3*polylog(3,I*exp(I*(d*x+c)))*x-3/8/a/d*f^3*ln(1+I*e xp(I*(d*x+c)))*x^3-9/4/a/d^3*f^3*polylog(3,-I*exp(I*(d*x+c)))*x+9/4*I*f^3* polylog(4,I*exp(I*(d*x+c)))/a/d^4+5*I/a/d^4*f^3*c*arctan(exp(I*(d*x+c)))-5 *I/a/d^3*e*f^2*arctan(exp(I*(d*x+c)))-1/4*I*(-6*I*d^3*f^3*x^3*exp(2*I*(d*x +c))-8*I*d^2*f^3*x^2*exp(3*I*(d*x+c))-8*I*d^2*e^2*f*exp(3*I*(d*x+c))-9*I*d ^2*f^3*x^2*exp(5*I*(d*x+c))+9*d^3*e*f^2*x^2*exp(I*(d*x+c))+9*d^3*e^2*f*x*e xp(I*(d*x+c))+I*d^2*f^3*x^2*exp(I*(d*x+c))-2*I*f^3*exp(5*I*(d*x+c))-4*I*f^ 3*exp(3*I*(d*x+c))+3*d^3*e^3*exp(5*I*(d*x+c))+2*d^3*e^3*exp(3*I*(d*x+c)...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2572 vs. \(2 (589) = 1178\).
Time = 0.43 (sec) , antiderivative size = 2572, normalized size of antiderivative = 3.68 \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
1/16*(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 - 4*(2*d ^2*f^3*x^2 + 4*d^2*e*f^2*x + 2*d^2*e^2*f + f^3)*cos(d*x + c)^3 - 2*(3*d^3* f^3*x^3 + 9*d^3*e*f^2*x^2 + 3*d^3*e^3 + 2*d*e*f^2 + (9*d^3*e^2*f + 2*d*f^3 )*x)*cos(d*x + c)^2 - 14*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + d^2*e^2*f)*cos(d*x + c) - 3*((3*I*d^2*f^3*x^2 + 6*I*d^2*e*f^2*x + 3*I*d^2*e^2*f + 4*I*f^3)*c os(d*x + c)^2*sin(d*x + c) + (3*I*d^2*f^3*x^2 + 6*I*d^2*e*f^2*x + 3*I*d^2* e^2*f + 4*I*f^3)*cos(d*x + c)^2)*dilog(I*cos(d*x + c) + sin(d*x + c)) + (( -9*I*d^2*f^3*x^2 - 18*I*d^2*e*f^2*x - 9*I*d^2*e^2*f - 28*I*f^3)*cos(d*x + c)^2*sin(d*x + c) + (-9*I*d^2*f^3*x^2 - 18*I*d^2*e*f^2*x - 9*I*d^2*e^2*f - 28*I*f^3)*cos(d*x + c)^2)*dilog(I*cos(d*x + c) - sin(d*x + c)) - 3*((-3*I *d^2*f^3*x^2 - 6*I*d^2*e*f^2*x - 3*I*d^2*e^2*f - 4*I*f^3)*cos(d*x + c)^2*s in(d*x + c) + (-3*I*d^2*f^3*x^2 - 6*I*d^2*e*f^2*x - 3*I*d^2*e^2*f - 4*I*f^ 3)*cos(d*x + c)^2)*dilog(-I*cos(d*x + c) + sin(d*x + c)) + ((9*I*d^2*f^3*x ^2 + 18*I*d^2*e*f^2*x + 9*I*d^2*e^2*f + 28*I*f^3)*cos(d*x + c)^2*sin(d*x + c) + (9*I*d^2*f^3*x^2 + 18*I*d^2*e*f^2*x + 9*I*d^2*e^2*f + 28*I*f^3)*cos( d*x + c)^2)*dilog(-I*cos(d*x + c) - sin(d*x + c)) + ((3*d^3*e^3 - 9*c*d^2* e^2*f + (9*c^2 + 28)*d*e*f^2 - (3*c^3 + 28*c)*f^3)*cos(d*x + c)^2*sin(d*x + c) + (3*d^3*e^3 - 9*c*d^2*e^2*f + (9*c^2 + 28)*d*e*f^2 - (3*c^3 + 28*c)* f^3)*cos(d*x + c)^2)*log(cos(d*x + c) + I*sin(d*x + c) + I) - 3*((d^3*e^3 - 3*c*d^2*e^2*f + (3*c^2 + 4)*d*e*f^2 - (c^3 + 4*c)*f^3)*cos(d*x + c)^2...
\[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{3} \sec ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{3} x^{3} \sec ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e f^{2} x^{2} \sec ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e^{2} f x \sec ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
(Integral(e**3*sec(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(f**3*x**3 *sec(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(3*e*f**2*x**2*sec(c + d *x)**3/(sin(c + d*x) + 1), x) + Integral(3*e**2*f*x*sec(c + d*x)**3/(sin(c + d*x) + 1), x))/a
Exception generated. \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sec \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Hanged} \]