Optimal. Leaf size=25 \[ \frac{b \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0280599, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4377, 12, 2606, 8, 3475} \[ \frac{b \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 4377
Rule 12
Rule 2606
Rule 8
Rule 3475
Rubi steps
\begin{align*} \int \sec (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx &=a \int \tan (c+d x) \, dx+\int b \sec (c+d x) \tan (c+d x) \, dx\\ &=-\frac{a \log (\cos (c+d x))}{d}+b \int \sec (c+d x) \tan (c+d x) \, dx\\ &=-\frac{a \log (\cos (c+d x))}{d}+\frac{b \operatorname{Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}\\ &=-\frac{a \log (\cos (c+d x))}{d}+\frac{b \sec (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0130665, size = 25, normalized size = 1. \[ \frac{b \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 25, normalized size = 1. \begin{align*}{\frac{b\sec \left ( dx+c \right ) }{d}}+{\frac{a\ln \left ( \sec \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11643, size = 43, normalized size = 1.72 \begin{align*} -\frac{a \log \left (-\sin \left (d x + c\right )^{2} + 1\right ) - \frac{2 \, b}{\cos \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.50081, size = 80, normalized size = 3.2 \begin{align*} -\frac{a \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - b}{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 \sin{\left (c + d x \right )} + b \tan{\left (c + d x \right )}\right ) \sec{\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 = 1.16685, size = 144, normalized size = 5.76 \begin{align*} \frac{a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{a + 2 \, b + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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