Optimal. Leaf size=28 \[ \frac{a \sec (c+d x)}{d}+\frac{b \sec ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.0501234, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {4377, 12, 2606, 30, 8} \[ \frac{a \sec (c+d x)}{d}+\frac{b \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 4377
Rule 12
Rule 2606
Rule 30
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx &=a \int \sec (c+d x) \tan (c+d x) \, dx+\int b \sec ^2(c+d x) \tan (c+d x) \, dx\\ &=b \int \sec ^2(c+d x) \tan (c+d x) \, dx+\frac{a \operatorname{Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}\\ &=\frac{a \sec (c+d x)}{d}+\frac{b \operatorname{Subst}(\int x \, dx,x,\sec (c+d x))}{d}\\ &=\frac{a \sec (c+d x)}{d}+\frac{b \sec ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0197068, size = 28, normalized size = 1. \[ \frac{a \sec (c+d x)}{d}+\frac{b \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 25, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{b \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{2}}+a\sec \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02535, size = 36, normalized size = 1.29 \begin{align*} \frac{b \tan \left (d x + c\right )^{2} + \frac{2 \, a}{\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 = 0.468432, size = 63, normalized size = 2.25 \begin{align*} \frac{2 \, a \cos \left (d x + c\right ) + b}{2 \, 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] 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 ^{2}{\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 = 1.1582, size = 96, normalized size = 3.43 \begin{align*} \frac{2 \,{\left (a + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{d{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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