Optimal. Leaf size=33 \[ \frac{a \sec ^2(c+d x)}{2 d}+\frac{b \sec ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0569483, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4377, 12, 2606, 30} \[ \frac{a \sec ^2(c+d x)}{2 d}+\frac{b \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4377
Rule 12
Rule 2606
Rule 30
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx &=a \int \sec ^2(c+d x) \tan (c+d x) \, dx+\int b \sec ^3(c+d x) \tan (c+d x) \, dx\\ &=b \int \sec ^3(c+d x) \tan (c+d x) \, dx+\frac{a \operatorname{Subst}(\int x \, dx,x,\sec (c+d x))}{d}\\ &=\frac{a \sec ^2(c+d x)}{2 d}+\frac{b \operatorname{Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{a \sec ^2(c+d x)}{2 d}+\frac{b \sec ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0225567, size = 33, normalized size = 1. \[ \frac{a \sec ^2(c+d x)}{2 d}+\frac{b \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 28, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{3}b}{3}}+{\frac{a \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05449, size = 43, normalized size = 1.3 \begin{align*} -\frac{\frac{3 \, a}{\sin \left (d x + c\right )^{2} - 1} - \frac{2 \, b}{\cos \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.471809, size = 66, normalized size = 2. \begin{align*} \frac{3 \, a \cos \left (d x + c\right ) + 2 \, b}{6 \, 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 (a \sin{\left (c + d x \right )} + b \tan{\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 = 1.16724, size = 131, normalized size = 3.97 \begin{align*} \frac{2 \,{\left (b - \frac{3 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{3 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{3 \, d{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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