Optimal. Leaf size=38 \[ \frac{2 a^4 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}-a^2 x \]
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Rubi [A] time = 0.0817753, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2670, 2680, 8} \[ \frac{2 a^4 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}-a^2 x \]
Antiderivative was successfully verified.
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Rule 2670
Rule 2680
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx &=a^4 \int \frac{\cos ^2(c+d x)}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac{2 a^4 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}-a^2 \int 1 \, dx\\ &=-a^2 x+\frac{2 a^4 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.0561172, size = 75, normalized size = 1.97 \[ \frac{2 a^2 \sqrt{\sin (c+d x)+1} \left (\sqrt{1-\sin (c+d x)} \sin ^{-1}\left (\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right )+\sqrt{\sin (c+d x)+1}\right ) \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 47, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( \tan \left ( dx+c \right ) -dx-c \right ) +2\,{\frac{{a}^{2}}{\cos \left ( dx+c \right ) }}+{a}^{2}\tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41559, size = 63, normalized size = 1.66 \begin{align*} -\frac{{\left (d x + c - \tan \left (d x + c\right )\right )} a^{2} - a^{2} \tan \left (d x + c\right ) - \frac{2 \, a^{2}}{\cos \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63995, size = 167, normalized size = 4.39 \begin{align*} -\frac{a^{2} d x - 2 \, a^{2} +{\left (a^{2} d x - 2 \, a^{2}\right )} \cos \left (d x + c\right ) -{\left (a^{2} d x + 2 \, a^{2}\right )} \sin \left (d x + c\right )}{d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + 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 \sin{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{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 = 1.1502, size = 45, normalized size = 1.18 \begin{align*} -\frac{{\left (d x + c\right )} a^{2} + \frac{4 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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