Optimal. Leaf size=26 \[ \frac{(a \sin (c+d x)+b \sec (c+d x))^3}{3 d} \]
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Rubi [A] time = 0.0429013, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.023, Rules used = {4385} \[ \frac{(a \sin (c+d x)+b \sec (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 4385
Rubi steps
\begin{align*} \int (b \sec (c+d x)+a \sin (c+d x))^2 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx &=\frac{(b \sec (c+d x)+a \sin (c+d x))^3}{3 d}\\ \end{align*}
Mathematica [A] time = 1.27789, size = 31, normalized size = 1.19 \[ \frac{\sec ^3(c+d x) (a \sin (2 (c+d x))+2 b)^3}{24 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.172, size = 118, normalized size = 4.5 \begin{align*}{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{2}b \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d\cos \left ( dx+c \right ) }}+{\frac{{a}^{2}b \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) }{d}}+{\frac{a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a{b}^{2}\sin \left ( dx+c \right ) }{d}}+{\frac{{b}^{3}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96345, size = 32, normalized size = 1.23 \begin{align*} \frac{{\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}^{3}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.60107, size = 217, normalized size = 8.35 \begin{align*} -\frac{3 \, a^{2} b \cos \left (d x + c\right )^{4} - 3 \, a^{2} b \cos \left (d x + c\right )^{2} - b^{3} +{\left (a^{3} \cos \left (d x + c\right )^{5} - a^{3} \cos \left (d x + c\right )^{3} - 3 \, a b^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 18.2675, size = 100, normalized size = 3.85 \begin{align*} \begin{cases} \frac{a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{a^{2} b \sin ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{d} + \frac{a b^{2} \sin{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{d} + \frac{b^{3} \sec ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + b \sec{\left (c \right )}\right )^{2} \left (a \cos{\left (c \right )} + b \tan{\left (c \right )} \sec{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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