Optimal. Leaf size=47 \[ \frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{C \tan (c+d x)}{d} \]
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Rubi [A] time = 0.075895, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3010, 2748, 3768, 3770, 3767, 8} \[ \frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3010
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx &=\int (B+C \cos (c+d x)) \sec ^3(c+d x) \, dx\\ &=B \int \sec ^3(c+d x) \, dx+C \int \sec ^2(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-\frac{C \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{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}\\ \end{align*}
Mathematica [A] time = 0.0156961, size = 47, normalized size = 1. \[ \frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 51, normalized size = 1.1 \begin{align*}{\frac{C\tan \left ( dx+c \right ) }{d}}+{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.36954, size = 78, normalized size = 1.66 \begin{align*} -\frac{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 )} - 4 \, C \tan \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63899, size = 198, normalized size = 4.21 \begin{align*} \frac{B \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - B \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (2 \, C \cos \left (d x + c\right ) + B\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \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 = 1.32777, size = 142, normalized size = 3.02 \begin{align*} \frac{B \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - B \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, 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 )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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