Optimal. Leaf size=205 \[ \frac {3 i \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 i \text {Li}_4\left (-e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 x \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 x \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i x^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 x^2 \tan (a+b x)}{2 b^2}+\frac {x^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {x^3 \tan ^2(a+b x)}{2 b}+\frac {3 i x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {i x^4}{4} \]
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Rubi [A] time = 0.29, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3720, 3719, 2190, 2279, 2391, 30, 2531, 6609, 2282, 6589} \[ -\frac {3 i x^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 i \text {Li}_4\left (-e^{2 i (a+b x)}\right )}{4 b^4}-\frac {3 x^2 \tan (a+b x)}{2 b^2}-\frac {3 x \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {x^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {x^3 \tan ^2(a+b x)}{2 b}+\frac {3 i x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {i x^4}{4} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2190
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 3719
Rule 3720
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int x^3 \tan ^3(a+b x) \, dx &=\frac {x^3 \tan ^2(a+b x)}{2 b}-\frac {3 \int x^2 \tan ^2(a+b x) \, dx}{2 b}-\int x^3 \tan (a+b x) \, dx\\ &=-\frac {i x^4}{4}-\frac {3 x^2 \tan (a+b x)}{2 b^2}+\frac {x^3 \tan ^2(a+b x)}{2 b}+2 i \int \frac {e^{2 i (a+b x)} x^3}{1+e^{2 i (a+b x)}} \, dx+\frac {3 \int x \tan (a+b x) \, dx}{b^2}+\frac {3 \int x^2 \, dx}{2 b}\\ &=\frac {3 i x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {i x^4}{4}+\frac {x^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {3 x^2 \tan (a+b x)}{2 b^2}+\frac {x^3 \tan ^2(a+b x)}{2 b}-\frac {(6 i) \int \frac {e^{2 i (a+b x)} x}{1+e^{2 i (a+b x)}} \, dx}{b^2}-\frac {3 \int x^2 \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b}\\ &=\frac {3 i x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {i x^4}{4}-\frac {3 x \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {x^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {3 i x^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 x^2 \tan (a+b x)}{2 b^2}+\frac {x^3 \tan ^2(a+b x)}{2 b}+\frac {3 \int \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b^3}+\frac {(3 i) \int x \text {Li}_2\left (-e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=\frac {3 i x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {i x^4}{4}-\frac {3 x \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {x^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {3 i x^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 x^2 \tan (a+b x)}{2 b^2}+\frac {x^3 \tan ^2(a+b x)}{2 b}-\frac {(3 i) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 \int \text {Li}_3\left (-e^{2 i (a+b x)}\right ) \, dx}{2 b^3}\\ &=\frac {3 i x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {i x^4}{4}-\frac {3 x \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {x^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i x^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 x^2 \tan (a+b x)}{2 b^2}+\frac {x^3 \tan ^2(a+b x)}{2 b}+\frac {(3 i) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{4 b^4}\\ &=\frac {3 i x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {i x^4}{4}-\frac {3 x \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {x^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i x^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i \text {Li}_4\left (-e^{2 i (a+b x)}\right )}{4 b^4}-\frac {3 x^2 \tan (a+b x)}{2 b^2}+\frac {x^3 \tan ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 6.67, size = 359, normalized size = 1.75 \[ -\frac {3 x^2 \sec (a) \sin (b x) \sec (a+b x)}{2 b^2}-\frac {3 \csc (a) \sec (a) \left (b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac {\cot (a) \left (i \text {Li}_2\left (e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )+i b x \left (-2 \tan ^{-1}(\cot (a))-\pi \right )-2 \left (b x-\tan ^{-1}(\cot (a))\right ) \log \left (1-e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )-2 \tan ^{-1}(\cot (a)) \log \left (\sin \left (b x-\tan ^{-1}(\cot (a))\right )\right )-\pi \log \left (1+e^{-2 i b x}\right )+\pi \log (\cos (b x))\right )}{\sqrt {\cot ^2(a)+1}}\right )}{2 b^4 \sqrt {\csc ^2(a) \left (\sin ^2(a)+\cos ^2(a)\right )}}+\frac {1}{8} i e^{i a} \sec (a) \left (\frac {3 e^{-2 i a} \left (1+e^{2 i a}\right ) \left (2 b^2 x^2 \text {Li}_2\left (-e^{-2 i (a+b x)}\right )-2 i b x \text {Li}_3\left (-e^{-2 i (a+b x)}\right )-\text {Li}_4\left (-e^{-2 i (a+b x)}\right )\right )}{b^4}-\frac {4 i \left (1+e^{-2 i a}\right ) x^3 \log \left (1+e^{-2 i (a+b x)}\right )}{b}+2 e^{-2 i a} x^4\right )+\frac {x^3 \sec ^2(a+b x)}{2 b}-\frac {1}{4} x^4 \tan (a) \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.57, size = 342, normalized size = 1.67 \[ \frac {4 \, b^{3} x^{3} \tan \left (b x + a\right )^{2} + 4 \, b^{3} x^{3} - 12 \, b^{2} x^{2} \tan \left (b x + a\right ) + 6 \, b x {\rm polylog}\left (3, \frac {\tan \left (b x + a\right )^{2} + 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + 6 \, b x {\rm polylog}\left (3, \frac {\tan \left (b x + a\right )^{2} - 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + {\left (6 i \, b^{2} x^{2} - 6 i\right )} {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) + {\left (-6 i \, b^{2} x^{2} + 6 i\right )} {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) + 4 \, {\left (b^{3} x^{3} - 3 \, b x\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) + 4 \, {\left (b^{3} x^{3} - 3 \, b x\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) - 3 i \, {\rm polylog}\left (4, \frac {\tan \left (b x + a\right )^{2} + 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + 3 i \, {\rm polylog}\left (4, \frac {\tan \left (b x + a\right )^{2} - 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right )}{8 \, b^{4}} \]
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 \[ \int x^{3} \tan \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 251, normalized size = 1.22 \[ -\frac {2 i a^{3} x}{b^{3}}+\frac {x^{2} \left (2 b x \,{\mathrm e}^{2 i \left (b x +a \right )}-3 i {\mathrm e}^{2 i \left (b x +a \right )}-3 i\right )}{b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{2}}+\frac {3 i \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{4}}+\frac {3 i \polylog \left (4, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{4 b^{4}}+\frac {3 i x^{2}}{b^{2}}-\frac {3 i a^{4}}{2 b^{4}}-\frac {3 i x^{2} \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{2}}-\frac {i x^{4}}{4}+\frac {3 i a^{2}}{b^{4}}+\frac {6 i a x}{b^{3}}+\frac {x^{3} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b}+\frac {3 x \polylog \left (3, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{3}}-\frac {3 x \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b^{3}}-\frac {6 a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.58, size = 1203, normalized size = 5.87 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\mathrm {tan}\left (a+b\,x\right )}^3 \,d x \]
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 \[ \int x^{3} \tan ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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