Optimal. Leaf size=117 \[ \frac{a (3 A+2 C) \tan (c+d x)}{3 d}+\frac{a (4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 A+3 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a C \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a C \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.170044, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {4077, 4047, 3768, 3770, 4046, 3767, 8} \[ \frac{a (3 A+2 C) \tan (c+d x)}{3 d}+\frac{a (4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 A+3 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a C \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a C \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4077
Rule 4047
Rule 3768
Rule 3770
Rule 4046
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{a C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} \int \sec ^2(c+d x) \left (4 a A+a (4 A+3 C) \sec (c+d x)+4 a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} \int \sec ^2(c+d x) \left (4 a A+4 a C \sec ^2(c+d x)\right ) \, dx+\frac{1}{4} (a (4 A+3 C)) \int \sec ^3(c+d x) \, dx\\ &=\frac{a (4 A+3 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a C \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{a C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{3} (a (3 A+2 C)) \int \sec ^2(c+d x) \, dx+\frac{1}{8} (a (4 A+3 C)) \int \sec (c+d x) \, dx\\ &=\frac{a (4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 A+3 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a C \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{a C \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{(a (3 A+2 C)) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{a (4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+2 C) \tan (c+d x)}{3 d}+\frac{a (4 A+3 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a C \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{a C \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.428452, size = 75, normalized size = 0.64 \[ \frac{a \left (3 (4 A+3 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (3 (4 A+3 C) \sec (c+d x)+24 (A+C)+8 C \tan ^2(c+d x)+6 C \sec ^3(c+d x)\right )\right )}{24 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 149, normalized size = 1.3 \begin{align*}{\frac{Aa\tan \left ( dx+c \right ) }{d}}+{\frac{2\,aC\tan \left ( dx+c \right ) }{3\,d}}+{\frac{aC \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{Aa\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{Aa\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{aC \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,aC\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,aC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.941002, size = 205, normalized size = 1.75 \begin{align*} \frac{16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a - 3 \, C a{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, A a{\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 )} + 48 \, A a \tan \left (d x + c\right )}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.509292, size = 335, normalized size = 2.86 \begin{align*} \frac{3 \,{\left (4 \, A + 3 \, C\right )} a \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (4 \, A + 3 \, C\right )} a \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (8 \,{\left (3 \, A + 2 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \,{\left (4 \, A + 3 \, C\right )} a \cos \left (d x + c\right )^{2} + 8 \, C a \cos \left (d x + c\right ) + 6 \, C a\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \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 \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int A \sec ^{3}{\left (c + d x \right )}\, dx + \int C \sec ^{4}{\left (c + d x \right )}\, dx + \int C \sec ^{5}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25697, size = 254, normalized size = 2.17 \begin{align*} \frac{3 \,{\left (4 \, A a + 3 \, C a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (4 \, A a + 3 \, C a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (12 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 9 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 60 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 49 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 84 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 31 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 39 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\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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