Optimal. Leaf size=85 \[ \frac{3 B \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 B \tan (c+d x) \sec (c+d x)}{8 d}+\frac{C \tan ^3(c+d x)}{3 d}+\frac{C \tan (c+d x)}{d} \]
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Rubi [A] time = 0.0913065, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3010, 2748, 3768, 3770, 3767} \[ \frac{3 B \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 B \tan (c+d x) \sec (c+d x)}{8 d}+\frac{C \tan ^3(c+d x)}{3 d}+\frac{C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3010
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx &=\int (B+C \cos (c+d x)) \sec ^5(c+d x) \, dx\\ &=B \int \sec ^5(c+d x) \, dx+C \int \sec ^4(c+d x) \, dx\\ &=\frac{B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} (3 B) \int \sec ^3(c+d x) \, dx-\frac{C \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{C \tan (c+d x)}{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{C \tan ^3(c+d x)}{3 d}+\frac{1}{8} (3 B) \int \sec (c+d x) \, dx\\ &=\frac{3 B \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{C \tan (c+d x)}{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{C \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.210047, size = 76, normalized size = 0.89 \[ \frac{B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 B \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )}{8 d}+\frac{C \left (\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 92, normalized size = 1.1 \begin{align*}{\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}}+{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.33661, size = 128, normalized size = 1.51 \begin{align*} \frac{16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C - 3 \, 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 )}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67855, size = 266, normalized size = 3.13 \begin{align*} \frac{9 \, B \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, B \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \, C \cos \left (d x + c\right )^{3} + 9 \, B \cos \left (d x + c\right )^{2} + 8 \, C \cos \left (d x + c\right ) + 6 \, B\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 = 1.37822, size = 221, normalized size = 2.6 \begin{align*} \frac{9 \, B \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 9 \, B \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (15 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 24 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 9 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 40 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 40 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, 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 )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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