Optimal. Leaf size=27 \[ A x+\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{C \tan (c+d x)}{d} \]
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Rubi [A] time = 0.0192748, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3770, 3767, 8} \[ A x+\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=A x+B \int \sec (c+d x) \, dx+C \int \sec ^2(c+d x) \, dx\\ &=A x+\frac{B \tanh ^{-1}(\sin (c+d x))}{d}-\frac{C \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=A x+\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{C \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0038755, size = 27, normalized size = 1. \[ A x+\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 35, normalized size = 1.3 \begin{align*} Ax+{\frac{B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{C\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.927017, size = 46, normalized size = 1.7 \begin{align*} A x + \frac{B \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right )}{d} + \frac{C \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.507408, size = 193, normalized size = 7.15 \begin{align*} \frac{2 \, A d x \cos \left (d x + c\right ) + B \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - B \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \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 (A + B \sec{\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20167, size = 81, normalized size = 3. \begin{align*} A x + \frac{B{\left (\log \left ({\left | \frac{1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) + 2 \right |}\right ) - \log \left ({\left | \frac{1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) - 2 \right |}\right )\right )}}{4 \, d} + \frac{C \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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