Optimal. Leaf size=43 \[ \frac{2 a^2 \cos (c+d x)}{d}-2 a^2 x+\frac{\sec (c+d x) (a \sin (c+d x)+a)^2}{d} \]
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Rubi [A] time = 0.0568371, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2855, 2638} \[ \frac{2 a^2 \cos (c+d x)}{d}-2 a^2 x+\frac{\sec (c+d x) (a \sin (c+d x)+a)^2}{d} \]
Antiderivative was successfully verified.
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Rule 2855
Rule 2638
Rubi steps
\begin{align*} \int \sec (c+d x) (a+a \sin (c+d x))^2 \tan (c+d x) \, dx &=\frac{\sec (c+d x) (a+a \sin (c+d x))^2}{d}-(2 a) \int (a+a \sin (c+d x)) \, dx\\ &=-2 a^2 x+\frac{\sec (c+d x) (a+a \sin (c+d x))^2}{d}-\left (2 a^2\right ) \int \sin (c+d x) \, dx\\ &=-2 a^2 x+\frac{2 a^2 \cos (c+d x)}{d}+\frac{\sec (c+d x) (a+a \sin (c+d x))^2}{d}\\ \end{align*}
Mathematica [B] time = 0.369367, size = 90, normalized size = 2.09 \[ \frac{(a \sin (c+d x)+a)^2 \left (-2 (c+d x)+\cos (c+d x)+\frac{4 \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}\right )}{d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 76, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{\cos \left ( dx+c \right ) }}+ \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) \right ) +2\,{a}^{2} \left ( \tan \left ( dx+c \right ) -dx-c \right ) +{\frac{{a}^{2}}{\cos \left ( dx+c \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70752, size = 77, normalized size = 1.79 \begin{align*} -\frac{2 \,{\left (d x + c - \tan \left (d x + c\right )\right )} a^{2} - a^{2}{\left (\frac{1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - \frac{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 [B] time = 1.08957, size = 230, normalized size = 5.35 \begin{align*} -\frac{2 \, a^{2} d x - a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} +{\left (2 \, a^{2} d x - 3 \, a^{2}\right )} \cos \left (d x + c\right ) -{\left (2 \, a^{2} d x - a^{2} \cos \left (d x + c\right ) + 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(-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 [B] time = 1.16254, size = 120, normalized size = 2.79 \begin{align*} -\frac{2 \,{\left ({\left (d x + c\right )} a^{2} + \frac{2 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1}\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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