Optimal. Leaf size=106 \[ \frac{a (B+C) \tan ^3(c+d x)}{3 d}+\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 B+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.216296, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184, Rules used = {3029, 2968, 3021, 2748, 3767, 3768, 3770} \[ \frac{a (B+C) \tan ^3(c+d x)}{3 d}+\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 B+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2968
Rule 3021
Rule 2748
Rule 3767
Rule 3768
Rule 3770
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 ^6(c+d x) \, dx &=\int (a+a \cos (c+d x)) (B+C \cos (c+d x)) \sec ^5(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 ^5(c+d x) \, dx\\ &=\frac{a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} \int (4 a (B+C)+a (3 B+4 C) \cos (c+d x)) \sec ^4(c+d x) \, dx\\ &=\frac{a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+(a (B+C)) \int \sec ^4(c+d x) \, dx+\frac{1}{4} (a (3 B+4 C)) \int \sec ^3(c+d x) \, dx\\ &=\frac{a (3 B+4 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{8} (a (3 B+4 C)) \int \sec (c+d x) \, dx-\frac{(a (B+C)) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{a (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (3 B+4 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{a (B+C) \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.359059, size = 77, normalized size = 0.73 \[ \frac{a \left (3 (3 B+4 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left (8 (B+C) (\cos (2 (c+d x))+2) \sec (c+d x)+6 B \sec ^2(c+d x)+9 B+12 C\right )\right )}{24 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 171, normalized size = 1.6 \begin{align*}{\frac{2\,aC\tan \left ( dx+c \right ) }{3\,d}}+{\frac{aC\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{Ba \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,Ba\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,Ba\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}}+{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18509, 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 )} B a + 16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a - 3 \, B 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 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7344, size = 339, normalized size = 3.2 \begin{align*} \frac{3 \,{\left (3 \, B + 4 \, C\right )} a \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (3 \, B + 4 \, C\right )} a \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (B + C\right )} a \cos \left (d x + c\right )^{3} + 3 \,{\left (3 \, B + 4 \, C\right )} a \cos \left (d x + c\right )^{2} + 8 \,{\left (B + C\right )} a \cos \left (d x + c\right ) + 6 \, B 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 = 1.4386, size = 254, normalized size = 2.4 \begin{align*} \frac{3 \,{\left (3 \, B 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 \, B 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 (9 \, B 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} - 49 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 28 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 31 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 52 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 39 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 36 \, 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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