Integrand size = 28, antiderivative size = 600 \[ \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f^3 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4} \]
12*I*f^2*(f*x+e)*polylog(2,I*exp(I*(d*x+c)))/a/d^3-6*f^2*(f*x+e)*arctanh(e xp(I*(d*x+c)))/a/d^3-3*(f*x+e)^3*arctanh(exp(I*(d*x+c)))/a/d+(f*x+e)^3*cot (1/2*c+1/4*Pi+1/2*d*x)/a/d+(f*x+e)^3*cot(d*x+c)/a/d-3/2*f*(f*x+e)^2*csc(d* x+c)/a/d^2-1/2*(f*x+e)^3*cot(d*x+c)*csc(d*x+c)/a/d-6*f*(f*x+e)^2*ln(1-I*ex p(I*(d*x+c)))/a/d^2-3*f*(f*x+e)^2*ln(1-exp(2*I*(d*x+c)))/a/d^2+9/2*I*f*(f* x+e)^2*polylog(2,-exp(I*(d*x+c)))/a/d^2-9*I*f^3*polylog(4,-exp(I*(d*x+c))) /a/d^4+3*I*f^2*(f*x+e)*polylog(2,exp(2*I*(d*x+c)))/a/d^3+3*I*f^3*polylog(2 ,-exp(I*(d*x+c)))/a/d^4-3*I*f^3*polylog(2,exp(I*(d*x+c)))/a/d^4+9*I*f^3*po lylog(4,exp(I*(d*x+c)))/a/d^4-9*f^2*(f*x+e)*polylog(3,-exp(I*(d*x+c)))/a/d ^3-12*f^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4+9*f^2*(f*x+e)*polylog(3,exp(I* (d*x+c)))/a/d^3-3/2*f^3*polylog(3,exp(2*I*(d*x+c)))/a/d^4-9/2*I*f*(f*x+e)^ 2*polylog(2,exp(I*(d*x+c)))/a/d^2+2*I*(f*x+e)^3/a/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1493\) vs. \(2(600)=1200\).
Time = 20.46 (sec) , antiderivative size = 1493, normalized size of antiderivative = 2.49 \[ \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \]
(3*e^3*Log[Tan[(c + d*x)/2]])/(2*a*d) + (3*e*f^2*Log[Tan[(c + d*x)/2]])/(a *d^3) + (9*e^2*f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) - c*Log[Tan[(c + d*x)/2]] + I*(PolyLog[2, -E^(I*(c + d*x))] - Poly Log[2, E^(I*(c + d*x))])))/(2*a*d^2) + (3*f^3*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) - c*Log[Tan[(c + d*x)/2]] + I*(PolyLo g[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])))/(a*d^4) + (E^(I*c) *f^3*Csc[c]*((2*d^3*x^3)/E^((2*I)*c) + (3*I)*d^2*(1 - E^((-2*I)*c))*x^2*Lo g[1 - E^((-I)*(c + d*x))] + (3*I)*d^2*(1 - E^((-2*I)*c))*x^2*Log[1 + E^((- I)*(c + d*x))] - 6*d*(1 - E^((-2*I)*c))*x*PolyLog[2, -E^((-I)*(c + d*x))] - 6*d*(1 - E^((-2*I)*c))*x*PolyLog[2, E^((-I)*(c + d*x))] + (6*I)*(1 - E^( (-2*I)*c))*PolyLog[3, -E^((-I)*(c + d*x))] + (6*I)*(1 - E^((-2*I)*c))*Poly Log[3, E^((-I)*(c + d*x))]))/(2*a*d^4) - (9*e*f^2*(d^2*x^2*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] - I*d*x*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] + I*d*x*PolyLog[2, Cos[c + d*x] + I*Sin[c + d*x]] + PolyLog[3, -Cos[c + d *x] - I*Sin[c + d*x]] - PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]]))/(a*d^3 ) + (3*f^3*(-2*d^3*x^3*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] + (3*I)*d^2* x^2*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] - (3*I)*d^2*x^2*PolyLog[2, Cos[c + d*x] + I*Sin[c + d*x]] - 6*d*x*PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] + 6*d*x*PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]] - (6*I)*PolyLog[ 4, -Cos[c + d*x] - I*Sin[c + d*x]] + (6*I)*PolyLog[4, Cos[c + d*x] + I*...
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.
| \(\displaystyle \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 5046 |
| \(\displaystyle \frac {\int (e+f x)^3 \csc ^3(c+d x)dx}{a}-\int \frac {(e+f x)^3 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e+f x)^3 \csc (c+d x)^3dx}{a}-\int \frac {(e+f x)^3 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle \frac {\frac {3 f^2 \int (e+f x) \csc (c+d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \csc (c+d x)dx-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\int \frac {(e+f x)^3 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f^2 \int (e+f x) \csc (c+d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \csc (c+d x)dx-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\int \frac {(e+f x)^3 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\int \frac {(e+f x)^3 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {3 f^2 \left (-\frac {f \int \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {f \int \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (-\frac {3 f \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\int \frac {(e+f x)^3 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {3 f^2 \left (\frac {i f \int e^{-i (c+d x)} \log \left (1-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i f \int e^{-i (c+d x)} \log \left (1+e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (-\frac {3 f \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\int \frac {(e+f x)^3 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (-\frac {3 f \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\int \frac {(e+f x)^3 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 5046 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\int (e+f x)^3 \csc ^2(c+d x)dx}{a}+\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\int (e+f x)^3 \csc (c+d x)^2dx}{a}+\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {3 f \int (e+f x)^2 \cot (c+d x)dx}{d}-\frac {(e+f x)^3 \cot (c+d x)}{d}}{a}+\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {3 f \int -(e+f x)^2 \tan \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x)^3 \cot (c+d x)}{d}}{a}+\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {3 f \int (e+f x)^2 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{d}-\frac {(e+f x)^3 \cot (c+d x)}{d}}{a}+\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \int \frac {e^{i (2 c+2 d x+\pi )} (e+f x)^2}{1+e^{i (2 c+2 d x+\pi )}}dx\right )}{d}}{a}+\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \int (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )dx}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {i f \int \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )dx}{2 d}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 5046 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}-\int \frac {(e+f x)^3}{\sin (c+d x) a+a}dx+\frac {\int (e+f x)^3 \csc (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}-\int \frac {(e+f x)^3}{\sin (c+d x) a+a}dx+\frac {\int (e+f x)^3 \csc (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}-\frac {\int (e+f x)^3 \csc ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{2 a}+\frac {\int (e+f x)^3 \csc (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}-\frac {\int (e+f x)^3 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {\int (e+f x)^3 \csc (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}+\frac {-\frac {3 f \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}-\frac {\int (e+f x)^3 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}+\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}-\frac {\int (e+f x)^3 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}+\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}-\frac {\frac {6 f \int (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}+\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}-\frac {\frac {6 f \int -(e+f x)^2 \tan \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )dx}{d}-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}+\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}-\frac {-\frac {6 f \int (e+f x)^2 \tan \left (\frac {1}{4} (2 c+3 \pi )+\frac {d x}{2}\right )dx}{d}-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}+\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \int \frac {e^{\frac {1}{2} i (2 c+2 d x+3 \pi )} (e+f x)^2}{1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}dx\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 d}}{a}+\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \int (e+f x) \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )dx}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {2 \text {arctanh}\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,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}}{a}+\frac {-\frac {\cot (c+d x) \csc (c+d x) (e+f x)^3}{2 d}-\frac {3 f \csc (c+d x) (e+f x)^2}{2 d^2}+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}+\frac {1}{2} \left (-\frac {2 \text {arctanh}\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,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}\right )}{a}-\frac {-\frac {\cot (c+d x) (e+f x)^3}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}-\frac {-\frac {2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) (e+f x)^3}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )dx}{d}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {2 \text {arctanh}\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,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}}{a}+\frac {-\frac {\cot (c+d x) \csc (c+d x) (e+f x)^3}{2 d}-\frac {3 f \csc (c+d x) (e+f x)^2}{2 d^2}+\frac {3 f^2 \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d^2}+\frac {1}{2} \left (-\frac {2 \text {arctanh}\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,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}\right )}{a}-\frac {-\frac {\cot (c+d x) (e+f x)^3}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}-\frac {-\frac {2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) (e+f x)^3}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {f \int e^{-\frac {1}{2} i (2 c+2 d x+3 \pi )} \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )de^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}{d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\) |
3.3.9.3.1 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_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Simp[(2*a)^n Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[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_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[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[(Csc[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. )*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a Int[(e + f*x)^m*Csc[c + d*x]^n, x], x] - Simp[b/a Int[(e + f*x)^m*(Csc[c + d*x]^(n - 1)/(a + b*S in[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ [n, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2325 vs. \(2 (540 ) = 1080\).
Time = 0.72 (sec) , antiderivative size = 2326, normalized size of antiderivative = 3.88
9/2/a/d*f^2*e*ln(1-exp(I*(d*x+c)))*x^2-6/a/d^3*f^2*e*ln(1-exp(I*(d*x+c)))* c-12/a/d^3*f^2*e*ln(1-I*exp(I*(d*x+c)))*c-6/a/d^2*f^2*e*ln(1-exp(I*(d*x+c) ))*x-12/a/d^2*f^2*e*ln(1-I*exp(I*(d*x+c)))*x-6/a/d^2*f^2*e*ln(exp(I*(d*x+c ))+1)*x-9/2/a/d^2*c*e^2*f*ln(exp(I*(d*x+c))-1)-24/a/d^3*e*f^2*c*ln(exp(I*( d*x+c)))+9/2/a/d^2*e^2*f*ln(1-exp(I*(d*x+c)))*c-9/2/a/d*e^2*f*ln(exp(I*(d* x+c))+1)*x+9/2/a/d*e^2*f*ln(1-exp(I*(d*x+c)))*x-9/2/a/d^3*c^2*f^2*e*ln(1-e xp(I*(d*x+c)))+9/2/a/d^3*c^2*f^2*e*ln(exp(I*(d*x+c))-1)+6/a/d^3*c*f^2*e*ln (exp(I*(d*x+c))-1)+6/a/d^3*c*f^2*e*ln(1+exp(2*I*(d*x+c)))-9/2/a/d*f^2*e*ln (exp(I*(d*x+c))+1)*x^2+9*I/a/d^2*e*f^2*polylog(2,-exp(I*(d*x+c)))*x-9*I/a/ d^2*e*f^2*polylog(2,exp(I*(d*x+c)))*x-12*I/a/d^3*e*f^2*c*arctan(exp(I*(d*x +c)))+24*I/a/d^2*e*f^2*c*x+3/2/a/d*e^3*ln(exp(I*(d*x+c))-1)-3/2/a/d*e^3*ln (exp(I*(d*x+c))+1)+(3*I*exp(2*I*(d*x+c))*f*e^2-3*I*f^3*x^2*exp(4*I*(d*x+c) )+3*I*d*e^3*exp(3*I*(d*x+c))-5*d*f^3*x^3*exp(2*I*(d*x+c))+3*d*f^3*x^3*exp( 4*I*(d*x+c))+6*e*f^2*x*exp(3*I*(d*x+c))-I*d*f^3*x^3*exp(I*(d*x+c))+9*d*e*f ^2*x^2*exp(4*I*(d*x+c))+9*d*e^2*f*x*exp(4*I*(d*x+c))-15*d*e*f^2*x^2*exp(2* I*(d*x+c))-6*e*f^2*x*exp(I*(d*x+c))-I*d*e^3*exp(I*(d*x+c))+9*I*d*e*f^2*x^2 *exp(3*I*(d*x+c))+9*I*d*e^2*f*x*exp(3*I*(d*x+c))+3*I*f^3*x^2*exp(2*I*(d*x+ c))-3*I*exp(4*I*(d*x+c))*f*e^2+12*d*e*f^2*x^2+12*d*e^2*f*x-3*f^3*x^2*exp(I *(d*x+c))-3*e^2*f*exp(I*(d*x+c))+3*f^3*x^2*exp(3*I*(d*x+c))+3*d*e^3*exp(4* I*(d*x+c))+3*exp(3*I*(d*x+c))*f*e^2+3*I*d*f^3*x^3*exp(3*I*(d*x+c))-5*d*...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 7842 vs. \(2 (522) = 1044\).
Time = 0.55 (sec) , antiderivative size = 7842, normalized size of antiderivative = 13.07 \[ \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
\[ \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{3} \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{3} x^{3} \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e f^{2} x^{2} \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e^{2} f x \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
(Integral(e**3*csc(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(f**3*x**3 *csc(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(3*e*f**2*x**2*csc(c + d *x)**3/(sin(c + d*x) + 1), x) + Integral(3*e**2*f*x*csc(c + d*x)**3/(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. 12815 vs. \(2 (522) = 1044\).
Time = 10.83 (sec) , antiderivative size = 12815, normalized size of antiderivative = 21.36 \[ \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
-1/8*(3*c*e^2*f*((3*sin(d*x + c)/(cos(d*x + c) + 1) + 20*sin(d*x + c)^2/(c os(d*x + c) + 1)^2 - 1)/(a*d*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a*d*sin (d*x + c)^3/(cos(d*x + c) + 1)^3) - (4*sin(d*x + c)/(cos(d*x + c) + 1) - s in(d*x + c)^2/(cos(d*x + c) + 1)^2)/(a*d) + 12*log(sin(d*x + c)/(cos(d*x + c) + 1))/(a*d)) + e^3*((4*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^ 2/(cos(d*x + c) + 1)^2)/a - (3*sin(d*x + c)/(cos(d*x + c) + 1) + 20*sin(d* x + c)^2/(cos(d*x + c) + 1)^2 - 1)/(a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a*sin(d*x + c)^3/(cos(d*x + c) + 1)^3) - 12*log(sin(d*x + c)/(cos(d*x + c) + 1))/a) + 8*(48*I*c^2*d*e*f^2 - 16*I*c^3*f^3 - 24*(-I*d^2*e^2*f + 2*I* c*d*e*f^2 - I*c^2*f^3 - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(5*d*x + 5* c) + (-I*d^2*e^2*f + 2*I*c*d*e*f^2 - I*c^2*f^3)*cos(4*d*x + 4*c) + 2*(d^2* e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(3*d*x + 3*c) + 2*(I*d^2*e^2*f - 2*I*c*d *e*f^2 + I*c^2*f^3)*cos(2*d*x + 2*c) - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3) *cos(d*x + c) + (-I*d^2*e^2*f + 2*I*c*d*e*f^2 - I*c^2*f^3)*sin(5*d*x + 5*c ) + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*sin(4*d*x + 4*c) + 2*(I*d^2*e^2*f - 2*I*c*d*e*f^2 + I*c^2*f^3)*sin(3*d*x + 3*c) - 2*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*sin(2*d*x + 2*c) + (-I*d^2*e^2*f + 2*I*c*d*e*f^2 - I*c^2*f^3)* sin(d*x + c))*arctan2(sin(d*x + c) + 1, cos(d*x + c)) - 24*(I*(d*x + c)^2* f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c) + ((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(5*d*x + 5*c) + (I*(d*x + c)^2*f^3 + 2*(I*d*e*f^2 ...
\[ \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \csc \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Hanged} \]