Optimal. Leaf size=122 \[ \frac{(4 A+5 C) \tan ^3(c+d x)}{15 d}+\frac{(4 A+5 C) \tan (c+d x)}{5 d}+\frac{A \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{3 B \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 B \tan (c+d x) \sec (c+d x)}{8 d} \]
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Rubi [A] time = 0.118319, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {3021, 2748, 3768, 3770, 3767} \[ \frac{(4 A+5 C) \tan ^3(c+d x)}{15 d}+\frac{(4 A+5 C) \tan (c+d x)}{5 d}+\frac{A \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{3 B \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 B \tan (c+d x) \sec (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 3021
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx &=\frac{A \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{5} \int (5 B+(4 A+5 C) \cos (c+d x)) \sec ^5(c+d x) \, dx\\ &=\frac{A \sec ^4(c+d x) \tan (c+d x)}{5 d}+B \int \sec ^5(c+d x) \, dx+\frac{1}{5} (4 A+5 C) \int \sec ^4(c+d x) \, dx\\ &=\frac{B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{A \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{4} (3 B) \int \sec ^3(c+d x) \, dx-\frac{(4 A+5 C) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac{(4 A+5 C) \tan (c+d x)}{5 d}+\frac{3 B \sec (c+d x) \tan (c+d x)}{8 d}+\frac{B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{A \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{(4 A+5 C) \tan ^3(c+d x)}{15 d}+\frac{1}{8} (3 B) \int \sec (c+d x) \, dx\\ &=\frac{3 B \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 A+5 C) \tan (c+d x)}{5 d}+\frac{3 B \sec (c+d x) \tan (c+d x)}{8 d}+\frac{B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{A \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{(4 A+5 C) \tan ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 0.560577, size = 80, normalized size = 0.66 \[ \frac{\tan (c+d x) \left (8 \left (5 (2 A+C) \tan ^2(c+d x)+3 A \tan ^4(c+d x)+15 (A+C)\right )+30 B \sec ^3(c+d x)+45 B \sec (c+d x)\right )+45 B \tanh ^{-1}(\sin (c+d x))}{120 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 144, normalized size = 1.2 \begin{align*}{\frac{8\,A\tan \left ( dx+c \right ) }{15\,d}}+{\frac{A \left ( \sec \left ( dx+c \right ) \right ) ^{4}\tan \left ( dx+c \right ) }{5\,d}}+{\frac{4\,A \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{15\,d}}+{\frac{B \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,B\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{2\,C\tan \left ( dx+c \right ) }{3\,d}}+{\frac{C\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 = 0.988888, size = 171, normalized size = 1.4 \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 + 80 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C - 15 \, 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 )}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01564, size = 329, normalized size = 2.7 \begin{align*} \frac{45 \, B \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 45 \, 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 )} \cos \left (d x + c\right )^{4} + 45 \, B \cos \left (d x + c\right )^{3} + 8 \,{\left (4 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{2} + 30 \, B \cos \left (d x + c\right ) + 24 \, 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 = 1.26249, size = 332, normalized size = 2.72 \begin{align*} \frac{45 \, B \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 45 \, B \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (120 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 75 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 120 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 160 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 30 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 320 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 464 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 400 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 160 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 30 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 320 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 120 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 75 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 120 \, C \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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