Optimal. Leaf size=63 \[ \frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{C \tan ^3(c+d x)}{3 d}+\frac{C \tan (c+d x)}{d} \]
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Rubi [A] time = 0.0575337, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {4047, 3768, 3770, 12, 3767} \[ \frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{C \tan ^3(c+d x)}{3 d}+\frac{C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 4047
Rule 3768
Rule 3770
Rule 12
Rule 3767
Rubi steps
\begin{align*} \int \sec ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=B \int \sec ^3(c+d x) \, dx+\int C \sec ^4(c+d x) \, dx\\ &=\frac{B \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} B \int \sec (c+d x) \, dx+C \int \sec ^4(c+d x) \, dx\\ &=\frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \sec (c+d x) \tan (c+d x)}{2 d}-\frac{C \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{C \tan (c+d x)}{d}+\frac{B \sec (c+d x) \tan (c+d x)}{2 d}+\frac{C \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.147771, size = 60, normalized size = 0.95 \[ \frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{C \left (\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 72, normalized size = 1.1 \begin{align*}{\frac{B\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{2\,C\tan \left ( dx+c \right ) }{3\,d}}+{\frac{C \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.929063, size = 95, normalized size = 1.51 \begin{align*} \frac{4 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C - 3 \, 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 )}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.497716, size = 236, normalized size = 3.75 \begin{align*} \frac{3 \, B \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, B \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (4 \, C \cos \left (d x + c\right )^{2} + 3 \, B \cos \left (d x + c\right ) + 2 \, C\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (B + C \sec{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28929, size = 165, normalized size = 2.62 \begin{align*} \frac{3 \, B \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, B \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (3 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 6 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 6 \, 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 )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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