Optimal. Leaf size=94 \[ \frac{5 i a^2 \sec ^3(c+d x)}{12 d}+\frac{5 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}+\frac{5 a^2 \tan (c+d x) \sec (c+d x)}{8 d} \]
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Rubi [A] time = 0.0766581, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3498, 3486, 3768, 3770} \[ \frac{5 i a^2 \sec ^3(c+d x)}{12 d}+\frac{5 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}+\frac{5 a^2 \tan (c+d x) \sec (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 3498
Rule 3486
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx &=\frac{i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}+\frac{1}{4} (5 a) \int \sec ^3(c+d x) (a+i a \tan (c+d x)) \, dx\\ &=\frac{5 i a^2 \sec ^3(c+d x)}{12 d}+\frac{i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}+\frac{1}{4} \left (5 a^2\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{5 i a^2 \sec ^3(c+d x)}{12 d}+\frac{5 a^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}+\frac{1}{8} \left (5 a^2\right ) \int \sec (c+d x) \, dx\\ &=\frac{5 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{5 i a^2 \sec ^3(c+d x)}{12 d}+\frac{5 a^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}\\ \end{align*}
Mathematica [B] time = 1.051, size = 215, normalized size = 2.29 \[ \frac{a^2 \sec ^4(c+d x) \left (-18 \sin (c+d x)+30 \sin (3 (c+d x))+128 i \cos (c+d x)-45 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-60 \cos (2 (c+d x)) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-15 \cos (4 (c+d x)) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+45 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{192 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 123, normalized size = 1.3 \begin{align*} -{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{2}\sin \left ( dx+c \right ) }{8\,d}}+{\frac{5\,{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{{\frac{2\,i}{3}}{a}^{2}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{a}^{2}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06383, size = 176, normalized size = 1.87 \begin{align*} -\frac{3 \, a^{2}{\left (\frac{2 \,{\left (\sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{2}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac{32 i \, a^{2}}{\cos \left (d x + c\right )^{3}}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.2719, size = 711, normalized size = 7.56 \begin{align*} \frac{-30 i \, a^{2} e^{\left (7 i \, d x + 7 i \, c\right )} + 146 i \, a^{2} e^{\left (5 i \, d x + 5 i \, c\right )} + 110 i \, a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} + 30 i \, a^{2} e^{\left (i \, d x + i \, c\right )} + 15 \,{\left (a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 15 \,{\left (a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{24 \,{\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int - \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 2 i \tan{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sec ^{3}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2133, size = 236, normalized size = 2.51 \begin{align*} \frac{15 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (9 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 48 i \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 33 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 48 i \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 33 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 16 i \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 9 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 16 i \, a^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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