Optimal. Leaf size=45 \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{a \sec (c+d x)}{d} \]
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Rubi [A] time = 0.103911, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2838, 2607, 30, 2606} \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{a \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2607
Rule 30
Rule 2606
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx &=a \int \sec ^2(c+d x) \tan ^2(c+d x) \, dx+a \int \sec (c+d x) \tan ^3(c+d x) \, dx\\ &=\frac{a \operatorname{Subst}\left (\int x^2 \, dx,x,\tan (c+d x)\right )}{d}+\frac{a \operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{a \sec (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}+\frac{a \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0373222, size = 45, normalized size = 1. \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{a \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 82, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ( a \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{3\,\cos \left ( dx+c \right ) }}-{\frac{ \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) }{3}} \right ) +{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{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 = 1.01503, size = 53, normalized size = 1.18 \begin{align*} \frac{a \tan \left (d x + c\right )^{3} - \frac{{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} 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 = 1.53109, size = 127, normalized size = 2.82 \begin{align*} \frac{a \cos \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right ) + 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 = 1.26336, size = 90, normalized size = 2. \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} - 12 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5 \, 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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