Optimal. Leaf size=259 \[ \frac {3 i d^3 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 i d^3 \text {Li}_4\left (-e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}+\frac {3 i d (c+d x)^2}{2 b^2}+\frac {(c+d x)^3}{2 b}-\frac {i (c+d x)^4}{4 d} \]
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Rubi [A] time = 0.36, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3720, 3719, 2190, 2279, 2391, 32, 2531, 6609, 2282, 6589} \[ \frac {3 d^2 (c+d x) \text {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i d (c+d x)^2 \text {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 i d^3 \text {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 i d^3 \text {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}+\frac {3 i d (c+d x)^2}{2 b^2}+\frac {(c+d x)^3}{2 b}-\frac {i (c+d x)^4}{4 d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2190
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 3719
Rule 3720
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int (c+d x)^3 \tan ^3(a+b x) \, dx &=\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {(3 d) \int (c+d x)^2 \tan ^2(a+b x) \, dx}{2 b}-\int (c+d x)^3 \tan (a+b x) \, dx\\ &=-\frac {i (c+d x)^4}{4 d}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}+2 i \int \frac {e^{2 i (a+b x)} (c+d x)^3}{1+e^{2 i (a+b x)}} \, dx+\frac {(3 d) \int (c+d x)^2 \, dx}{2 b}+\frac {\left (3 d^2\right ) \int (c+d x) \tan (a+b x) \, dx}{b^2}\\ &=\frac {3 i d (c+d x)^2}{2 b^2}+\frac {(c+d x)^3}{2 b}-\frac {i (c+d x)^4}{4 d}+\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {(3 d) \int (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b}-\frac {\left (6 i d^2\right ) \int \frac {e^{2 i (a+b x)} (c+d x)}{1+e^{2 i (a+b x)}} \, dx}{b^2}\\ &=\frac {3 i d (c+d x)^2}{2 b^2}+\frac {(c+d x)^3}{2 b}-\frac {i (c+d x)^4}{4 d}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}+\frac {\left (3 i d^2\right ) \int (c+d x) \text {Li}_2\left (-e^{2 i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (3 d^3\right ) \int \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b^3}\\ &=\frac {3 i d (c+d x)^2}{2 b^2}+\frac {(c+d x)^3}{2 b}-\frac {i (c+d x)^4}{4 d}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {\left (3 i d^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4}-\frac {\left (3 d^3\right ) \int \text {Li}_3\left (-e^{2 i (a+b x)}\right ) \, dx}{2 b^3}\\ &=\frac {3 i d (c+d x)^2}{2 b^2}+\frac {(c+d x)^3}{2 b}-\frac {i (c+d x)^4}{4 d}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i d^3 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}+\frac {\left (3 i d^3\right ) \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 d (c+d x)^2}{2 b^2}+\frac {(c+d x)^3}{2 b}-\frac {i (c+d x)^4}{4 d}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i d^3 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i d^3 \text {Li}_4\left (-e^{2 i (a+b x)}\right )}{4 b^4}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [B] time = 6.89, size = 803, normalized size = 3.10 \[ \frac {\sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a)) c^3}{b \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {3 d \csc (a) \left (b^2 e^{-i \tan ^{-1}(\cot (a))} x^2-\frac {\cot (a) \left (i b x \left (-2 \tan ^{-1}(\cot (a))-\pi \right )-\pi \log \left (1+e^{-2 i b x}\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 )+\pi \log (\cos (b x))-2 \tan ^{-1}(\cot (a)) \log \left (\sin \left (b x-\tan ^{-1}(\cot (a))\right )\right )+i \text {Li}_2\left (e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )\right )}{\sqrt {\cot ^2(a)+1}}\right ) \sec (a) c^2}{2 b^2 \sqrt {\csc ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}}+\frac {i d^2 e^{-i a} \left (2 b^2 \left (2 b x-3 i \left (1+e^{2 i a}\right ) \log \left (1+e^{-2 i (a+b x)}\right )\right ) x^2+6 b \left (1+e^{2 i a}\right ) \text {Li}_2\left (-e^{-2 i (a+b x)}\right ) x-3 i \left (1+e^{2 i a}\right ) \text {Li}_3\left (-e^{-2 i (a+b x)}\right )\right ) \sec (a) c}{4 b^3}-\frac {3 d^2 \sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a)) c}{b^3 \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}+\frac {1}{8} i d^3 e^{i a} \left (2 e^{-2 i a} x^4-\frac {4 i \left (1+e^{-2 i a}\right ) \log \left (1+e^{-2 i (a+b x)}\right ) x^3}{b}+\frac {3 e^{-2 i a} \left (1+e^{2 i a}\right ) \left (2 b^2 \text {Li}_2\left (-e^{-2 i (a+b x)}\right ) x^2-2 i b \text {Li}_3\left (-e^{-2 i (a+b x)}\right ) x-\text {Li}_4\left (-e^{-2 i (a+b x)}\right )\right )}{b^4}\right ) \sec (a)-\frac {3 \sec (a) \sec (a+b x) \left (x^2 \sin (b x) d^3+2 c x \sin (b x) d^2+c^2 \sin (b x) d\right )}{2 b^2}-\frac {1}{4} x \left (4 c^3+6 d x c^2+4 d^2 x^2 c+d^3 x^3\right ) \tan (a)-\frac {3 d^3 \csc (a) \left (b^2 e^{-i \tan ^{-1}(\cot (a))} x^2-\frac {\cot (a) \left (i b x \left (-2 \tan ^{-1}(\cot (a))-\pi \right )-\pi \log \left (1+e^{-2 i b x}\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 )+\pi \log (\cos (b x))-2 \tan ^{-1}(\cot (a)) \log \left (\sin \left (b x-\tan ^{-1}(\cot (a))\right )\right )+i \text {Li}_2\left (e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )\right )}{\sqrt {\cot ^2(a)+1}}\right ) \sec (a)}{2 b^4 \sqrt {\csc ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.45, size = 590, normalized size = 2.28 \[ \frac {4 \, b^{3} d^{3} x^{3} + 12 \, b^{3} c d^{2} x^{2} + 12 \, b^{3} c^{2} d x - 3 i \, d^{3} {\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 \, d^{3} {\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 ) + 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \tan \left (b x + a\right )^{2} + {\left (6 i \, b^{2} d^{3} x^{2} + 12 i \, b^{2} c d^{2} x + 6 i \, b^{2} c^{2} d - 6 i \, d^{3}\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} d^{3} x^{2} - 12 i \, b^{2} c d^{2} x - 6 i \, b^{2} c^{2} d + 6 i \, d^{3}\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} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (b^{3} c^{2} d - b d^{3}\right )} 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} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (b^{3} c^{2} d - b d^{3}\right )} 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 ) + 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\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 (b d^{3} x + b c d^{2}\right )} {\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 ) - 12 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \tan \left (b x + a\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 \[ \int {\left (d x + c\right )}^{3} \tan \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 720, normalized size = 2.78 \[ \frac {3 d^{3} \polylog \left (3, -{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{2 b^{3}}+\frac {d^{3} \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) x^{3}}{b}+\frac {6 i c \,d^{2} a^{2} x}{b^{2}}-\frac {6 i c^{2} d a x}{b}+\frac {2 d^{3} a^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {3 i a^{4} d^{3}}{2 b^{4}}-i c \,d^{2} x^{3}-\frac {3 i c^{2} d \,x^{2}}{2}+\frac {3 c \,d^{2} \polylog \left (3, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{3}}+\frac {3 i d^{3} \polylog \left (4, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{4 b^{4}}+\frac {c^{3} \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b}-\frac {3 i c \,d^{2} \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{b^{2}}+i c^{3} x -\frac {i d^{3} x^{4}}{4}-\frac {2 c^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b}+\frac {2 b \,d^{3} x^{3} {\mathrm e}^{2 i \left (b x +a \right )}-3 i d^{3} x^{2} {\mathrm e}^{2 i \left (b x +a \right )}+6 b c \,d^{2} x^{2} {\mathrm e}^{2 i \left (b x +a \right )}-6 i c \,d^{2} x \,{\mathrm e}^{2 i \left (b x +a \right )}+6 b \,c^{2} d x \,{\mathrm e}^{2 i \left (b x +a \right )}-3 i c^{2} d \,{\mathrm e}^{2 i \left (b x +a \right )}-3 i d^{3} x^{2}+2 b \,c^{3} {\mathrm e}^{2 i \left (b x +a \right )}-6 i c \,d^{2} x -3 i c^{2} d}{b^{2} \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right )^{2}}-\frac {3 d^{2} c \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b^{3}}+\frac {6 d^{2} c \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {3 i d^{3} x^{2}}{b^{2}}+\frac {3 i d^{3} a^{2}}{b^{4}}-\frac {3 d^{3} \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {3 i d^{3} \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{4}}-\frac {6 d^{3} a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {6 c \,d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {3 i c^{2} d \,a^{2}}{b^{2}}-\frac {2 i a^{3} d^{3} x}{b^{3}}+\frac {4 i c \,d^{2} a^{3}}{b^{3}}+\frac {6 c^{2} d a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {6 i d^{3} a x}{b^{3}}-\frac {3 i d^{3} \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right ) x^{2}}{2 b^{2}}-\frac {3 i c^{2} d \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{2}}+\frac {3 c^{2} d \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{b}+\frac {3 c \,d^{2} \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) x^{2}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.19, size = 2405, normalized size = 9.29 \[ \text {result too large to display} \]
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 \[ \int {\mathrm {tan}\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^3 \,d x \]
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 \[ \int \left (c + d x\right )^{3} \tan ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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