Optimal. Leaf size=140 \[ \frac{a (4 A+5 C) \tan ^3(c+d x)}{15 d}+\frac{a (4 A+5 C) \tan (c+d x)}{5 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{b (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b (3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A b \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.203593, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {3032, 3021, 2748, 3767, 3768, 3770} \[ \frac{a (4 A+5 C) \tan ^3(c+d x)}{15 d}+\frac{a (4 A+5 C) \tan (c+d x)}{5 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{b (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b (3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A b \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3032
Rule 3021
Rule 2748
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx &=\frac{a A \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{5} \int \left (5 A b+a (4 A+5 C) \cos (c+d x)+5 b 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{a A \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{20} \int (4 a (4 A+5 C)+5 b (3 A+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}+\frac{a A \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{4} (b (3 A+4 C)) \int \sec ^3(c+d x) \, dx+\frac{1}{5} (a (4 A+5 C)) \int \sec ^4(c+d x) \, dx\\ &=\frac{b (3 A+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 A \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{8} (b (3 A+4 C)) \int \sec (c+d x) \, dx-\frac{(a (4 A+5 C)) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac{b (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 A+5 C) \tan (c+d x)}{5 d}+\frac{b (3 A+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 A \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{a (4 A+5 C) \tan ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 0.78606, size = 96, normalized size = 0.69 \[ \frac{\tan (c+d x) \left (8 a \left (5 (2 A+C) \tan ^2(c+d x)+3 A \tan ^4(c+d x)+15 (A+C)\right )+15 b (3 A+4 C) \sec (c+d x)+30 A b \sec ^3(c+d x)\right )+15 b (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{120 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 192, normalized size = 1.4 \begin{align*}{\frac{Ab \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,Ab\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,Ab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{Cb\tan \left ( dx+c \right ) \sec \left ( dx+c \right ) }{2\,d}}+{\frac{Cb\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{8\,A\tan \left ( dx+c \right ) a}{15\,d}}+{\frac{aA \left ( \sec \left ( dx+c \right ) \right ) ^{4}\tan \left ( dx+c \right ) }{5\,d}}+{\frac{4\,aA \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{15\,d}}+{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00314, size = 236, normalized size = 1.69 \begin{align*} \frac{16 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a + 80 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a - 15 \, A b{\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 )} - 60 \, C 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 )}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44778, size = 389, normalized size = 2.78 \begin{align*} \frac{15 \,{\left (3 \, A + 4 \, C\right )} b \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (3 \, A + 4 \, C\right )} b \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (4 \, A + 5 \, C\right )} a \cos \left (d x + c\right )^{4} + 15 \,{\left (3 \, A + 4 \, C\right )} b \cos \left (d x + c\right )^{3} + 8 \,{\left (4 \, A + 5 \, C\right )} a \cos \left (d x + c\right )^{2} + 30 \, A b \cos \left (d x + c\right ) + 24 \, A a\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \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 = 1.50369, size = 451, normalized size = 3.22 \begin{align*} \frac{15 \,{\left (3 \, A b + 4 \, C b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \,{\left (3 \, A b + 4 \, C b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (120 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 120 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 75 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 60 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 160 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 320 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 30 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 120 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 464 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 400 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 160 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 320 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 30 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 120 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 120 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 120 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 75 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 60 \, C 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 )}^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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