Optimal. Leaf size=117 \[ \frac{a (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{b (2 A+3 C) \tan (c+d x)}{3 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.190869, 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 = {3032, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac{a (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{b (2 A+3 C) \tan (c+d x)}{3 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3032
Rule 3021
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx &=\frac{a A \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} \int \left (4 A b+a (3 A+4 C) \cos (c+d x)+4 b C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{A b \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{a A \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{12} \int (3 a (3 A+4 C)+4 b (2 A+3 C) \cos (c+d x)) \sec ^3(c+d x) \, dx\\ &=\frac{A b \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{a A \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{3} (b (2 A+3 C)) \int \sec ^2(c+d x) \, dx+\frac{1}{4} (a (3 A+4 C)) \int \sec ^3(c+d x) \, dx\\ &=\frac{a (3 A+4 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{A b \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{a A \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{8} (a (3 A+4 C)) \int \sec (c+d x) \, dx-\frac{(b (2 A+3 C)) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{a (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a (3 A+4 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{A b \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{a A \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.446148, size = 80, normalized size = 0.68 \[ \frac{\tan (c+d x) \left (3 a (3 A+4 C) \sec (c+d x)+6 a A \sec ^3(c+d x)+8 b \left (A \tan ^2(c+d x)+3 (A+C)\right )\right )+3 a (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{24 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 149, normalized size = 1.3 \begin{align*}{\frac{2\,Ab\tan \left ( dx+c \right ) }{3\,d}}+{\frac{Ab \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{Cb\tan \left ( dx+c \right ) }{d}}+{\frac{aA \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,aA\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,aA\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{aC\tan \left ( dx+c \right ) \sec \left ( dx+c \right ) }{2\,d}}+{\frac{aC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992948, 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 )} A b - 3 \, A 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 \, C 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 \, C b \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 = 1.41702, size = 335, normalized size = 2.86 \begin{align*} \frac{3 \,{\left (3 \, A + 4 \, C\right )} a \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (3 \, A + 4 \, C\right )} a \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (8 \,{\left (2 \, A + 3 \, C\right )} b \cos \left (d x + c\right )^{3} + 3 \,{\left (3 \, A + 4 \, C\right )} a \cos \left (d x + c\right )^{2} + 8 \, A b \cos \left (d x + c\right ) + 6 \, A 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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2762, size = 410, normalized size = 3.5 \begin{align*} \frac{3 \,{\left (3 \, A a + 4 \, C a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (3 \, A a + 4 \, C a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (15 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 12 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 24 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 24 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 9 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 12 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 40 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 72 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 40 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 72 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 12 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, C b \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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