Optimal. Leaf size=127 \[ -\frac {2 i \text {Li}_2\left (e^{2 i a x}\right )}{a^5}-\frac {\cot (a x)}{a^5}+\frac {4 x \log \left (1-e^{2 i a x}\right )}{a^4}-\frac {x \csc ^2(a x)}{a^4}-\frac {2 i x^2}{a^3}-\frac {2 x^2 \cot (a x)}{a^3}-\frac {x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac {x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))} \]
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Rubi [A] time = 0.18, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {4600, 4186, 3767, 8, 4184, 3717, 2190, 2279, 2391} \[ -\frac {2 i \text {PolyLog}\left (2,e^{2 i a x}\right )}{a^5}-\frac {2 i x^2}{a^3}-\frac {2 x^2 \cot (a x)}{a^3}-\frac {x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac {x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac {4 x \log \left (1-e^{2 i a x}\right )}{a^4}-\frac {\cot (a x)}{a^5}-\frac {x \csc ^2(a x)}{a^4} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 3767
Rule 4184
Rule 4186
Rule 4600
Rubi steps
\begin {align*} \int \frac {x^4 \csc ^2(a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx &=\frac {x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac {3 \int x^2 \csc ^4(a x) \, dx}{a^2}\\ &=-\frac {x \csc ^2(a x)}{a^4}-\frac {x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac {x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac {\int \csc ^2(a x) \, dx}{a^4}+\frac {2 \int x^2 \csc ^2(a x) \, dx}{a^2}\\ &=-\frac {2 x^2 \cot (a x)}{a^3}-\frac {x \csc ^2(a x)}{a^4}-\frac {x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac {x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac {\operatorname {Subst}(\int 1 \, dx,x,\cot (a x))}{a^5}+\frac {4 \int x \cot (a x) \, dx}{a^3}\\ &=-\frac {2 i x^2}{a^3}-\frac {\cot (a x)}{a^5}-\frac {2 x^2 \cot (a x)}{a^3}-\frac {x \csc ^2(a x)}{a^4}-\frac {x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac {x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac {(8 i) \int \frac {e^{2 i a x} x}{1-e^{2 i a x}} \, dx}{a^3}\\ &=-\frac {2 i x^2}{a^3}-\frac {\cot (a x)}{a^5}-\frac {2 x^2 \cot (a x)}{a^3}-\frac {x \csc ^2(a x)}{a^4}-\frac {x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac {4 x \log \left (1-e^{2 i a x}\right )}{a^4}+\frac {x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac {4 \int \log \left (1-e^{2 i a x}\right ) \, dx}{a^4}\\ &=-\frac {2 i x^2}{a^3}-\frac {\cot (a x)}{a^5}-\frac {2 x^2 \cot (a x)}{a^3}-\frac {x \csc ^2(a x)}{a^4}-\frac {x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac {4 x \log \left (1-e^{2 i a x}\right )}{a^4}+\frac {x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i a x}\right )}{a^5}\\ &=-\frac {2 i x^2}{a^3}-\frac {\cot (a x)}{a^5}-\frac {2 x^2 \cot (a x)}{a^3}-\frac {x \csc ^2(a x)}{a^4}-\frac {x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac {4 x \log \left (1-e^{2 i a x}\right )}{a^4}-\frac {2 i \text {Li}_2\left (e^{2 i a x}\right )}{a^5}+\frac {x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}\\ \end {align*}
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Mathematica [A] time = 1.05, size = 102, normalized size = 0.80 \[ \frac {a^3 \left (-x^2\right ) \cot (a x)-2 i a \left (a^2 x^2+\text {Li}_2\left (e^{2 i a x}\right )\right )+\frac {\left (a^2 x^2+1\right )^2 \sin (a x)}{x (a x \cos (a x)-\sin (a x))}+a^2 x+4 a^2 x \log \left (1-e^{2 i a x}\right )+\frac {1}{x}}{a^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.00, size = 406, normalized size = 3.20 \[ \frac {a^{3} x^{3} - {\left (2 \, a^{3} x^{3} + a x\right )} \cos \left (a x\right )^{2} + {\left (2 \, a^{2} x^{2} + 1\right )} \cos \left (a x\right ) \sin \left (a x\right ) + a x + {\left (-2 i \, a x \cos \left (a x\right ) \sin \left (a x\right ) - 2 i \, \cos \left (a x\right )^{2} + 2 i\right )} {\rm Li}_2\left (\cos \left (a x\right ) + i \, \sin \left (a x\right )\right ) + {\left (2 i \, a x \cos \left (a x\right ) \sin \left (a x\right ) + 2 i \, \cos \left (a x\right )^{2} - 2 i\right )} {\rm Li}_2\left (\cos \left (a x\right ) - i \, \sin \left (a x\right )\right ) + {\left (2 i \, a x \cos \left (a x\right ) \sin \left (a x\right ) + 2 i \, \cos \left (a x\right )^{2} - 2 i\right )} {\rm Li}_2\left (-\cos \left (a x\right ) + i \, \sin \left (a x\right )\right ) + {\left (-2 i \, a x \cos \left (a x\right ) \sin \left (a x\right ) - 2 i \, \cos \left (a x\right )^{2} + 2 i\right )} {\rm Li}_2\left (-\cos \left (a x\right ) - i \, \sin \left (a x\right )\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2} - a x\right )} \log \left (\cos \left (a x\right ) + i \, \sin \left (a x\right ) + 1\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2} - a x\right )} \log \left (\cos \left (a x\right ) - i \, \sin \left (a x\right ) + 1\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2} - a x\right )} \log \left (-\cos \left (a x\right ) + i \, \sin \left (a x\right ) + 1\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2} - a x\right )} \log \left (-\cos \left (a x\right ) - i \, \sin \left (a x\right ) + 1\right )}{a^{6} x \cos \left (a x\right ) \sin \left (a x\right ) + a^{5} \cos \left (a x\right )^{2} - a^{5}} \]
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 \[ \int \frac {x^{4} \csc \left (a x\right )^{2}}{{\left (a x \cos \left (a x\right ) - \sin \left (a x\right )\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.10, size = 172, normalized size = 1.35 \[ -\frac {2 i \left (2 i a^{2} x^{2} {\mathrm e}^{2 i a x}+2 x^{3} a^{3}-2 i a^{2} x^{2}-a x \,{\mathrm e}^{2 i a x}+i {\mathrm e}^{2 i a x}+a x -i\right )}{\left ({\mathrm e}^{2 i a x}-1\right ) \left (a x \,{\mathrm e}^{2 i a x}+i {\mathrm e}^{2 i a x}+a x -i\right ) a^{5}}-\frac {4 i x^{2}}{a^{3}}+\frac {4 x \ln \left ({\mathrm e}^{i a x}+1\right )}{a^{4}}-\frac {4 i \polylog \left (2, -{\mathrm e}^{i a x}\right )}{a^{5}}+\frac {4 x \ln \left (1-{\mathrm e}^{i a x}\right )}{a^{4}}-\frac {4 i \polylog \left (2, {\mathrm e}^{i a x}\right )}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 608, normalized size = 4.79 \[ -\frac {2 \, a x + {\left (4 \, a^{2} x^{2} + 8 i \, a x \cos \left (2 \, a x\right ) - 8 \, a x \sin \left (2 \, a x\right ) - 4 i \, a x - {\left (4 \, a^{2} x^{2} + 4 i \, a x\right )} \cos \left (4 \, a x\right ) + 4 \, {\left (-i \, a^{2} x^{2} + a x\right )} \sin \left (4 \, a x\right )\right )} \arctan \left (\sin \left (a x\right ), \cos \left (a x\right ) + 1\right ) - {\left (4 \, a^{2} x^{2} + 8 i \, a x \cos \left (2 \, a x\right ) - 8 \, a x \sin \left (2 \, a x\right ) - 4 i \, a x - {\left (4 \, a^{2} x^{2} + 4 i \, a x\right )} \cos \left (4 \, a x\right ) - 4 \, {\left (i \, a^{2} x^{2} - a x\right )} \sin \left (4 \, a x\right )\right )} \arctan \left (\sin \left (a x\right ), -\cos \left (a x\right ) + 1\right ) + 4 \, {\left (a^{3} x^{3} + i \, a^{2} x^{2}\right )} \cos \left (4 \, a x\right ) - {\left (4 i \, a^{2} x^{2} + 2 \, a x - 2 i\right )} \cos \left (2 \, a x\right ) - {\left (4 \, a x - {\left (4 \, a x + 4 i\right )} \cos \left (4 \, a x\right ) - 4 \, {\left (i \, a x - 1\right )} \sin \left (4 \, a x\right ) + 8 i \, \cos \left (2 \, a x\right ) - 8 \, \sin \left (2 \, a x\right ) - 4 i\right )} {\rm Li}_2\left (-e^{\left (i \, a x\right )}\right ) - {\left (4 \, a x - {\left (4 \, a x + 4 i\right )} \cos \left (4 \, a x\right ) - 4 \, {\left (i \, a x - 1\right )} \sin \left (4 \, a x\right ) + 8 i \, \cos \left (2 \, a x\right ) - 8 \, \sin \left (2 \, a x\right ) - 4 i\right )} {\rm Li}_2\left (e^{\left (i \, a x\right )}\right ) - {\left (2 i \, a^{2} x^{2} - 4 \, a x \cos \left (2 \, a x\right ) - 4 i \, a x \sin \left (2 \, a x\right ) + 2 \, a x - 2 \, {\left (i \, a^{2} x^{2} - a x\right )} \cos \left (4 \, a x\right ) + {\left (2 \, a^{2} x^{2} + 2 i \, a x\right )} \sin \left (4 \, a x\right )\right )} \log \left (\cos \left (a x\right )^{2} + \sin \left (a x\right )^{2} + 2 \, \cos \left (a x\right ) + 1\right ) - {\left (2 i \, a^{2} x^{2} - 4 \, a x \cos \left (2 \, a x\right ) - 4 i \, a x \sin \left (2 \, a x\right ) + 2 \, a x - 2 \, {\left (i \, a^{2} x^{2} - a x\right )} \cos \left (4 \, a x\right ) + {\left (2 \, a^{2} x^{2} + 2 i \, a x\right )} \sin \left (4 \, a x\right )\right )} \log \left (\cos \left (a x\right )^{2} + \sin \left (a x\right )^{2} - 2 \, \cos \left (a x\right ) + 1\right ) - {\left (-4 i \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right )} \sin \left (4 \, a x\right ) + {\left (4 \, a^{2} x^{2} - 2 i \, a x - 2\right )} \sin \left (2 \, a x\right ) - 2 i}{{\left (i \, a x + {\left (-i \, a x + 1\right )} \cos \left (4 \, a x\right ) + {\left (a x + i\right )} \sin \left (4 \, a x\right ) - 2 \, \cos \left (2 \, a x\right ) - 2 i \, \sin \left (2 \, a x\right ) + 1\right )} a^{5}} \]
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 \[ \int \frac {x^4}{{\sin \left (a\,x\right )}^2\,{\left (\sin \left (a\,x\right )-a\,x\,\cos \left (a\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {x^{4} \csc ^{2}{\left (a x \right )}}{\left (a x \cos {\left (a x \right )} - \sin {\left (a x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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