Optimal. Leaf size=86 \[ \frac{a (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A b \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.172846, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {3032, 3021, 2748, 3767, 8, 3770} \[ \frac{a (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A b \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3032
Rule 3021
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx &=\frac{a A \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{3} \int \left (3 A b+a (2 A+3 C) \cos (c+d x)+3 b C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{A b \sec (c+d x) \tan (c+d x)}{2 d}+\frac{a A \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{6} \int (2 a (2 A+3 C)+3 b (A+2 C) \cos (c+d x)) \sec ^2(c+d x) \, dx\\ &=\frac{A b \sec (c+d x) \tan (c+d x)}{2 d}+\frac{a A \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{2} (b (A+2 C)) \int \sec (c+d x) \, dx+\frac{1}{3} (a (2 A+3 C)) \int \sec ^2(c+d x) \, dx\\ &=\frac{b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A b \sec (c+d x) \tan (c+d x)}{2 d}+\frac{a A \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac{(a (2 A+3 C)) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (2 A+3 C) \tan (c+d x)}{3 d}+\frac{A b \sec (c+d x) \tan (c+d x)}{2 d}+\frac{a A \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.323874, size = 59, normalized size = 0.69 \[ \frac{\tan (c+d x) \left (2 a A \tan ^2(c+d x)+6 a (A+C)+3 A b \sec (c+d x)\right )+3 b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 108, normalized size = 1.3 \begin{align*}{\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{Cb\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{2\,A\tan \left ( dx+c \right ) a}{3\,d}}+{\frac{aA \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{3\,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 = 1.01852, size = 144, normalized size = 1.67 \begin{align*} \frac{4 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a - 3 \, A b{\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 )} + 6 \, C b{\left (\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 = 1.51209, size = 285, normalized size = 3.31 \begin{align*} \frac{3 \,{\left (A + 2 \, C\right )} b \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (A + 2 \, C\right )} b \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (2 \,{\left (2 \, A + 3 \, C\right )} a \cos \left (d x + c\right )^{2} + 3 \, A b \cos \left (d x + c\right ) + 2 \, A 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 [B] time = 1.58034, size = 248, normalized size = 2.88 \begin{align*} \frac{3 \,{\left (A b + 2 \, C b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (A b + 2 \, C b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (6 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 4 \, 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} + 6 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, A 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 )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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