Optimal. Leaf size=193 \[ -\frac {d^2 \text {Li}_3\left (-i e^{i (a+b x)}\right )}{b^3}+\frac {d^2 \text {Li}_3\left (i e^{i (a+b x)}\right )}{b^3}+\frac {d^2 \tanh ^{-1}(\sin (a+b x))}{b^3}+\frac {i d (c+d x) \text {Li}_2\left (-i e^{i (a+b x)}\right )}{b^2}-\frac {i d (c+d x) \text {Li}_2\left (i e^{i (a+b x)}\right )}{b^2}-\frac {d (c+d x) \sec (a+b x)}{b^2}-\frac {i (c+d x)^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {(c+d x)^2 \tan (a+b x) \sec (a+b x)}{2 b} \]
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Rubi [A] time = 0.14, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4186, 3770, 4181, 2531, 2282, 6589} \[ \frac {i d (c+d x) \text {Li}_2\left (-i e^{i (a+b x)}\right )}{b^2}-\frac {i d (c+d x) \text {Li}_2\left (i e^{i (a+b x)}\right )}{b^2}-\frac {d (c+d x) \sec (a+b x)}{b^2}-\frac {d^2 \text {Li}_3\left (-i e^{i (a+b x)}\right )}{b^3}+\frac {d^2 \text {Li}_3\left (i e^{i (a+b x)}\right )}{b^3}+\frac {d^2 \tanh ^{-1}(\sin (a+b x))}{b^3}-\frac {i (c+d x)^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {(c+d x)^2 \tan (a+b x) \sec (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 3770
Rule 4181
Rule 4186
Rule 6589
Rubi steps
\begin {align*} \int (c+d x)^2 \sec ^3(a+b x) \, dx &=-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \sec (a+b x) \tan (a+b x)}{2 b}+\frac {1}{2} \int (c+d x)^2 \sec (a+b x) \, dx+\frac {d^2 \int \sec (a+b x) \, dx}{b^2}\\ &=-\frac {i (c+d x)^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {d^2 \tanh ^{-1}(\sin (a+b x))}{b^3}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \sec (a+b x) \tan (a+b x)}{2 b}-\frac {d \int (c+d x) \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b}+\frac {d \int (c+d x) \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b}\\ &=-\frac {i (c+d x)^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {d^2 \tanh ^{-1}(\sin (a+b x))}{b^3}+\frac {i d (c+d x) \text {Li}_2\left (-i e^{i (a+b x)}\right )}{b^2}-\frac {i d (c+d x) \text {Li}_2\left (i e^{i (a+b x)}\right )}{b^2}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \sec (a+b x) \tan (a+b x)}{2 b}-\frac {\left (i d^2\right ) \int \text {Li}_2\left (-i e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (i d^2\right ) \int \text {Li}_2\left (i e^{i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {i (c+d x)^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {d^2 \tanh ^{-1}(\sin (a+b x))}{b^3}+\frac {i d (c+d x) \text {Li}_2\left (-i e^{i (a+b x)}\right )}{b^2}-\frac {i d (c+d x) \text {Li}_2\left (i e^{i (a+b x)}\right )}{b^2}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \sec (a+b x) \tan (a+b x)}{2 b}-\frac {d^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}+\frac {d^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}\\ &=-\frac {i (c+d x)^2 \tan ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {d^2 \tanh ^{-1}(\sin (a+b x))}{b^3}+\frac {i d (c+d x) \text {Li}_2\left (-i e^{i (a+b x)}\right )}{b^2}-\frac {i d (c+d x) \text {Li}_2\left (i e^{i (a+b x)}\right )}{b^2}-\frac {d^2 \text {Li}_3\left (-i e^{i (a+b x)}\right )}{b^3}+\frac {d^2 \text {Li}_3\left (i e^{i (a+b x)}\right )}{b^3}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \sec (a+b x) \tan (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.92, size = 184, normalized size = 0.95 \[ \frac {-2 i b^2 (c+d x)^2 \tan ^{-1}\left (e^{i (a+b x)}\right )+b^2 (c+d x)^2 \tan (a+b x) \sec (a+b x)+2 i b d (c+d x) \text {Li}_2\left (-i e^{i (a+b x)}\right )-2 i b d (c+d x) \text {Li}_2\left (i e^{i (a+b x)}\right )-2 b d (c+d x) \sec (a+b x)-2 d^2 \text {Li}_3\left (-i e^{i (a+b x)}\right )+2 d^2 \text {Li}_3\left (i e^{i (a+b x)}\right )+2 d^2 \tanh ^{-1}(\sin (a+b x))}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [C] time = 1.24, size = 795, normalized size = 4.12 \[ -\frac {2 \, d^{2} \cos \left (b x + a\right )^{2} {\rm polylog}\left (3, i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - 2 \, d^{2} \cos \left (b x + a\right )^{2} {\rm polylog}\left (3, i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + 2 \, d^{2} \cos \left (b x + a\right )^{2} {\rm polylog}\left (3, -i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - 2 \, d^{2} \cos \left (b x + a\right )^{2} {\rm polylog}\left (3, -i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) - {\left (-2 i \, b d^{2} x - 2 i \, b c d\right )} \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 ) - {\left (-2 i \, b d^{2} x - 2 i \, b c d\right )} \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 ) - {\left (2 i \, b d^{2} x + 2 i \, b c d\right )} \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 ) - {\left (2 i \, b d^{2} x + 2 i \, b c d\right )} \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 ) - {\left (b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} + 2\right )} d^{2}\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 ) + {\left (b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} + 2\right )} d^{2}\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 ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\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 ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\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 ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\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 ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\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 ) - {\left (b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} + 2\right )} d^{2}\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 ) + {\left (b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} + 2\right )} d^{2}\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 ) + 4 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \sin \left (b x + a\right )}{4 \, b^{3} \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 \[ \int {\left (d x + c\right )}^{2} \sec \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 584, normalized size = 3.03 \[ -\frac {i c^{2} \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b}-\frac {i d^{2} a^{2} \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {2 i c d a \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {2 i d^{2} \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {c d \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}-\frac {c d \ln \left (1+i {\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}-\frac {i c d \polylog \left (2, i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {i d^{2} \polylog \left (2, i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}-\frac {d^{2} \ln \left (1+i {\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{2 b}+\frac {a^{2} d^{2} \ln \left (1+i {\mathrm e}^{i \left (b x +a \right )}\right )}{2 b^{3}}-\frac {a^{2} d^{2} \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right )}{2 b^{3}}+\frac {d^{2} \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{2 b}+\frac {d^{2} \polylog \left (3, i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {c d \ln \left (1+i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}-\frac {i \left (d^{2} x^{2} b \,{\mathrm e}^{3 i \left (b x +a \right )}+2 c d x b \,{\mathrm e}^{3 i \left (b x +a \right )}+c^{2} b \,{\mathrm e}^{3 i \left (b x +a \right )}-d^{2} x^{2} b \,{\mathrm e}^{i \left (b x +a \right )}-2 c d x b \,{\mathrm e}^{i \left (b x +a \right )}-2 i d^{2} x \,{\mathrm e}^{3 i \left (b x +a \right )}-c^{2} b \,{\mathrm e}^{i \left (b x +a \right )}-2 i c d \,{\mathrm e}^{3 i \left (b x +a \right )}-2 i d^{2} x \,{\mathrm e}^{i \left (b x +a \right )}-2 i c d \,{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{2}}+\frac {i c d \polylog \left (2, -i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {c d \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}-\frac {d^{2} \polylog \left (3, -i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {i d^{2} \polylog \left (2, -i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.70, size = 1893, normalized size = 9.81 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.01 \[ \text {Hanged} \]
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 \[ \int \left (c + d x\right )^{2} \sec ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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