Optimal. Leaf size=49 \[ \frac{a b \cos (c+d x)}{d}+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{d}+b^2 (-x) \]
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Rubi [A] time = 0.0484278, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2691, 2638} \[ \frac{a b \cos (c+d x)}{d}+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{d}+b^2 (-x) \]
Antiderivative was successfully verified.
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Rule 2691
Rule 2638
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac{\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{d}-\int \left (b^2+a b \sin (c+d x)\right ) \, dx\\ &=-b^2 x+\frac{\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{d}-(a b) \int \sin (c+d x) \, dx\\ &=-b^2 x+\frac{a b \cos (c+d x)}{d}+\frac{\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0586616, size = 55, normalized size = 1.12 \[ \frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \sec (c+d x)}{d}-\frac{b^2 \tan ^{-1}(\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 46, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({a}^{2}\tan \left ( dx+c \right ) +2\,{\frac{ab}{\cos \left ( dx+c \right ) }}+{b}^{2} \left ( \tan \left ( dx+c \right ) -dx-c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46466, size = 62, normalized size = 1.27 \begin{align*} -\frac{{\left (d x + c - \tan \left (d x + c\right )\right )} b^{2} - a^{2} \tan \left (d x + c\right ) - \frac{2 \, a b}{\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 = 2.09162, size = 104, normalized size = 2.12 \begin{align*} -\frac{b^{2} d x \cos \left (d x + c\right ) - 2 \, a b -{\left (a^{2} + b^{2}\right )} \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \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*} \int \left (a + b \sin{\left (c + d x \right )}\right )^{2} \sec ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11494, size = 85, normalized size = 1.73 \begin{align*} -\frac{{\left (d x + c\right )} b^{2} + \frac{2 \,{\left (a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, a b\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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