Optimal. Leaf size=131 \[ \frac {a^2 (c+d x)^2}{2 d}-\frac {4 i a b (c+d x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {2 i a b d \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {2 i a b d \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}+\frac {b^2 (c+d x) \tan (e+f x)}{f}+\frac {b^2 d \log (\cos (e+f x))}{f^2} \]
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Rubi [A] time = 0.12, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4190, 4181, 2279, 2391, 4184, 3475} \[ \frac {2 i a b d \text {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {2 i a b d \text {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}+\frac {a^2 (c+d x)^2}{2 d}-\frac {4 i a b (c+d x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {b^2 (c+d x) \tan (e+f x)}{f}+\frac {b^2 d \log (\cos (e+f x))}{f^2} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 3475
Rule 4181
Rule 4184
Rule 4190
Rubi steps
\begin {align*} \int (c+d x) (a+b \sec (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)+2 a b (c+d x) \sec (e+f x)+b^2 (c+d x) \sec ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^2}{2 d}+(2 a b) \int (c+d x) \sec (e+f x) \, dx+b^2 \int (c+d x) \sec ^2(e+f x) \, dx\\ &=\frac {a^2 (c+d x)^2}{2 d}-\frac {4 i a b (c+d x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {b^2 (c+d x) \tan (e+f x)}{f}-\frac {(2 a b d) \int \log \left (1-i e^{i (e+f x)}\right ) \, dx}{f}+\frac {(2 a b d) \int \log \left (1+i e^{i (e+f x)}\right ) \, dx}{f}-\frac {\left (b^2 d\right ) \int \tan (e+f x) \, dx}{f}\\ &=\frac {a^2 (c+d x)^2}{2 d}-\frac {4 i a b (c+d x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {b^2 d \log (\cos (e+f x))}{f^2}+\frac {b^2 (c+d x) \tan (e+f x)}{f}+\frac {(2 i a b d) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^2}-\frac {(2 i a b d) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^2}\\ &=\frac {a^2 (c+d x)^2}{2 d}-\frac {4 i a b (c+d x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {b^2 d \log (\cos (e+f x))}{f^2}+\frac {2 i a b d \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {2 i a b d \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}+\frac {b^2 (c+d x) \tan (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 151, normalized size = 1.15 \[ \frac {2 a^2 c f^2 x+a^2 d f^2 x^2+4 a b c f \tanh ^{-1}(\sin (e+f x))+4 i a b d \text {Li}_2\left (-i e^{i (e+f x)}\right )-4 i a b d \text {Li}_2\left (i e^{i (e+f x)}\right )-8 i a b d f x \tan ^{-1}\left (e^{i (e+f x)}\right )+2 b^2 c f \tan (e+f x)+2 b^2 d f x \tan (e+f x)+2 b^2 d \log (\cos (e+f x))}{2 f^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.65, size = 505, normalized size = 3.85 \[ \frac {-2 i \, a b d \cos \left (f x + e\right ) {\rm Li}_2\left (i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) - 2 i \, a b d \cos \left (f x + e\right ) {\rm Li}_2\left (i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) + 2 i \, a b d \cos \left (f x + e\right ) {\rm Li}_2\left (-i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) + 2 i \, a b d \cos \left (f x + e\right ) {\rm Li}_2\left (-i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) - {\left (2 \, a b d e - 2 \, a b c f - b^{2} d\right )} \cos \left (f x + e\right ) \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) + {\left (2 \, a b d e - 2 \, a b c f + b^{2} d\right )} \cos \left (f x + e\right ) \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + i\right ) + 2 \, {\left (a b d f x + a b d e\right )} \cos \left (f x + e\right ) \log \left (i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (a b d f x + a b d e\right )} \cos \left (f x + e\right ) \log \left (i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (a b d f x + a b d e\right )} \cos \left (f x + e\right ) \log \left (-i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (a b d f x + a b d e\right )} \cos \left (f x + e\right ) \log \left (-i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right ) - {\left (2 \, a b d e - 2 \, a b c f - b^{2} d\right )} \cos \left (f x + e\right ) \log \left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) + {\left (2 \, a b d e - 2 \, a b c f + b^{2} d\right )} \cos \left (f x + e\right ) \log \left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + i\right ) + {\left (a^{2} d f^{2} x^{2} + 2 \, a^{2} c f^{2} x\right )} \cos \left (f x + e\right ) + 2 \, {\left (b^{2} d f x + b^{2} c f\right )} \sin \left (f x + e\right )}{2 \, f^{2} \cos \left (f x + e\right )} \]
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 \[ \int {\left (d x + c\right )} {\left (b \sec \left (f x + e\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.52, size = 271, normalized size = 2.07 \[ \frac {a^{2} d \,x^{2}}{2}-\frac {a^{2} d \,e^{2}}{2 f^{2}}+a^{2} c x +\frac {a^{2} c e}{f}-\frac {2 a b d \ln \left (i {\mathrm e}^{i \left (f x +e \right )}+1\right ) x}{f}+\frac {2 a b d \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{f}-\frac {2 a b d \ln \left (i {\mathrm e}^{i \left (f x +e \right )}+1\right ) e}{f^{2}}+\frac {2 a b d \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) e}{f^{2}}-\frac {2 i a b d \dilog \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {2 i a b d \dilog \left (i {\mathrm e}^{i \left (f x +e \right )}+1\right )}{f^{2}}+\frac {2 a b c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}-\frac {2 a b d e \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f^{2}}+\frac {b^{2} d \tan \left (f x +e \right ) x}{f}+\frac {b^{2} d \ln \left (\cos \left (f x +e \right )\right )}{f^{2}}+\frac {c \,b^{2} \tan \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int {\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^2\,\left (c+d\,x\right ) \,d x \]
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 \[ \int \left (a + b \sec {\left (e + f x \right )}\right )^{2} \left (c + d x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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