Integrand size = 12, antiderivative size = 205 \[ \int x^3 \tan ^3(a+b x) \, dx=\frac {3 i x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {i x^4}{4}-\frac {3 x \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {x^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4}-\frac {3 x^2 \tan (a+b x)}{2 b^2}+\frac {x^3 \tan ^2(a+b x)}{2 b} \]
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Time = 0.36 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3801, 3800, 2221, 2317, 2438, 30, 2611, 6744, 2320, 6724} \[ \int x^3 \tan ^3(a+b x) \, dx=\frac {3 i \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 i \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 x \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 x \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 x^2 \tan (a+b x)}{2 b^2}+\frac {x^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {x^3 \tan ^2(a+b x)}{2 b}+\frac {3 i x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {i x^4}{4} \]
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Rule 30
Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3800
Rule 3801
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \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 \operatorname {PolyLog}\left (2,-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 \operatorname {PolyLog}\left (2,-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 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \operatorname {PolyLog}\left (3,-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) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 \int \operatorname {PolyLog}\left (3,-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 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \operatorname {PolyLog}\left (3,-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) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i \operatorname {PolyLog}\left (4,-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*}
Time = 6.54 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.80 \[ \int x^3 \tan ^3(a+b x) \, dx=\frac {i e^{i a} \left (2 b^4 e^{-2 i a} x^4-4 i b^3 \left (1+e^{-2 i a}\right ) x^3 \log \left (1+e^{-2 i (a+b x)}\right )+6 b^2 \left (1+e^{-2 i a}\right ) x^2 \operatorname {PolyLog}\left (2,-e^{-2 i (a+b x)}\right )-6 i b \left (1+e^{-2 i a}\right ) x \operatorname {PolyLog}\left (3,-e^{-2 i (a+b x)}\right )-3 \left (1+e^{-2 i a}\right ) \operatorname {PolyLog}\left (4,-e^{-2 i (a+b x)}\right )\right ) \sec (a)}{8 b^4}+\frac {x^3 \sec ^2(a+b x)}{2 b}-\frac {3 \csc (a) \left (b^2 e^{-i \arctan (\cot (a))} x^2-\frac {\cot (a) \left (i b x (-\pi -2 \arctan (\cot (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x-\arctan (\cot (a))) \log \left (1-e^{2 i (b x-\arctan (\cot (a)))}\right )+\pi \log (\cos (b x))-2 \arctan (\cot (a)) \log (\sin (b x-\arctan (\cot (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x-\arctan (\cot (a)))}\right )\right )}{\sqrt {1+\cot ^2(a)}}\right ) \sec (a)}{2 b^4 \sqrt {\csc ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}}-\frac {3 x^2 \sec (a) \sec (a+b x) \sin (b x)}{2 b^2}-\frac {1}{4} x^4 \tan (a) \]
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Time = 0.36 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.22
method | result | size |
risch | \(\frac {3 i \operatorname {Li}_{2}\left (-{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{4}}+\frac {x^{2} \left (2 b x \,{\mathrm e}^{2 i \left (b x +a \right )}-3 i {\mathrm e}^{2 i \left (b x +a \right )}-3 i\right )}{b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{2}}+\frac {3 i x^{2}}{b^{2}}-\frac {i x^{4}}{4}+\frac {3 i \operatorname {Li}_{4}\left (-{\mathrm e}^{2 i \left (b x +a \right )}\right )}{4 b^{4}}-\frac {3 i a^{4}}{2 b^{4}}-\frac {3 i x^{2} \operatorname {Li}_{2}\left (-{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{2}}+\frac {6 i a x}{b^{3}}-\frac {2 i a^{3} x}{b^{3}}-\frac {6 a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {3 i a^{2}}{b^{4}}+\frac {x^{3} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b}+\frac {3 x \,\operatorname {Li}_{3}\left (-{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{3}}-\frac {3 x \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b^{3}}\) | \(251\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (163) = 326\).
Time = 0.27 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.68 \[ \int x^3 \tan ^3(a+b x) \, dx=\frac {4 \, b^{3} x^{3} \tan \left (b x + a\right )^{2} + 4 \, b^{3} x^{3} - 12 \, b^{2} x^{2} \tan \left (b x + a\right ) + 6 \, b x {\rm polylog}\left (3, \frac {\tan \left (b x + a\right )^{2} + 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + 6 \, b x {\rm polylog}\left (3, \frac {\tan \left (b x + a\right )^{2} - 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) - 6 \, {\left (-i \, b^{2} x^{2} + i\right )} {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) - 6 \, {\left (i \, b^{2} x^{2} - i\right )} {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) + 4 \, {\left (b^{3} x^{3} - 3 \, b x\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) + 4 \, {\left (b^{3} x^{3} - 3 \, b x\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) - 3 i \, {\rm polylog}\left (4, \frac {\tan \left (b x + a\right )^{2} + 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + 3 i \, {\rm polylog}\left (4, \frac {\tan \left (b x + a\right )^{2} - 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right )}{8 \, b^{4}} \]
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\[ \int x^3 \tan ^3(a+b x) \, dx=\int x^{3} \tan ^{3}{\left (a + b x \right )}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1205 vs. \(2 (163) = 326\).
Time = 0.47 (sec) , antiderivative size = 1205, normalized size of antiderivative = 5.88 \[ \int x^3 \tan ^3(a+b x) \, dx=\text {Too large to display} \]
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\[ \int x^3 \tan ^3(a+b x) \, dx=\int { x^{3} \tan \left (b x + a\right )^{3} \,d x } \]
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Timed out. \[ \int x^3 \tan ^3(a+b x) \, dx=\int x^3\,{\mathrm {tan}\left (a+b\,x\right )}^3 \,d x \]
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