Optimal. Leaf size=125 \[ -\frac{a (-3 (A+B+C)+A+B) \tan (c+d x)}{3 d}+\frac{a (3 A+4 (B+C)) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 (B+C)) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a (A+B) \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.253843, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.18, Rules used = {3031, 3021, 2748, 3768, 3770, 3767, 8} \[ -\frac{a (-3 (A+B+C)+A+B) \tan (c+d x)}{3 d}+\frac{a (3 A+4 (B+C)) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 (B+C)) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a (A+B) \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3031
Rule 3021
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+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 (A+B)-a (3 A+4 (B+C)) \cos (c+d x)-4 a C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{a (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 (B+C))+4 a (A+B-3 (A+B+C)) \cos (c+d x)) \sec ^3(c+d x) \, dx\\ &=\frac{a (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}{4} (a (3 A+4 (B+C))) \int \sec ^3(c+d x) \, dx-\frac{1}{3} (a (A+B-3 (A+B+C))) \int \sec ^2(c+d x) \, dx\\ &=\frac{a (3 A+4 (B+C)) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a (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 (B+C))) \int \sec (c+d x) \, dx+\frac{(a (A+B-3 (A+B+C))) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{a (3 A+4 (B+C)) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{a (A+B-3 (A+B+C)) \tan (c+d x)}{3 d}+\frac{a (3 A+4 (B+C)) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a (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 = 0.592432, size = 84, normalized size = 0.67 \[ \frac{a \left (3 (3 A+4 (B+C)) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (8 \left ((A+B) \tan ^2(c+d x)+3 (A+B+C)\right )+3 (3 A+4 (B+C)) \sec (c+d x)+6 A \sec ^3(c+d x)\right )\right )}{24 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.072, size = 223, normalized size = 1.8 \begin{align*}{\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\,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 ) \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\,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{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}}+{\frac{aC\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02244, size = 294, normalized size = 2.35 \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 )} - 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 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 = 2.04004, size = 375, normalized size = 3. \begin{align*} \frac{3 \,{\left (3 \, A + 4 \, 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 \, A + 4 \, B + 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 + 2 \, B + 3 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \,{\left (3 \, A + 4 \, B + 4 \, C\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 [B] time = 1.28072, size = 343, normalized size = 2.74 \begin{align*} \frac{3 \,{\left (3 \, A a + 4 \, 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 \, A a + 4 \, 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 \, 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} + 12 \, C 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} - 60 \, C 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} + 84 \, C 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 ) - 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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