Optimal. Leaf size=115 \[ \frac {3 i d^3 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}+\frac {3 i d (c+d x)^2}{2 b^2} \]
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Rubi [A] time = 0.17, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4409, 4184, 3719, 2190, 2279, 2391} \[ \frac {3 i d^3 \text {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}+\frac {3 i d (c+d x)^2}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4184
Rule 4409
Rubi steps
\begin {align*} \int (c+d x)^3 \sec ^2(a+b x) \tan (a+b x) \, dx &=\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac {(3 d) \int (c+d x)^2 \sec ^2(a+b x) \, dx}{2 b}\\ &=\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\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 \sec ^2(a+b x)}{2 b}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}-\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 {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 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 {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}-\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 {3 i d (c+d x)^2}{2 b^2}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d^3 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}\\ \end {align*}
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Mathematica [B] time = 6.38, size = 286, normalized size = 2.49 \[ -\frac {3 c d^2 \sec (a) (b x \sin (a)+\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x)))}{b^3 \left (\sin ^2(a)+\cos ^2(a)\right )}-\frac {3 \sec (a) \sec (a+b x) \left (c^2 d \sin (b x)+2 c d^2 x \sin (b x)+d^3 x^2 \sin (b x)\right )}{2 b^2}-\frac {3 d^3 \csc (a) \sec (a) \left (b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac {\cot (a) \left (i \text {Li}_2\left (e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )+i b x \left (-2 \tan ^{-1}(\cot (a))-\pi \right )-2 \left (b x-\tan ^{-1}(\cot (a))\right ) \log \left (1-e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )-2 \tan ^{-1}(\cot (a)) \log \left (\sin \left (b x-\tan ^{-1}(\cot (a))\right )\right )-\pi \log \left (1+e^{-2 i b x}\right )+\pi \log (\cos (b x))\right )}{\sqrt {\cot ^2(a)+1}}\right )}{2 b^4 \sqrt {\csc ^2(a) \left (\sin ^2(a)+\cos ^2(a)\right )}}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.53, size = 540, normalized size = 4.70 \[ \frac {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} - 3 i \, d^{3} \cos \left (b x + a\right )^{2} {\rm Li}_2\left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) + 3 i \, d^{3} \cos \left (b x + a\right )^{2} {\rm Li}_2\left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + 3 i \, d^{3} \cos \left (b x + a\right )^{2} {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - 3 i \, d^{3} \cos \left (b x + a\right )^{2} {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) - 3 \, {\left (b c d^{2} - a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) - 3 \, {\left (b c d^{2} - a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - 3 \, {\left (b d^{3} x + a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (b d^{3} x + a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (b d^{3} x + a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (b d^{3} x + a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (b c d^{2} - a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) - 3 \, {\left (b c d^{2} - a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right )}{2 \, b^{4} \cos \left (b x + a\right )^{2}} \]
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 \[ \int {\left (d x + c\right )}^{3} \sec \left (b x + a\right )^{2} \tan \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 301, normalized size = 2.62 \[ \frac {2 b \,d^{3} x^{3} {\mathrm e}^{2 i \left (b x +a \right )}-3 i d^{3} x^{2} {\mathrm e}^{2 i \left (b x +a \right )}+6 b c \,d^{2} x^{2} {\mathrm e}^{2 i \left (b x +a \right )}-6 i c \,d^{2} x \,{\mathrm e}^{2 i \left (b x +a \right )}+6 b \,c^{2} d x \,{\mathrm e}^{2 i \left (b x +a \right )}-3 i c^{2} d \,{\mathrm e}^{2 i \left (b x +a \right )}-3 i d^{3} x^{2}+2 b \,c^{3} {\mathrm e}^{2 i \left (b x +a \right )}-6 i c \,d^{2} x -3 i c^{2} d}{b^{2} \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right )^{2}}-\frac {3 d^{2} c \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b^{3}}+\frac {6 d^{2} c \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {3 i d^{3} x^{2}}{b^{2}}+\frac {6 i d^{3} a x}{b^{3}}+\frac {3 i d^{3} a^{2}}{b^{4}}-\frac {3 d^{3} \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {3 i d^{3} \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{4}}-\frac {6 d^{3} a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.63, size = 667, normalized size = 5.80 \[ -\frac {6 \, b^{2} c^{2} d + {\left (6 \, b d^{3} x + 6 \, b c d^{2} + 6 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (4 \, b x + 4 \, a\right ) + 12 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (6 i \, b d^{3} x + 6 i \, b c d^{2}\right )} \sin \left (4 \, b x + 4 \, a\right ) + {\left (12 i \, b d^{3} x + 12 i \, b c d^{2}\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 ) - 6 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x\right )} \cos \left (4 \, b x + 4 \, a\right ) + {\left (4 i \, b^{3} d^{3} x^{3} + 4 i \, b^{3} c^{3} + 6 \, b^{2} c^{2} d + {\left (12 i \, b^{3} c d^{2} - 6 \, b^{2} d^{3}\right )} x^{2} + {\left (12 i \, b^{3} c^{2} d - 12 \, b^{2} c d^{2}\right )} x\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left (3 \, d^{3} \cos \left (4 \, b x + 4 \, a\right ) + 6 \, d^{3} \cos \left (2 \, b x + 2 \, a\right ) + 3 i \, d^{3} \sin \left (4 \, b x + 4 \, a\right ) + 6 i \, d^{3} \sin \left (2 \, b x + 2 \, a\right ) + 3 \, d^{3}\right )} {\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + {\left (-3 i \, b d^{3} x - 3 i \, b c d^{2} + {\left (-3 i \, b d^{3} x - 3 i \, b c d^{2}\right )} \cos \left (4 \, b x + 4 \, a\right ) + {\left (-6 i \, b d^{3} x - 6 i \, b c d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right ) + 3 \, {\left (b d^{3} x + b c d^{2}\right )} \sin \left (4 \, b x + 4 \, a\right ) + 6 \, {\left (b d^{3} x + b c d^{2}\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 ) + {\left (-6 i \, b^{2} d^{3} x^{2} - 12 i \, b^{2} c d^{2} x\right )} \sin \left (4 \, b x + 4 \, a\right ) - {\left (4 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c^{3} - 6 i \, b^{2} c^{2} d + 6 \, {\left (2 \, b^{3} c d^{2} + i \, b^{2} d^{3}\right )} x^{2} + 12 \, {\left (b^{3} c^{2} d + i \, b^{2} c d^{2}\right )} x\right )} \sin \left (2 \, b x + 2 \, a\right )}{-2 i \, b^{4} \cos \left (4 \, b x + 4 \, a\right ) - 4 i \, b^{4} \cos \left (2 \, b x + 2 \, a\right ) + 2 \, b^{4} \sin \left (4 \, b x + 4 \, a\right ) + 4 \, b^{4} \sin \left (2 \, b x + 2 \, a\right ) - 2 i \, b^{4}} \]
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 \[ \int \frac {\mathrm {tan}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^3}{{\cos \left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \left (c + d x\right )^{3} \tan {\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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