Optimal. Leaf size=86 \[ \frac{a (2 B+3 C) \tan (c+d x)}{3 d}+\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (B+C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.209661, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3029, 2968, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac{a (2 B+3 C) \tan (c+d x)}{3 d}+\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (B+C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2968
Rule 3021
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+a \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx &=\int (a+a \cos (c+d x)) (B+C \cos (c+d x)) \sec ^4(c+d x) \, dx\\ &=\int \left (a B+(a B+a C) \cos (c+d x)+a 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{1}{3} \int (3 a (B+C)+a (2 B+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}+(a (B+C)) \int \sec ^3(c+d x) \, dx+\frac{1}{3} (a (2 B+3 C)) \int \sec ^2(c+d x) \, dx\\ &=\frac{a (B+C) \sec (c+d x) \tan (c+d x)}{2 d}+\frac{a B \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{2} (a (B+C)) \int \sec (c+d x) \, dx-\frac{(a (2 B+3 C)) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (2 B+3 C) \tan (c+d x)}{3 d}+\frac{a (B+C) \sec (c+d x) \tan (c+d x)}{2 d}+\frac{a B \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.312306, size = 56, normalized size = 0.65 \[ \frac{a \left (3 (B+C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (3 (B+C) \sec (c+d x)+2 B \tan ^2(c+d x)+6 (B+C)\right )\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 128, normalized size = 1.5 \begin{align*}{\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}}+{\frac{2\,Ba\tan \left ( dx+c \right ) }{3\,d}}+{\frac{Ba \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{aC\tan \left ( dx+c \right ) }{d}}+{\frac{Ba\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{Ba\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 = 1.29784, size = 171, normalized size = 1.99 \begin{align*} \frac{4 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a - 3 \, 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 )} - 3 \, 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 )} + 12 \, C a \tan \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68686, size = 288, normalized size = 3.35 \begin{align*} \frac{3 \,{\left (B + C\right )} a \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (B + C\right )} a \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (2 \,{\left (2 \, B + 3 \, C\right )} a \cos \left (d x + c\right )^{2} + 3 \,{\left (B + C\right )} a \cos \left (d x + c\right ) + 2 \, B a\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \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 = 1.64673, size = 208, normalized size = 2.42 \begin{align*} \frac{3 \,{\left (B a + C a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (B a + C a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (3 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 4 \, B 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} + 9 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 9 \, 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 )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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