Optimal. Leaf size=104 \[ \frac {i \text {Li}_2\left (-e^{i a x}\right )}{a^4}-\frac {i \text {Li}_2\left (e^{i a x}\right )}{a^4}-\frac {\csc (a x)}{a^4}-\frac {2 x \tanh ^{-1}\left (e^{i a x}\right )}{a^3}-\frac {x \cot (a x) \csc (a x)}{a^3}+\frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))} \]
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Rubi [A] time = 0.09, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4600, 4185, 4183, 2279, 2391} \[ \frac {i \text {PolyLog}\left (2,-e^{i a x}\right )}{a^4}-\frac {i \text {PolyLog}\left (2,e^{i a x}\right )}{a^4}+\frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac {\csc (a x)}{a^4}-\frac {2 x \tanh ^{-1}\left (e^{i a x}\right )}{a^3}-\frac {x \cot (a x) \csc (a x)}{a^3} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4183
Rule 4185
Rule 4600
Rubi steps
\begin {align*} \int \frac {x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx &=\frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac {2 \int x \csc ^3(a x) \, dx}{a^2}\\ &=-\frac {\csc (a x)}{a^4}-\frac {x \cot (a x) \csc (a x)}{a^3}+\frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac {\int x \csc (a x) \, dx}{a^2}\\ &=-\frac {2 x \tanh ^{-1}\left (e^{i a x}\right )}{a^3}-\frac {\csc (a x)}{a^4}-\frac {x \cot (a x) \csc (a x)}{a^3}+\frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac {\int \log \left (1-e^{i a x}\right ) \, dx}{a^3}+\frac {\int \log \left (1+e^{i a x}\right ) \, dx}{a^3}\\ &=-\frac {2 x \tanh ^{-1}\left (e^{i a x}\right )}{a^3}-\frac {\csc (a x)}{a^4}-\frac {x \cot (a x) \csc (a x)}{a^3}+\frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac {i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i a x}\right )}{a^4}-\frac {i \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i a x}\right )}{a^4}\\ &=-\frac {2 x \tanh ^{-1}\left (e^{i a x}\right )}{a^3}-\frac {\csc (a x)}{a^4}-\frac {x \cot (a x) \csc (a x)}{a^3}+\frac {i \text {Li}_2\left (-e^{i a x}\right )}{a^4}-\frac {i \text {Li}_2\left (e^{i a x}\right )}{a^4}+\frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}\\ \end {align*}
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Mathematica [A] time = 1.00, size = 157, normalized size = 1.51 \[ \frac {a^2 x^2 \csc (a x)+a^2 x^2 \log \left (1-e^{i a x}\right ) \cot (a x)-a^2 x^2 \log \left (1+e^{i a x}\right ) \cot (a x)+i \text {Li}_2\left (-e^{i a x}\right ) (a x \cot (a x)-1)-i \text {Li}_2\left (e^{i a x}\right ) (a x \cot (a x)-1)-a x \log \left (1-e^{i a x}\right )+a x \log \left (1+e^{i a x}\right )+\csc (a x)}{a^4 (a x \cot (a x)-1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.96, size = 295, normalized size = 2.84 \[ \frac {2 \, a^{2} x^{2} - {\left (i \, a x \cos \left (a x\right ) - i \, \sin \left (a x\right )\right )} {\rm Li}_2\left (\cos \left (a x\right ) + i \, \sin \left (a x\right )\right ) - {\left (-i \, a x \cos \left (a x\right ) + i \, \sin \left (a x\right )\right )} {\rm Li}_2\left (\cos \left (a x\right ) - i \, \sin \left (a x\right )\right ) - {\left (i \, a x \cos \left (a x\right ) - i \, \sin \left (a x\right )\right )} {\rm Li}_2\left (-\cos \left (a x\right ) + i \, \sin \left (a x\right )\right ) - {\left (-i \, a x \cos \left (a x\right ) + i \, \sin \left (a x\right )\right )} {\rm Li}_2\left (-\cos \left (a x\right ) - i \, \sin \left (a x\right )\right ) - {\left (a^{2} x^{2} \cos \left (a x\right ) - a x \sin \left (a x\right )\right )} \log \left (\cos \left (a x\right ) + i \, \sin \left (a x\right ) + 1\right ) - {\left (a^{2} x^{2} \cos \left (a x\right ) - a x \sin \left (a x\right )\right )} \log \left (\cos \left (a x\right ) - i \, \sin \left (a x\right ) + 1\right ) + {\left (a^{2} x^{2} \cos \left (a x\right ) - a x \sin \left (a x\right )\right )} \log \left (-\cos \left (a x\right ) + i \, \sin \left (a x\right ) + 1\right ) + {\left (a^{2} x^{2} \cos \left (a x\right ) - a x \sin \left (a x\right )\right )} \log \left (-\cos \left (a x\right ) - i \, \sin \left (a x\right ) + 1\right ) + 2}{2 \, {\left (a^{5} x \cos \left (a x\right ) - a^{4} \sin \left (a x\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \csc \left (a x\right )}{{\left (a x \cos \left (a x\right ) - \sin \left (a x\right )\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.47, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \csc \left (a x \right )}{\left (a x \cos \left (a x \right )-\sin \left (a x \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3}{\sin \left (a\,x\right )\,{\left (\sin \left (a\,x\right )-a\,x\,\cos \left (a\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \csc {\left (a x \right )}}{\left (a x \cos {\left (a x \right )} - \sin {\left (a x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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