Integrand size = 28, antiderivative size = 392 \[ \int \frac {(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 i (e+f x)^2}{a d}-\frac {3 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {f^2 \text {arctanh}(\cos (c+d x))}{a d^3}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {f (e+f x) \csc (c+d x)}{a d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}+\frac {4 i f^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {3 f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3} \]
2*I*(f*x+e)^2/a/d-3*(f*x+e)^2*arctanh(exp(I*(d*x+c)))/a/d-f^2*arctanh(cos( d*x+c))/a/d^3+(f*x+e)^2*cot(1/2*c+1/4*Pi+1/2*d*x)/a/d+(f*x+e)^2*cot(d*x+c) /a/d-f*(f*x+e)*csc(d*x+c)/a/d^2-1/2*(f*x+e)^2*cot(d*x+c)*csc(d*x+c)/a/d-4* f*(f*x+e)*ln(1-I*exp(I*(d*x+c)))/a/d^2-2*f*(f*x+e)*ln(1-exp(2*I*(d*x+c)))/ a/d^2+3*I*f*(f*x+e)*polylog(2,-exp(I*(d*x+c)))/a/d^2+4*I*f^2*polylog(2,I*e xp(I*(d*x+c)))/a/d^3-3*I*f*(f*x+e)*polylog(2,exp(I*(d*x+c)))/a/d^2+I*f^2*p olylog(2,exp(2*I*(d*x+c)))/a/d^3-3*f^2*polylog(3,-exp(I*(d*x+c)))/a/d^3+3* f^2*polylog(3,exp(I*(d*x+c)))/a/d^3
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(951\) vs. \(2(392)=784\).
Time = 12.37 (sec) , antiderivative size = 951, normalized size of antiderivative = 2.43 \[ \int \frac {(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {8 \left (2 i d^2 e f x+i d^2 f^2 x^2-3 d^2 e^2 \text {arctanh}(\cos (c+d x)+i \sin (c+d x))-2 f^2 \text {arctanh}(\cos (c+d x)+i \sin (c+d x))-6 d^2 e f x \text {arctanh}(\cos (c+d x)+i \sin (c+d x))-3 d^2 f^2 x^2 \text {arctanh}(\cos (c+d x)+i \sin (c+d x))+2 d^2 e f x \cot (c)+d^2 f^2 x^2 \cot (c)-2 d e f \log (1-\cos (2 (c+d x))-i \sin (2 (c+d x)))-2 d f^2 x \log (1-\cos (2 (c+d x))-i \sin (2 (c+d x)))+3 i d f (e+f x) \operatorname {PolyLog}(2,-\cos (c+d x)-i \sin (c+d x))-3 i d f (e+f x) \operatorname {PolyLog}(2,\cos (c+d x)+i \sin (c+d x))+i f^2 \operatorname {PolyLog}(2,\cos (2 (c+d x))+i \sin (2 (c+d x)))-3 f^2 \operatorname {PolyLog}(3,-\cos (c+d x)-i \sin (c+d x))+3 f^2 \operatorname {PolyLog}(3,\cos (c+d x)+i \sin (c+d x))\right )+\frac {32 d^2 f (\cos (c)+i \sin (c)) \left (\frac {(e+f x)^2 (\cos (c)-i \sin (c))}{2 f}-\frac {(e+f x) \log (1+i \cos (c+d x)+\sin (c+d x)) (1+i \cos (c)+\sin (c))}{d}+\frac {f \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (1+\sin (c)))}{d^2}\right )}{\cos (c)+i (1+\sin (c))}-\frac {d (e+f x) \csc (c) \csc ^2(c+d x) \left (2 f \cos \left (\frac {d x}{2}\right )+2 f \cos \left (\frac {3 d x}{2}\right )+5 d e \cos \left (c-\frac {d x}{2}\right )+5 d f x \cos \left (c-\frac {d x}{2}\right )-d e \cos \left (c+\frac {d x}{2}\right )-d f x \cos \left (c+\frac {d x}{2}\right )-2 f \cos \left (2 c+\frac {d x}{2}\right )+d e \cos \left (c+\frac {3 d x}{2}\right )+d f x \cos \left (c+\frac {3 d x}{2}\right )-2 f \cos \left (2 c+\frac {3 d x}{2}\right )-3 d e \cos \left (3 c+\frac {3 d x}{2}\right )-3 d f x \cos \left (3 c+\frac {3 d x}{2}\right )-4 d e \cos \left (c+\frac {5 d x}{2}\right )-4 d f x \cos \left (c+\frac {5 d x}{2}\right )+2 d e \cos \left (3 c+\frac {5 d x}{2}\right )+2 d f x \cos \left (3 c+\frac {5 d x}{2}\right )+d e \sin \left (\frac {d x}{2}\right )+d f x \sin \left (\frac {d x}{2}\right )+d e \sin \left (\frac {3 d x}{2}\right )+d f x \sin \left (\frac {3 d x}{2}\right )+2 f \sin \left (c-\frac {d x}{2}\right )+2 f \sin \left (c+\frac {d x}{2}\right )+3 d e \sin \left (2 c+\frac {d x}{2}\right )+3 d f x \sin \left (2 c+\frac {d x}{2}\right )+2 f \sin \left (c+\frac {3 d x}{2}\right )+d e \sin \left (2 c+\frac {3 d x}{2}\right )+d f x \sin \left (2 c+\frac {3 d x}{2}\right )-2 f \sin \left (3 c+\frac {3 d x}{2}\right )-2 d e \sin \left (2 c+\frac {5 d x}{2}\right )-2 d f x \sin \left (2 c+\frac {5 d x}{2}\right )\right )}{\left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}}{8 a d^3} \]
(8*((2*I)*d^2*e*f*x + I*d^2*f^2*x^2 - 3*d^2*e^2*ArcTanh[Cos[c + d*x] + I*S in[c + d*x]] - 2*f^2*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] - 6*d^2*e*f*x* ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] - 3*d^2*f^2*x^2*ArcTanh[Cos[c + d*x ] + I*Sin[c + d*x]] + 2*d^2*e*f*x*Cot[c] + d^2*f^2*x^2*Cot[c] - 2*d*e*f*Lo g[1 - Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)]] - 2*d*f^2*x*Log[1 - Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)]] + (3*I)*d*f*(e + f*x)*PolyLog[2, -Cos[c + d* x] - I*Sin[c + d*x]] - (3*I)*d*f*(e + f*x)*PolyLog[2, Cos[c + d*x] + I*Sin [c + d*x]] + I*f^2*PolyLog[2, Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]] - 3*f ^2*PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] + 3*f^2*PolyLog[3, Cos[c + d *x] + I*Sin[c + d*x]]) + (32*d^2*f*(Cos[c] + I*Sin[c])*(((e + f*x)^2*(Cos[ c] - I*Sin[c]))/(2*f) - ((e + f*x)*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]* (1 + I*Cos[c] + Sin[c]))/d + (f*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x ]]*(Cos[c] - I*(1 + Sin[c])))/d^2))/(Cos[c] + I*(1 + Sin[c])) - (d*(e + f* x)*Csc[c]*Csc[c + d*x]^2*(2*f*Cos[(d*x)/2] + 2*f*Cos[(3*d*x)/2] + 5*d*e*Co s[c - (d*x)/2] + 5*d*f*x*Cos[c - (d*x)/2] - d*e*Cos[c + (d*x)/2] - d*f*x*C os[c + (d*x)/2] - 2*f*Cos[2*c + (d*x)/2] + d*e*Cos[c + (3*d*x)/2] + d*f*x* Cos[c + (3*d*x)/2] - 2*f*Cos[2*c + (3*d*x)/2] - 3*d*e*Cos[3*c + (3*d*x)/2] - 3*d*f*x*Cos[3*c + (3*d*x)/2] - 4*d*e*Cos[c + (5*d*x)/2] - 4*d*f*x*Cos[c + (5*d*x)/2] + 2*d*e*Cos[3*c + (5*d*x)/2] + 2*d*f*x*Cos[3*c + (5*d*x)/2] + d*e*Sin[(d*x)/2] + d*f*x*Sin[(d*x)/2] + d*e*Sin[(3*d*x)/2] + d*f*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.
| \(\displaystyle \int \frac {(e+f x)^2 \csc ^3(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 5046 |
| \(\displaystyle \frac {\int (e+f x)^2 \csc ^3(c+d x)dx}{a}-\int \frac {(e+f x)^2 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e+f x)^2 \csc (c+d x)^3dx}{a}-\int \frac {(e+f x)^2 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle \frac {\frac {f^2 \int \csc (c+d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^2 \csc (c+d x)dx-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\int \frac {(e+f x)^2 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f^2 \int \csc (c+d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^2 \csc (c+d x)dx-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\int \frac {(e+f x)^2 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {1}{2} \int (e+f x)^2 \csc (c+d x)dx-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\int \frac {(e+f x)^2 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\int \frac {(e+f x)^2 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (-\frac {2 f \int (e+f x) \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {2 f \int (e+f x) \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\int \frac {(e+f x)^2 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\int \frac {(e+f x)^2 \csc ^2(c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 5046 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\int (e+f x)^2 \csc ^2(c+d x)dx}{a}+\int \frac {(e+f x)^2 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\int (e+f x)^2 \csc (c+d x)^2dx}{a}+\int \frac {(e+f x)^2 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {2 f \int (e+f x) \cot (c+d x)dx}{d}-\frac {(e+f x)^2 \cot (c+d x)}{d}}{a}+\int \frac {(e+f x)^2 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {2 f \int -\left ((e+f x) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx}{d}-\frac {(e+f x)^2 \cot (c+d x)}{d}}{a}+\int \frac {(e+f x)^2 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {2 f \int (e+f x) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{d}-\frac {(e+f x)^2 \cot (c+d x)}{d}}{a}+\int \frac {(e+f x)^2 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \int \frac {e^{i (2 c+2 d x+\pi )} (e+f x)}{1+e^{i (2 c+2 d x+\pi )}}dx\right )}{d}}{a}+\int \frac {(e+f x)^2 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {i f \int \log \left (1+e^{i (2 c+2 d x+\pi )}\right )dx}{2 d}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\int \frac {(e+f x)^2 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {f \int e^{-i (2 c+2 d x+\pi )} \log \left (1+e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\int \frac {(e+f x)^2 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}+\int \frac {(e+f x)^2 \csc (c+d x)}{\sin (c+d x) a+a}dx-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 5046 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\int \frac {(e+f x)^2}{\sin (c+d x) a+a}dx+\frac {\int (e+f x)^2 \csc (c+d x)dx}{a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\int \frac {(e+f x)^2}{\sin (c+d x) a+a}dx+\frac {\int (e+f x)^2 \csc (c+d x)dx}{a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\int (e+f x)^2 \csc ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{2 a}+\frac {\int (e+f x)^2 \csc (c+d x)dx}{a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\int (e+f x)^2 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {\int (e+f x)^2 \csc (c+d x)dx}{a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}+\frac {-\frac {2 f \int (e+f x) \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {2 f \int (e+f x) \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {\int (e+f x)^2 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {\int (e+f x)^2 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\int (e+f x)^2 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {4 f \int (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {4 f \int -\left ((e+f x) \tan \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )\right )dx}{d}-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {4 f \int (e+f x) \tan \left (\frac {1}{4} (2 c+3 \pi )+\frac {d x}{2}\right )dx}{d}-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle \frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 f \left (\frac {i (e+f x)^2}{2 f}-2 i \int \frac {e^{\frac {1}{2} i (2 c+2 d x+3 \pi )} (e+f x)}{1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}dx\right )}{d}}{2 a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {i f \int \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )dx}{d}-\frac {i (e+f x) \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {\frac {1}{2} \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f^2 \text {arctanh}(\cos (c+d x))}{d^3}-\frac {f (e+f x) \csc (c+d x)}{d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {f \int e^{-\frac {1}{2} i (2 c+2 d x+3 \pi )} \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )de^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}{d^2}-\frac {i (e+f x) \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}-\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
3.3.10.3.1 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[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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. 1256 vs. \(2 (359 ) = 718\).
Time = 0.52 (sec) , antiderivative size = 1257, normalized size of antiderivative = 3.21
3/2/d^3/a*c^2*f^2*ln(exp(I*(d*x+c))-1)-3/2/d/a*f^2*ln(exp(I*(d*x+c))+1)*x^ 2+3/2/d/a*f^2*ln(1-exp(I*(d*x+c)))*x^2+1/a/d^3*f^2*ln(exp(I*(d*x+c))-1)+3* I/a/d^2*e*f*polylog(2,-exp(I*(d*x+c)))-3*I/a/d^2*f^2*polylog(2,exp(I*(d*x+ c)))*x+3*I/a/d^2*f^2*polylog(2,-exp(I*(d*x+c)))*x-1/a/d^3*f^2*ln(exp(I*(d* x+c))+1)-2/a/d^2*e*f*ln(exp(I*(d*x+c))-1)-2/a/d^2*e*f*ln(exp(I*(d*x+c))+1) -2/a/d^2*f^2*ln(1-exp(I*(d*x+c)))*x-2/a/d^2*f^2*ln(exp(I*(d*x+c))+1)*x-2/a /d^3*f^2*ln(1-exp(I*(d*x+c)))*c+2/a/d^3*c*f^2*ln(exp(I*(d*x+c))-1)+4*I/a/d ^3*f^2*c^2+4*I/a/d*f^2*x^2+2*I/a/d^3*f^2*polylog(2,exp(I*(d*x+c)))-4*I/a/d ^3*f^2*c*arctan(exp(I*(d*x+c)))+8*I/a/d^2*c*f^2*x+4*I/a/d^2*e*f*arctan(exp (I*(d*x+c)))-3*I/a/d^2*e*f*polylog(2,exp(I*(d*x+c)))+(3*d*f^2*x^2*exp(4*I* (d*x+c))+6*d*e*f*x*exp(4*I*(d*x+c))+3*d*e^2*exp(4*I*(d*x+c))-5*d*f^2*x^2*e xp(2*I*(d*x+c))+6*I*d*e*f*x*exp(3*I*(d*x+c))-10*d*e*f*x*exp(2*I*(d*x+c))+2 *f^2*x*exp(3*I*(d*x+c))+3*I*d*f^2*x^2*exp(3*I*(d*x+c))-2*I*d*e*f*x*exp(I*( d*x+c))-5*d*e^2*exp(2*I*(d*x+c))+4*d*f^2*x^2+2*e*f*exp(3*I*(d*x+c))+2*I*ex p(2*I*(d*x+c))*f*e+2*I*f^2*x*exp(2*I*(d*x+c))-2*I*f^2*x*exp(4*I*(d*x+c))+8 *d*e*f*x-2*f^2*x*exp(I*(d*x+c))-I*d*e^2*exp(I*(d*x+c))-I*d*f^2*x^2*exp(I*( d*x+c))+4*d*e^2-2*e*f*exp(I*(d*x+c))-2*I*exp(4*I*(d*x+c))*f*e+3*I*d*e^2*ex p(3*I*(d*x+c)))/(exp(2*I*(d*x+c))-1)^2/d^2/(exp(I*(d*x+c))+I)/a+2*I*f^2*po lylog(2,-exp(I*(d*x+c)))/a/d^3+4*I*f^2*polylog(2,I*exp(I*(d*x+c)))/a/d^3-2 /a/d^2*f*e*ln(1+exp(2*I*(d*x+c)))+8/a/d^2*f*e*ln(exp(I*(d*x+c)))-4/a/d^...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4026 vs. \(2 (348) = 696\).
Time = 0.45 (sec) , antiderivative size = 4026, normalized size of antiderivative = 10.27 \[ \int \frac {(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
-1/4*(4*d^2*f^2*x^2 + 4*d^2*e^2 - 8*(d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2)* cos(d*x + c)^3 - 4*d*e*f - 2*(3*d^2*f^2*x^2 + 3*d^2*e^2 - 2*d*e*f + 2*(3*d ^2*e*f - d*f^2)*x)*cos(d*x + c)^2 + 4*(2*d^2*e*f - d*f^2)*x + 6*(d^2*f^2*x ^2 + 2*d^2*e*f*x + d^2*e^2)*cos(d*x + c) + 2*(-3*I*d*f^2*x + (3*I*d*f^2*x + 3*I*d*e*f - 2*I*f^2)*cos(d*x + c)^3 - 3*I*d*e*f + (3*I*d*f^2*x + 3*I*d*e *f - 2*I*f^2)*cos(d*x + c)^2 + 2*I*f^2 + (-3*I*d*f^2*x - 3*I*d*e*f + 2*I*f ^2)*cos(d*x + c) + (-3*I*d*f^2*x - 3*I*d*e*f + (3*I*d*f^2*x + 3*I*d*e*f - 2*I*f^2)*cos(d*x + c)^2 + 2*I*f^2)*sin(d*x + c))*dilog(cos(d*x + c) + I*si n(d*x + c)) + 2*(3*I*d*f^2*x + (-3*I*d*f^2*x - 3*I*d*e*f + 2*I*f^2)*cos(d* x + c)^3 + 3*I*d*e*f + (-3*I*d*f^2*x - 3*I*d*e*f + 2*I*f^2)*cos(d*x + c)^2 - 2*I*f^2 + (3*I*d*f^2*x + 3*I*d*e*f - 2*I*f^2)*cos(d*x + c) + (3*I*d*f^2 *x + 3*I*d*e*f + (-3*I*d*f^2*x - 3*I*d*e*f + 2*I*f^2)*cos(d*x + c)^2 - 2*I *f^2)*sin(d*x + c))*dilog(cos(d*x + c) - I*sin(d*x + c)) + 8*(-I*f^2*cos(d *x + c)^3 - I*f^2*cos(d*x + c)^2 + I*f^2*cos(d*x + c) + I*f^2 + (-I*f^2*co s(d*x + c)^2 + I*f^2)*sin(d*x + c))*dilog(I*cos(d*x + c) - sin(d*x + c)) + 8*(I*f^2*cos(d*x + c)^3 + I*f^2*cos(d*x + c)^2 - I*f^2*cos(d*x + c) - I*f ^2 + (I*f^2*cos(d*x + c)^2 - I*f^2)*sin(d*x + c))*dilog(-I*cos(d*x + c) - sin(d*x + c)) + 2*(-3*I*d*f^2*x + (3*I*d*f^2*x + 3*I*d*e*f + 2*I*f^2)*cos( d*x + c)^3 - 3*I*d*e*f + (3*I*d*f^2*x + 3*I*d*e*f + 2*I*f^2)*cos(d*x + c)^ 2 - 2*I*f^2 + (-3*I*d*f^2*x - 3*I*d*e*f - 2*I*f^2)*cos(d*x + c) + (-3*I...
\[ \int \frac {(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{2} \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{2} x^{2} \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {2 e f x \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
(Integral(e**2*csc(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(f**2*x**2 *csc(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(2*e*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. 6160 vs. \(2 (348) = 696\).
Time = 2.29 (sec) , antiderivative size = 6160, normalized size of antiderivative = 15.71 \[ \int \frac {(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
-1/8*(2*c*e*f*((3*sin(d*x + c)/(cos(d*x + c) + 1) + 20*sin(d*x + c)^2/(cos (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) - sin (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^2*((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*(16*I*c^2*f^2 - 16*(-I*d*e*f + I*c*f^2 - (d*e*f - c*f^2)*cos (5*d*x + 5*c) + (-I*d*e*f + I*c*f^2)*cos(4*d*x + 4*c) + 2*(d*e*f - c*f^2)* cos(3*d*x + 3*c) + 2*(I*d*e*f - I*c*f^2)*cos(2*d*x + 2*c) - (d*e*f - c*f^2 )*cos(d*x + c) + (-I*d*e*f + I*c*f^2)*sin(5*d*x + 5*c) + (d*e*f - c*f^2)*s in(4*d*x + 4*c) + 2*(I*d*e*f - I*c*f^2)*sin(3*d*x + 3*c) - 2*(d*e*f - c*f^ 2)*sin(2*d*x + 2*c) + (-I*d*e*f + I*c*f^2)*sin(d*x + c))*arctan2(sin(d*x + c) + 1, cos(d*x + c)) - 16*((d*x + c)*f^2*cos(5*d*x + 5*c) + I*(d*x + c)* f^2*cos(4*d*x + 4*c) - 2*(d*x + c)*f^2*cos(3*d*x + 3*c) - 2*I*(d*x + c)*f^ 2*cos(2*d*x + 2*c) + (d*x + c)*f^2*cos(d*x + c) + I*(d*x + c)*f^2*sin(5*d* x + 5*c) - (d*x + c)*f^2*sin(4*d*x + 4*c) - 2*I*(d*x + c)*f^2*sin(3*d*x + 3*c) + 2*(d*x + c)*f^2*sin(2*d*x + 2*c) + I*(d*x + c)*f^2*sin(d*x + c) + I *(d*x + c)*f^2)*arctan2(cos(d*x + c), sin(d*x + c) + 1) - 2*(-3*I*(d*x ...
\[ \int \frac {(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \csc \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Hanged} \]