Optimal. Leaf size=98 \[ -\frac {i x^3}{b}-\frac {x^4}{4}+\frac {3 x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i x \text {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 \text {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {x^3 \tan (a+b x)}{b} \]
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Rubi [A]
time = 0.11, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {3801, 3800,
2221, 2611, 2320, 6724, 30} \begin {gather*} \frac {3 \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i x \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}+\frac {x^3 \tan (a+b x)}{b}-\frac {i x^3}{b}-\frac {x^4}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 3801
Rule 6724
Rubi steps
\begin {align*} \int x^3 \tan ^2(a+b x) \, dx &=\frac {x^3 \tan (a+b x)}{b}-\frac {3 \int x^2 \tan (a+b x) \, dx}{b}-\int x^3 \, dx\\ &=-\frac {i x^3}{b}-\frac {x^4}{4}+\frac {x^3 \tan (a+b x)}{b}+\frac {(6 i) \int \frac {e^{2 i (a+b x)} x^2}{1+e^{2 i (a+b x)}} \, dx}{b}\\ &=-\frac {i x^3}{b}-\frac {x^4}{4}+\frac {3 x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}+\frac {x^3 \tan (a+b x)}{b}-\frac {6 \int x \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {i x^3}{b}-\frac {x^4}{4}+\frac {3 x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i x \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac {x^3 \tan (a+b x)}{b}+\frac {(3 i) \int \text {Li}_2\left (-e^{2 i (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {i x^3}{b}-\frac {x^4}{4}+\frac {3 x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i x \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac {x^3 \tan (a+b x)}{b}+\frac {3 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4}\\ &=-\frac {i x^3}{b}-\frac {x^4}{4}+\frac {3 x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i x \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {x^3 \tan (a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 1.20, size = 122, normalized size = 1.24 \begin {gather*} -\frac {x^4}{4}+\frac {2 b^2 x^2 \left (-\frac {2 i b e^{2 i a} x}{1+e^{2 i a}}+3 \log \left (1+e^{2 i (a+b x)}\right )\right )-6 i b x \text {PolyLog}\left (2,-e^{2 i (a+b x)}\right )+3 \text {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {x^3 \sec (a) \sec (a+b x) \sin (b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 133, normalized size = 1.36
method | result | size |
risch | \(-\frac {x^{4}}{4}+\frac {2 i x^{3}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}-\frac {2 i x^{3}}{b}+\frac {6 i a^{2} x}{b^{3}}+\frac {4 i a^{3}}{b^{4}}+\frac {3 x^{2} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b^{2}}-\frac {3 i x \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b^{3}}+\frac {3 \polylog \left (3, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{4}}-\frac {6 a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 639 vs. \(2 (83) = 166\).
time = 0.56, size = 639, normalized size = 6.52 \begin {gather*} \frac {2 \, {\left (b x + a - \tan \left (b x + a\right )\right )} a^{3} - \frac {3 \, {\left ({\left (b x + a\right )}^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (b x + a\right )}^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, {\left (b x + a\right )}^{2} \cos \left (2 \, b x + 2 \, a\right ) + {\left (b x + a\right )}^{2} - {\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 4 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2}}{\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1} + \frac {2 \, {\left (i \, {\left (b x + a\right )}^{4} - 4 i \, {\left (b x + a\right )}^{3} a + 12 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a + {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left (-i \, {\left (b x + a\right )}^{2} + 2 i \, {\left (b x + a\right )} a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \arctan \left (\sin \left (2 \, b x + 2 \, a\right ), \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + {\left (i \, {\left (b x + a\right )}^{4} - 4 \, {\left (b x + a\right )}^{3} {\left (i \, a + 2\right )} + 24 \, {\left (b x + a\right )}^{2} a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 12 \, {\left (b x \cos \left (2 \, b x + 2 \, a\right ) + i \, b x \sin \left (2 \, b x + 2 \, a\right ) + b x\right )} {\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 6 \, {\left (i \, {\left (b x + a\right )}^{2} - 2 i \, {\left (b x + a\right )} a + {\left (i \, {\left (b x + a\right )}^{2} - 2 i \, {\left (b x + a\right )} a\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 6 \, {\left (i \, \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right ) + i\right )} {\rm Li}_{3}(-e^{\left (2 i \, b x + 2 i \, a\right )}) - {\left ({\left (b x + a\right )}^{4} - 4 \, {\left (b x + a\right )}^{3} {\left (a - 2 i\right )} - 24 i \, {\left (b x + a\right )}^{2} a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )}}{-4 i \, \cos \left (2 \, b x + 2 \, a\right ) + 4 \, \sin \left (2 \, b x + 2 \, a\right ) - 4 i}}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 225 vs. \(2 (83) = 166\).
time = 0.37, size = 225, normalized size = 2.30 \begin {gather*} -\frac {b^{4} x^{4} - 4 \, b^{3} x^{3} \tan \left (b x + a\right ) - 6 \, b^{2} x^{2} \log \left (-\frac {2 \, {\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) - 6 \, b^{2} x^{2} \log \left (-\frac {2 \, {\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) - 6 i \, b x {\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 ) + 6 i \, b x {\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 ) - 3 \, {\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 ) - 3 \, {\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 )}{4 \, b^{4}} \end {gather*}
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 \begin {gather*} \int x^{3} \tan ^{2}{\left (a + b x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^3\,{\mathrm {tan}\left (a+b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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