Optimal. Leaf size=57 \[ -\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}+a^2 x \]
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Rubi [A] time = 0.0821145, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3788, 2635, 8, 4044, 3013} \[ -\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}+a^2 x \]
Antiderivative was successfully verified.
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Rule 3788
Rule 2635
Rule 8
Rule 4044
Rule 3013
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+a \sec (c+d x))^2 \, dx &=\left (2 a^2\right ) \int \cos ^2(c+d x) \, dx+\int \cos ^3(c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 \cos (c+d x) \sin (c+d x)}{d}+a^2 \int 1 \, dx+\int \cos (c+d x) \left (a^2+a^2 \cos ^2(c+d x)\right ) \, dx\\ &=a^2 x+\frac{a^2 \cos (c+d x) \sin (c+d x)}{d}-\frac{\operatorname{Subst}\left (\int \left (2 a^2-a^2 x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=a^2 x+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \cos (c+d x) \sin (c+d x)}{d}-\frac{a^2 \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0853281, size = 41, normalized size = 0.72 \[ \frac{a^2 (21 \sin (c+d x)+6 \sin (2 (c+d x))+\sin (3 (c+d x))+12 d x)}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 64, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{2} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2 \right ) \sin \left ( dx+c \right ) }{3}}+2\,{a}^{2} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{a}^{2}\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989147, size = 82, normalized size = 1.44 \begin{align*} -\frac{2 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{2} - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} - 6 \, a^{2} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69427, size = 113, normalized size = 1.98 \begin{align*} \frac{3 \, a^{2} d x +{\left (a^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} \cos \left (d x + c\right ) + 5 \, a^{2}\right )} \sin \left (d x + c\right )}{3 \, d} \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 = 1.35726, size = 108, normalized size = 1.89 \begin{align*} \frac{3 \,{\left (d x + c\right )} a^{2} + \frac{2 \,{\left (3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 8 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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