Optimal. Leaf size=26 \[ \frac{2 a \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{d} \]
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Rubi [A] time = 0.0575865, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {2673} \[ \frac{2 a \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{d} \]
Antiderivative was successfully verified.
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Rule 2673
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac{2 a \sec (c+d x) \sqrt{a+a \sin (c+d x)}}{d}\\ \end{align*}
Mathematica [B] time = 0.157883, size = 67, normalized size = 2.58 \[ \frac{2 (a (\sin (c+d x)+1))^{3/2}}{d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.083, size = 37, normalized size = 1.4 \begin{align*} 2\,{\frac{{a}^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) \sqrt{a+a\sin \left ( dx+c \right ) }d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.63246, size = 132, normalized size = 5.08 \begin{align*} -\frac{2 \,{\left (a^{\frac{3}{2}} + \frac{2 \, a^{\frac{3}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{\frac{3}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )}}{d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61122, size = 63, normalized size = 2.42 \begin{align*} \frac{2 \, \sqrt{a \sin \left (d x + c\right ) + a} a}{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(-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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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