Optimal. Leaf size=33 \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \sec ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0838112, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2838, 2606, 30, 2607} \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2606
Rule 30
Rule 2607
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx &=a \int \sec ^3(c+d x) \tan (c+d x) \, dx+a \int \sec ^2(c+d x) \tan ^2(c+d x) \, dx\\ &=\frac{a \operatorname{Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{d}+\frac{a \operatorname{Subst}\left (\int x^2 \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{a \sec ^3(c+d x)}{3 d}+\frac{a \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0212681, size = 33, normalized size = 1. \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 36, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{a}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04008, size = 35, normalized size = 1.06 \begin{align*} \frac{a \tan \left (d x + c\right )^{3} + \frac{a}{\cos \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56659, size = 127, normalized size = 3.85 \begin{align*} \frac{a \cos \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) - 2 \, a}{3 \,{\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\right )}} \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 [A] time = 1.24004, size = 72, normalized size = 2.18 \begin{align*} \frac{\frac{3 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1} - \frac{3 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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