Integrand size = 26, antiderivative size = 216 \[ \int \frac {(e+f x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x) \cot (c+d x)}{a d}-\frac {f \csc (c+d x)}{2 a d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}-\frac {f \log (\sin (c+d x))}{a d^2}+\frac {3 i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}-\frac {3 i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2} \]
-3*(f*x+e)*arctanh(exp(I*(d*x+c)))/a/d+(f*x+e)*cot(1/2*c+1/4*Pi+1/2*d*x)/a /d+(f*x+e)*cot(d*x+c)/a/d-1/2*f*csc(d*x+c)/a/d^2-1/2*(f*x+e)*cot(d*x+c)*cs c(d*x+c)/a/d-2*f*ln(sin(1/2*c+1/4*Pi+1/2*d*x))/a/d^2-f*ln(sin(d*x+c))/a/d^ 2+3/2*I*f*polylog(2,-exp(I*(d*x+c)))/a/d^2-3/2*I*f*polylog(2,exp(I*(d*x+c) ))/a/d^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(492\) vs. \(2(216)=432\).
Time = 8.35 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.28 \[ \int \frac {(e+f x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (-d (e+f x) \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )-16 d (e+f x) \sin \left (\frac {1}{2} (c+d x)\right )+8 f (c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 (-f+2 d (e+f x)) \cot \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-16 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 d e \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-12 c f \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-8 f (\log (\cos (c+d x))+\log (\tan (c+d x))) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 f \left ((c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 (f+2 d (e+f x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \tan \left (\frac {1}{2} (c+d x)\right )+d (e+f x) \sec \left (\frac {1}{2} (c+d x)\right ) \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )\right )}{8 a d^2 (1+\sin (c+d x))} \]
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-(d*(e + f*x)*(1 + Cot[(c + d*x)/2 ])*Csc[(c + d*x)/2]) - 16*d*(e + f*x)*Sin[(c + d*x)/2] + 8*f*(c + d*x)*(Co s[(c + d*x)/2] + Sin[(c + d*x)/2]) + 2*(-f + 2*d*(e + f*x))*Cot[(c + d*x)/ 2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 16*f*Log[Cos[(c + d*x)/2] + Sin [(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*d*e*Log[Tan[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 12*c*f*Log[Tan[(c + d*x) /2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 8*f*(Log[Cos[c + d*x]] + Log[ Tan[c + d*x]])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*f*((c + d*x)*(Lo g[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) + I*(PolyLog[2, -E^(I*( c + d*x))] - PolyLog[2, E^(I*(c + d*x))]))*(Cos[(c + d*x)/2] + Sin[(c + d* x)/2]) - 2*(f + 2*d*(e + f*x))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*Tan[( c + d*x)/2] + d*(e + f*x)*Sec[(c + d*x)/2]*(1 + Tan[(c + d*x)/2])))/(8*a*d ^2*(1 + Sin[c + d*x]))
Time = 2.10 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.32, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {5046, 3042, 4673, 3042, 4671, 2715, 2838, 5046, 3042, 4672, 3042, 25, 3956, 5046, 3042, 3799, 3042, 4671, 2715, 2838, 4672, 3042, 25, 3956}
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.
| \(\displaystyle \int \frac {(e+f x) \csc ^3(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 5046 |
| \(\displaystyle \frac {\int (e+f x) \csc ^3(c+d x)dx}{a}-\int \frac {(e+f x) \csc ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e+f x) \csc (c+d x)^3dx}{a}-\int \frac {(e+f x) \csc ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4673 |
| \(\displaystyle \frac {\frac {1}{2} \int (e+f x) \csc (c+d x)dx-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\int \frac {(e+f x) \csc ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \int (e+f x) \csc (c+d x)dx-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\int \frac {(e+f x) \csc ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\int \frac {(e+f x) \csc ^2(c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (-\frac {f \int \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {f \int \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\int \frac {(e+f x) \csc ^2(c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (\frac {i f \int e^{-i (c+d x)} \log \left (1-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i f \int e^{-i (c+d x)} \log \left (1+e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\int \frac {(e+f x) \csc ^2(c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 5046 |
| \(\displaystyle -\frac {\int (e+f x) \csc ^2(c+d x)dx}{a}+\int \frac {(e+f x) \csc (c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int (e+f x) \csc (c+d x)^2dx}{a}+\int \frac {(e+f x) \csc (c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {\frac {f \int \cot (c+d x)dx}{d}-\frac {(e+f x) \cot (c+d x)}{d}}{a}+\int \frac {(e+f x) \csc (c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {f \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x) \cot (c+d x)}{d}}{a}+\int \frac {(e+f x) \csc (c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {f \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{d}-\frac {(e+f x) \cot (c+d x)}{d}}{a}+\int \frac {(e+f x) \csc (c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \int \frac {(e+f x) \csc (c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 5046 |
| \(\displaystyle -\int \frac {e+f x}{\sin (c+d x) a+a}dx+\frac {\int (e+f x) \csc (c+d x)dx}{a}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {e+f x}{\sin (c+d x) a+a}dx+\frac {\int (e+f x) \csc (c+d x)dx}{a}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle -\frac {\int (e+f x) \csc ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{2 a}+\frac {\int (e+f x) \csc (c+d x)dx}{a}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int (e+f x) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {\int (e+f x) \csc (c+d x)dx}{a}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {-\frac {f \int \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {f \int \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {\int (e+f x) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {i f \int e^{-i (c+d x)} \log \left (1-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i f \int e^{-i (c+d x)} \log \left (1+e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {\int (e+f x) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {\int (e+f x) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}}{a}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {\frac {2 f \int \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}}{a}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 f \int -\tan \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )dx}{d}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}}{a}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {2 f \int \tan \left (\frac {1}{4} (2 c+3 \pi )+\frac {d x}{2}\right )dx}{d}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}}{a}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}}{a}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \csc (c+d x)}{2 d^2}-\frac {(e+f x) \cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\frac {4 f \log \left (-\cos \left (\frac {c}{2}+\frac {d x}{2}-\frac {\pi }{4}\right )\right )}{d^2}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}-\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
-1/2*((-2*(e + f*x)*Cot[c/2 + Pi/4 + (d*x)/2])/d + (4*f*Log[-Cos[c/2 - Pi/ 4 + (d*x)/2]])/d^2)/a - (-(((e + f*x)*Cot[c + d*x])/d) + (f*Log[-Sin[c + d *x]])/d^2)/a + ((-2*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/d + (I*f*PolyLog[2 , -E^(I*(c + d*x))])/d^2 - (I*f*PolyLog[2, E^(I*(c + d*x))])/d^2)/a + (-1/ 2*(f*Csc[c + d*x])/d^2 - ((e + f*x)*Cot[c + d*x]*Csc[c + d*x])/(2*d) + ((- 2*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/d + (I*f*PolyLog[2, -E^(I*(c + d*x)) ])/d^2 - (I*f*PolyLog[2, E^(I*(c + d*x))])/d^2)/2)/a
3.3.11.3.1 Defintions of rubi rules used
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((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[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], 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_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S imp[b^2*((n - 2)/(n - 1)) Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
Int[(Csc[(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. 486 vs. \(2 (188 ) = 376\).
Time = 0.48 (sec) , antiderivative size = 487, normalized size of antiderivative = 2.25
| method | result | size |
| risch | \(\frac {3 d f x \,{\mathrm e}^{4 i \left (d x +c \right )}+3 d e \,{\mathrm e}^{4 i \left (d x +c \right )}-5 d f x \,{\mathrm e}^{2 i \left (d x +c \right )}+3 i d f x \,{\mathrm e}^{3 i \left (d x +c \right )}-5 d e \,{\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{3 i \left (d x +c \right )} f +3 i d e \,{\mathrm e}^{3 i \left (d x +c \right )}-i f \,{\mathrm e}^{4 i \left (d x +c \right )}+4 d x f -i d f x \,{\mathrm e}^{i \left (d x +c \right )}+4 d e -{\mathrm e}^{i \left (d x +c \right )} f -i d e \,{\mathrm e}^{i \left (d x +c \right )}+i {\mathrm e}^{2 i \left (d x +c \right )} f}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} d^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a}+\frac {3 f \ln \left (1-{\mathrm e}^{i \left (d x +c \right )}\right ) x}{2 d a}-\frac {3 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) x}{2 d a}+\frac {2 i f \arctan \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}-\frac {3 c f \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d^{2} a}+\frac {3 e \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d a}-\frac {3 e \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d a}+\frac {3 f \ln \left (1-{\mathrm e}^{i \left (d x +c \right )}\right ) c}{2 d^{2} a}-\frac {3 i f \,\operatorname {Li}_{2}\left ({\mathrm e}^{i \left (d x +c \right )}\right )}{2 a \,d^{2}}+\frac {3 i f \,\operatorname {Li}_{2}\left (-{\mathrm e}^{i \left (d x +c \right )}\right )}{2 a \,d^{2}}-\frac {f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a \,d^{2}}+\frac {4 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{d^{2} a}-\frac {f \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a \,d^{2}}-\frac {f \ln \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )}{a \,d^{2}}\) | \(487\) |
(3*d*f*x*exp(4*I*(d*x+c))+3*d*e*exp(4*I*(d*x+c))-5*d*f*x*exp(2*I*(d*x+c))+ 3*I*d*f*x*exp(3*I*(d*x+c))-5*d*e*exp(2*I*(d*x+c))+exp(3*I*(d*x+c))*f+3*I*d *e*exp(3*I*(d*x+c))-I*f*exp(4*I*(d*x+c))+4*d*x*f-I*d*f*x*exp(I*(d*x+c))+4* d*e-exp(I*(d*x+c))*f-I*d*e*exp(I*(d*x+c))+I*exp(2*I*(d*x+c))*f)/(exp(2*I*( d*x+c))-1)^2/d^2/(exp(I*(d*x+c))+I)/a+3/2/d/a*f*ln(1-exp(I*(d*x+c)))*x-3/2 /d/a*f*ln(exp(I*(d*x+c))+1)*x+2*I/d^2/a*f*arctan(exp(I*(d*x+c)))-3/2/d^2/a *c*f*ln(exp(I*(d*x+c))-1)+3/2/d/a*e*ln(exp(I*(d*x+c))-1)-3/2/d/a*e*ln(exp( I*(d*x+c))+1)+3/2/d^2/a*f*ln(1-exp(I*(d*x+c)))*c-3/2*I*f*polylog(2,exp(I*( d*x+c)))/a/d^2+3/2*I*f*polylog(2,-exp(I*(d*x+c)))/a/d^2-1/a/d^2*f*ln(exp(I *(d*x+c))+1)+4/d^2/a*f*ln(exp(I*(d*x+c)))-1/a/d^2*f*ln(exp(I*(d*x+c))-1)-1 /a/d^2*f*ln(1+exp(2*I*(d*x+c)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1359 vs. \(2 (184) = 368\).
Time = 0.33 (sec) , antiderivative size = 1359, normalized size of antiderivative = 6.29 \[ \int \frac {(e+f x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
1/4*(8*(d*f*x + d*e)*cos(d*x + c)^3 - 4*d*f*x + 2*(3*d*f*x + 3*d*e - f)*co s(d*x + c)^2 - 4*d*e - 6*(d*f*x + d*e)*cos(d*x + c) - 3*(I*f*cos(d*x + c)^ 3 + I*f*cos(d*x + c)^2 - I*f*cos(d*x + c) + (I*f*cos(d*x + c)^2 - I*f)*sin (d*x + c) - I*f)*dilog(cos(d*x + c) + I*sin(d*x + c)) - 3*(-I*f*cos(d*x + c)^3 - I*f*cos(d*x + c)^2 + I*f*cos(d*x + c) + (-I*f*cos(d*x + c)^2 + I*f) *sin(d*x + c) + I*f)*dilog(cos(d*x + c) - I*sin(d*x + c)) - 3*(I*f*cos(d*x + c)^3 + I*f*cos(d*x + c)^2 - I*f*cos(d*x + c) + (I*f*cos(d*x + c)^2 - I* f)*sin(d*x + c) - I*f)*dilog(-cos(d*x + c) + I*sin(d*x + c)) - 3*(-I*f*cos (d*x + c)^3 - I*f*cos(d*x + c)^2 + I*f*cos(d*x + c) + (-I*f*cos(d*x + c)^2 + I*f)*sin(d*x + c) + I*f)*dilog(-cos(d*x + c) - I*sin(d*x + c)) - ((3*d* f*x + 3*d*e + 2*f)*cos(d*x + c)^3 - 3*d*f*x + (3*d*f*x + 3*d*e + 2*f)*cos( d*x + c)^2 - 3*d*e - (3*d*f*x + 3*d*e + 2*f)*cos(d*x + c) - (3*d*f*x - (3* d*f*x + 3*d*e + 2*f)*cos(d*x + c)^2 + 3*d*e + 2*f)*sin(d*x + c) - 2*f)*log (cos(d*x + c) + I*sin(d*x + c) + 1) - ((3*d*f*x + 3*d*e + 2*f)*cos(d*x + c )^3 - 3*d*f*x + (3*d*f*x + 3*d*e + 2*f)*cos(d*x + c)^2 - 3*d*e - (3*d*f*x + 3*d*e + 2*f)*cos(d*x + c) - (3*d*f*x - (3*d*f*x + 3*d*e + 2*f)*cos(d*x + c)^2 + 3*d*e + 2*f)*sin(d*x + c) - 2*f)*log(cos(d*x + c) - I*sin(d*x + c) + 1) + ((3*d*e - (3*c + 2)*f)*cos(d*x + c)^3 + (3*d*e - (3*c + 2)*f)*cos( d*x + c)^2 - 3*d*e + (3*c + 2)*f - (3*d*e - (3*c + 2)*f)*cos(d*x + c) + (( 3*d*e - (3*c + 2)*f)*cos(d*x + c)^2 - 3*d*e + (3*c + 2)*f)*sin(d*x + c)...
\[ \int \frac {(e+f x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f x \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
(Integral(e*csc(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(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. 2080 vs. \(2 (184) = 368\).
Time = 0.78 (sec) , antiderivative size = 2080, normalized size of antiderivative = 9.63 \[ \int \frac {(e+f x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
(16*d*f*x*cos(5*d*x + 5*c) + 16*I*d*f*x*sin(5*d*x + 5*c) - 16*I*d*e - 8*(f *cos(5*d*x + 5*c) + I*f*cos(4*d*x + 4*c) - 2*f*cos(3*d*x + 3*c) - 2*I*f*co s(2*d*x + 2*c) + f*cos(d*x + c) + I*f*sin(5*d*x + 5*c) - f*sin(4*d*x + 4*c ) - 2*I*f*sin(3*d*x + 3*c) + 2*f*sin(2*d*x + 2*c) + I*f*sin(d*x + c) + I*f )*arctan2(cos(c) + sin(d*x), cos(d*x) + sin(c)) + 2*(-3*I*d*f*x - 3*I*d*e - (3*d*f*x + 3*d*e + 2*f)*cos(5*d*x + 5*c) + (-3*I*d*f*x - 3*I*d*e - 2*I*f )*cos(4*d*x + 4*c) + 2*(3*d*f*x + 3*d*e + 2*f)*cos(3*d*x + 3*c) + 2*(3*I*d *f*x + 3*I*d*e + 2*I*f)*cos(2*d*x + 2*c) - (3*d*f*x + 3*d*e + 2*f)*cos(d*x + c) + (-3*I*d*f*x - 3*I*d*e - 2*I*f)*sin(5*d*x + 5*c) + (3*d*f*x + 3*d*e + 2*f)*sin(4*d*x + 4*c) + 2*(3*I*d*f*x + 3*I*d*e + 2*I*f)*sin(3*d*x + 3*c ) - 2*(3*d*f*x + 3*d*e + 2*f)*sin(2*d*x + 2*c) + (-3*I*d*f*x - 3*I*d*e - 2 *I*f)*sin(d*x + c) - 2*I*f)*arctan2(sin(d*x + c), cos(d*x + c) + 1) + 2*(3 *I*d*e + (3*d*e - 2*f)*cos(5*d*x + 5*c) + (3*I*d*e - 2*I*f)*cos(4*d*x + 4* c) - 2*(3*d*e - 2*f)*cos(3*d*x + 3*c) + 2*(-3*I*d*e + 2*I*f)*cos(2*d*x + 2 *c) + (3*d*e - 2*f)*cos(d*x + c) + (3*I*d*e - 2*I*f)*sin(5*d*x + 5*c) - (3 *d*e - 2*f)*sin(4*d*x + 4*c) + 2*(-3*I*d*e + 2*I*f)*sin(3*d*x + 3*c) + 2*( 3*d*e - 2*f)*sin(2*d*x + 2*c) + (3*I*d*e - 2*I*f)*sin(d*x + c) - 2*I*f)*ar ctan2(sin(d*x + c), cos(d*x + c) - 1) - 6*(d*f*x*cos(5*d*x + 5*c) + I*d*f* x*cos(4*d*x + 4*c) - 2*d*f*x*cos(3*d*x + 3*c) - 2*I*d*f*x*cos(2*d*x + 2*c) + d*f*x*cos(d*x + c) + I*d*f*x*sin(5*d*x + 5*c) - d*f*x*sin(4*d*x + 4*...
\[ \int \frac {(e+f x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \csc \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Hanged} \]