Optimal. Leaf size=169 \[ \frac {d^2 \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {d^2 \log (\cos (a+b x))}{b^3}-\frac {i d (c+d x) \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac {d (c+d x) \tan (a+b x)}{b^2}+\frac {(c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b}+\frac {c d x}{b}+\frac {d^2 x^2}{2 b}-\frac {i (c+d x)^3}{3 d} \]
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Rubi [A] time = 0.22, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3720, 3475, 3719, 2190, 2531, 2282, 6589} \[ -\frac {i d (c+d x) \text {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}+\frac {d^2 \text {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {d (c+d x) \tan (a+b x)}{b^2}-\frac {d^2 \log (\cos (a+b x))}{b^3}+\frac {(c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b}+\frac {c d x}{b}+\frac {d^2 x^2}{2 b}-\frac {i (c+d x)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3475
Rule 3719
Rule 3720
Rule 6589
Rubi steps
\begin {align*} \int (c+d x)^2 \tan ^3(a+b x) \, dx &=\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b}-\frac {d \int (c+d x) \tan ^2(a+b x) \, dx}{b}-\int (c+d x)^2 \tan (a+b x) \, dx\\ &=-\frac {i (c+d x)^3}{3 d}-\frac {d (c+d x) \tan (a+b x)}{b^2}+\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b}+2 i \int \frac {e^{2 i (a+b x)} (c+d x)^2}{1+e^{2 i (a+b x)}} \, dx+\frac {d \int (c+d x) \, dx}{b}+\frac {d^2 \int \tan (a+b x) \, dx}{b^2}\\ &=\frac {c d x}{b}+\frac {d^2 x^2}{2 b}-\frac {i (c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {d^2 \log (\cos (a+b x))}{b^3}-\frac {d (c+d x) \tan (a+b x)}{b^2}+\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b}-\frac {(2 d) \int (c+d x) \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b}\\ &=\frac {c d x}{b}+\frac {d^2 x^2}{2 b}-\frac {i (c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {d^2 \log (\cos (a+b x))}{b^3}-\frac {i d (c+d x) \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac {d (c+d x) \tan (a+b x)}{b^2}+\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b}+\frac {\left (i d^2\right ) \int \text {Li}_2\left (-e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=\frac {c d x}{b}+\frac {d^2 x^2}{2 b}-\frac {i (c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {d^2 \log (\cos (a+b x))}{b^3}-\frac {i d (c+d x) \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac {d (c+d x) \tan (a+b x)}{b^2}+\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b}+\frac {d^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^3}\\ &=\frac {c d x}{b}+\frac {d^2 x^2}{2 b}-\frac {i (c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {d^2 \log (\cos (a+b x))}{b^3}-\frac {i d (c+d x) \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}+\frac {d^2 \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {d (c+d x) \tan (a+b x)}{b^2}+\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [B] time = 6.66, size = 454, normalized size = 2.69 \[ -\frac {d^2 \sec (a) (b x \sin (a)+\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x)))}{b^3 \left (\sin ^2(a)+\cos ^2(a)\right )}+\frac {\sec (a) \sec (a+b x) \left (d^2 (-x) \sin (b x)-c d \sin (b x)\right )}{b^2}+\frac {c d \csc (a) \sec (a) \left (b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac {\cot (a) \left (i \text {Li}_2\left (e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )+i b x \left (-2 \tan ^{-1}(\cot (a))-\pi \right )-2 \left (b x-\tan ^{-1}(\cot (a))\right ) \log \left (1-e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )-2 \tan ^{-1}(\cot (a)) \log \left (\sin \left (b x-\tan ^{-1}(\cot (a))\right )\right )-\pi \log \left (1+e^{-2 i b x}\right )+\pi \log (\cos (b x))\right )}{\sqrt {\cot ^2(a)+1}}\right )}{b^2 \sqrt {\csc ^2(a) \left (\sin ^2(a)+\cos ^2(a)\right )}}+\frac {i e^{-i a} d^2 \sec (a) \left (2 b^2 x^2 \left (2 b x-3 i \left (1+e^{2 i a}\right ) \log \left (1+e^{-2 i (a+b x)}\right )\right )+6 \left (1+e^{2 i a}\right ) b x \text {Li}_2\left (-e^{-2 i (a+b x)}\right )-3 i \left (1+e^{2 i a}\right ) \text {Li}_3\left (-e^{-2 i (a+b x)}\right )\right )}{12 b^3}+\frac {c^2 \sec (a) (b x \sin (a)+\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x)))}{b \left (\sin ^2(a)+\cos ^2(a)\right )}+\frac {(c+d x)^2 \sec ^2(a+b x)}{2 b}-\frac {1}{3} x \tan (a) \left (3 c^2+3 c d x+d^2 x^2\right ) \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.45, size = 352, normalized size = 2.08 \[ \frac {2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + d^{2} {\rm polylog}\left (3, \frac {\tan \left (b x + a\right )^{2} + 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + d^{2} {\rm polylog}\left (3, \frac {\tan \left (b x + a\right )^{2} - 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \tan \left (b x + a\right )^{2} + {\left (2 i \, b d^{2} x + 2 i \, b c d\right )} {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) + {\left (-2 i \, b d^{2} x - 2 i \, b c d\right )} {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - d^{2}\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - d^{2}\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) - 4 \, {\left (b d^{2} x + b c d\right )} \tan \left (b x + a\right )}{4 \, b^{3}} \]
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 \[ \int {\left (d x + c\right )}^{2} \tan \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 400, normalized size = 2.37 \[ -\frac {i d^{2} x^{3}}{3}-\frac {4 i c d a x}{b}+\frac {4 i d^{2} a^{3}}{3 b^{3}}+\frac {2 b \,d^{2} x^{2} {\mathrm e}^{2 i \left (b x +a \right )}-2 i d^{2} x \,{\mathrm e}^{2 i \left (b x +a \right )}+4 b c d x \,{\mathrm e}^{2 i \left (b x +a \right )}-2 i c d \,{\mathrm e}^{2 i \left (b x +a \right )}+2 b \,c^{2} {\mathrm e}^{2 i \left (b x +a \right )}-2 i d^{2} x -2 i c d}{b^{2} \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right )^{2}}-\frac {i d^{2} \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{b^{2}}+i c^{2} x +\frac {2 c d \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{b}+\frac {d^{2} \polylog \left (3, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{3}}-\frac {d^{2} \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b^{3}}+\frac {2 d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {2 d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {c^{2} \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b}-\frac {2 c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b}+\frac {2 i d^{2} a^{2} x}{b^{2}}-\frac {i c d \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b^{2}}+\frac {d^{2} \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) x^{2}}{b}-i c d \,x^{2}+\frac {4 c d a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {2 i c d \,a^{2}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.67, size = 1226, normalized size = 7.25 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tan}\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^2 \,d x \]
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 \[ \int \left (c + d x\right )^{2} \tan ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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