Optimal. Leaf size=134 \[ \frac {a^2 (c+d x)^2}{2 d}-\frac {4 i a^2 (c+d x) \text {ArcTan}\left (e^{i (e+f x)}\right )}{f}+\frac {a^2 d \log (\cos (e+f x))}{f^2}+\frac {2 i a^2 d \text {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {2 i a^2 d \text {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}+\frac {a^2 (c+d x) \tan (e+f x)}{f} \]
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Rubi [A]
time = 0.09, antiderivative size = 134, 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 = {4275, 4266,
2317, 2438, 4269, 3556} \begin {gather*} -\frac {4 i a^2 (c+d x) \text {ArcTan}\left (e^{i (e+f x)}\right )}{f}+\frac {a^2 (c+d x) \tan (e+f x)}{f}+\frac {a^2 (c+d x)^2}{2 d}+\frac {2 i a^2 d \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {2 i a^2 d \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}+\frac {a^2 d \log (\cos (e+f x))}{f^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 3556
Rule 4266
Rule 4269
Rule 4275
Rubi steps
\begin {align*} \int (c+d x) (a+a \sec (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)+2 a^2 (c+d x) \sec (e+f x)+a^2 (c+d x) \sec ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^2}{2 d}+a^2 \int (c+d x) \sec ^2(e+f x) \, dx+\left (2 a^2\right ) \int (c+d x) \sec (e+f x) \, dx\\ &=\frac {a^2 (c+d x)^2}{2 d}-\frac {4 i a^2 (c+d x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {a^2 (c+d x) \tan (e+f x)}{f}-\frac {\left (a^2 d\right ) \int \tan (e+f x) \, dx}{f}-\frac {\left (2 a^2 d\right ) \int \log \left (1-i e^{i (e+f x)}\right ) \, dx}{f}+\frac {\left (2 a^2 d\right ) \int \log \left (1+i e^{i (e+f x)}\right ) \, dx}{f}\\ &=\frac {a^2 (c+d x)^2}{2 d}-\frac {4 i a^2 (c+d x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {a^2 d \log (\cos (e+f x))}{f^2}+\frac {a^2 (c+d x) \tan (e+f x)}{f}+\frac {\left (2 i a^2 d\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^2}-\frac {\left (2 i a^2 d\right ) \text {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^2 (c+d x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {a^2 d \log (\cos (e+f x))}{f^2}+\frac {2 i a^2 d \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {2 i a^2 d \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}+\frac {a^2 (c+d x) \tan (e+f x)}{f}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(330\) vs. \(2(134)=268\).
time = 5.57, size = 330, normalized size = 2.46 \begin {gather*} \frac {a^2 (1+\cos (e+f x))^2 \sec ^4\left (\frac {1}{2} (e+f x)\right ) \left (8 c f \tanh ^{-1}\left (\sin (e)+\cos (e) \tan \left (\frac {f x}{2}\right )\right )+8 d \text {ArcTan}(\cot (e)) \tanh ^{-1}\left (\sin (e)+\cos (e) \tan \left (\frac {f x}{2}\right )\right )-\frac {4 d \csc (e) \left ((f x-\text {ArcTan}(\cot (e))) \left (\log \left (1-e^{i (f x-\text {ArcTan}(\cot (e)))}\right )-\log \left (1+e^{i (f x-\text {ArcTan}(\cot (e)))}\right )\right )+i \text {PolyLog}\left (2,-e^{i (f x-\text {ArcTan}(\cot (e)))}\right )-i \text {PolyLog}\left (2,e^{i (f x-\text {ArcTan}(\cot (e)))}\right )\right )}{\sqrt {\csc ^2(e)}}+\frac {2 f (c+d x) \sin \left (\frac {f x}{2}\right )}{\left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {2 f (c+d x) \sin \left (\frac {f x}{2}\right )}{\left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}-2 d f x \tan (e)+f x (2 c f+d f x+2 d \tan (e))+2 d (\log (\cos (e+f x))+f x \tan (e))\right )}{8 f^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.56, size = 235, normalized size = 1.75
method | result | size |
derivativedivides | \(\frac {a^{2} c \tan \left (f x +e \right )-\frac {a^{2} d e \tan \left (f x +e \right )}{f}+\frac {a^{2} d \left (\left (f x +e \right ) \tan \left (f x +e \right )+\ln \left (\cos \left (f x +e \right )\right )\right )}{f}+2 a^{2} c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )-\frac {2 a^{2} d e \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+\frac {2 a^{2} d \left (-\left (f x +e \right ) \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right )+\left (f x +e \right ) \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right )+i \dilog \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right )-i \dilog \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right )\right )}{f}+a^{2} c \left (f x +e \right )-\frac {a^{2} d e \left (f x +e \right )}{f}+\frac {a^{2} d \left (f x +e \right )^{2}}{2 f}}{f}\) | \(235\) |
default | \(\frac {a^{2} c \tan \left (f x +e \right )-\frac {a^{2} d e \tan \left (f x +e \right )}{f}+\frac {a^{2} d \left (\left (f x +e \right ) \tan \left (f x +e \right )+\ln \left (\cos \left (f x +e \right )\right )\right )}{f}+2 a^{2} c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )-\frac {2 a^{2} d e \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+\frac {2 a^{2} d \left (-\left (f x +e \right ) \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right )+\left (f x +e \right ) \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right )+i \dilog \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right )-i \dilog \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right )\right )}{f}+a^{2} c \left (f x +e \right )-\frac {a^{2} d e \left (f x +e \right )}{f}+\frac {a^{2} d \left (f x +e \right )^{2}}{2 f}}{f}\) | \(235\) |
risch | \(\frac {a^{2} d \,x^{2}}{2}+a^{2} c x +\frac {2 i a^{2} \left (d x +c \right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}-\frac {2 a^{2} d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {a^{2} d \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f^{2}}-\frac {4 i a^{2} c \arctan \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f}+\frac {4 i a^{2} d e \arctan \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {2 a^{2} \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right ) d x}{f}-\frac {2 a^{2} \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right ) d e}{f^{2}}+\frac {2 a^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) d x}{f}+\frac {2 a^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) d e}{f^{2}}+\frac {2 i a^{2} d \dilog \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {2 i a^{2} d \dilog \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}\) | \(274\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 572 vs. \(2 (119) = 238\).
time = 2.93, size = 572, normalized size = 4.27 \begin {gather*} \frac {-2 i \, a^{2} 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^{2} 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^{2} 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^{2} 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^{2} c f - 2 \, a^{2} d e + a^{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^{2} c f - 2 \, a^{2} d e - a^{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^{2} d f x + a^{2} 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^{2} d f x + a^{2} 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^{2} d f x + a^{2} 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^{2} d f x + a^{2} 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^{2} c f - 2 \, a^{2} d e + a^{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^{2} c f - 2 \, a^{2} d e - a^{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 (a^{2} d f x + a^{2} c f\right )} \sin \left (f x + e\right )}{2 \, f^{2} \cos \left (f x + e\right )} \end {gather*}
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 \begin {gather*} a^{2} \left (\int c\, dx + \int 2 c \sec {\left (e + f x \right )}\, dx + \int c \sec ^{2}{\left (e + f x \right )}\, dx + \int d x\, dx + \int 2 d x \sec {\left (e + f x \right )}\, dx + \int d x \sec ^{2}{\left (e + f x \right )}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2\,\left (c+d\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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