Optimal. Leaf size=106 \[ \frac{a (A+B) \tan ^3(c+d x)}{3 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a (3 A+4 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.164655, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2968, 3021, 2748, 3767, 3768, 3770} \[ \frac{a (A+B) \tan ^3(c+d x)}{3 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a (3 A+4 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 2968
Rule 3021
Rule 2748
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx &=\int \left (a A+(a A+a B) \cos (c+d x)+a B \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 (4 a (A+B)+a (3 A+4 B) \cos (c+d x)) \sec ^4(c+d x) \, dx\\ &=\frac{a A \sec ^3(c+d x) \tan (c+d x)}{4 d}+(a (A+B)) \int \sec ^4(c+d x) \, dx+\frac{1}{4} (a (3 A+4 B)) \int \sec ^3(c+d x) \, dx\\ &=\frac{a (3 A+4 B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a A \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{8} (a (3 A+4 B)) \int \sec (c+d x) \, dx-\frac{(a (A+B)) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{a (3 A+4 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a (3 A+4 B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a A \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{a (A+B) \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.363648, size = 77, normalized size = 0.73 \[ \frac{a \left (3 (3 A+4 B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left (8 (A+B) (\cos (2 (c+d x))+2) \sec (c+d x)+6 A \sec ^2(c+d x)+9 A+12 B\right )\right )}{24 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.151, size = 171, normalized size = 1.6 \begin{align*}{\frac{2\,aA\tan \left ( dx+c \right ) }{3\,d}}+{\frac{aA \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{aB\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{aB\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,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{2\,aB\tan \left ( dx+c \right ) }{3\,d}}+{\frac{aB\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982084, size = 220, normalized size = 2.08 \begin{align*} \frac{16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a + 16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a - 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 \, B 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 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40998, size = 339, normalized size = 3.2 \begin{align*} \frac{3 \,{\left (3 \, A + 4 \, B\right )} a \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (3 \, A + 4 \, B\right )} a \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (A + B\right )} a \cos \left (d x + c\right )^{3} + 3 \,{\left (3 \, A + 4 \, B\right )} a \cos \left (d x + c\right )^{2} + 8 \,{\left (A + B\right )} a \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 [A] time = 1.27073, size = 254, normalized size = 2.4 \begin{align*} \frac{3 \,{\left (3 \, A a + 4 \, B 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 \, B a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (9 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 12 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 49 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 28 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 31 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 52 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 39 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 36 \, B 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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