Optimal. Leaf size=45 \[ \frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 x}{2} \]
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Rubi [A] time = 0.0602976, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3788, 2637, 4045, 8} \[ \frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 x}{2} \]
Antiderivative was successfully verified.
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Rule 3788
Rule 2637
Rule 4045
Rule 8
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \, dx &=\left (2 a^2\right ) \int \cos (c+d x) \, dx+\int \cos ^2(c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} \left (3 a^2\right ) \int 1 \, dx\\ &=\frac{3 a^2 x}{2}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0393088, size = 34, normalized size = 0.76 \[ \frac{a^2 (6 (c+d x)+8 \sin (c+d x)+\sin (2 (c+d x)))}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 52, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +2\,{a}^{2}\sin \left ( dx+c \right ) +{a}^{2} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09854, size = 65, normalized size = 1.44 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} + 4 \,{\left (d x + c\right )} a^{2} + 8 \, a^{2} \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70518, size = 82, normalized size = 1.82 \begin{align*} \frac{3 \, a^{2} d x +{\left (a^{2} \cos \left (d x + c\right ) + 4 \, a^{2}\right )} \sin \left (d x + c\right )}{2 \, d} \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*} a^{2} \left (\int 2 \cos ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cos ^{2}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32744, size = 86, normalized size = 1.91 \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 )^{3} + 5 \, 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 )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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