Optimal. Leaf size=58 \[ \frac{4 \sqrt{a \sin (c+d x)+a}}{d^2}-\frac{2 x \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (c+d x)+a}}{d} \]
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Rubi [A] time = 0.0681968, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3319, 3296, 2638} \[ \frac{4 \sqrt{a \sin (c+d x)+a}}{d^2}-\frac{2 x \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (c+d x)+a}}{d} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x \sqrt{a+a \sin (c+d x)} \, dx &=\left (\csc \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right ) \sqrt{a+a \sin (c+d x)}\right ) \int x \sin \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right ) \, dx\\ &=-\frac{2 x \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right ) \sqrt{a+a \sin (c+d x)}}{d}+\frac{\left (2 \csc \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right ) \sqrt{a+a \sin (c+d x)}\right ) \int \cos \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right ) \, dx}{d}\\ &=\frac{4 \sqrt{a+a \sin (c+d x)}}{d^2}-\frac{2 x \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right ) \sqrt{a+a \sin (c+d x)}}{d}\\ \end{align*}
Mathematica [A] time = 0.154155, size = 76, normalized size = 1.31 \[ -\frac{2 \sqrt{a (\sin (c+d x)+1)} \left ((d x-2) \cos \left (\frac{1}{2} (c+d x)\right )-(d x+2) \sin \left (\frac{1}{2} (c+d x)\right )\right )}{d^2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.053, size = 93, normalized size = 1.6 \begin{align*}{\frac{-i\sqrt{2} \left ( -idx+dx{{\rm e}^{i \left ( dx+c \right ) }}+2\,i{{\rm e}^{i \left ( dx+c \right ) }}-2 \right ) \left ({{\rm e}^{i \left ( dx+c \right ) }}+i \right ) }{ \left ({{\rm e}^{2\,i \left ( dx+c \right ) }}-1+2\,i{{\rm e}^{i \left ( dx+c \right ) }} \right ){d}^{2}}\sqrt{-a \left ( -2-2\,\sin \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (d x + c\right ) + a} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 x \sqrt{a \left (\sin{\left (c + d x \right )} + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (d x + c\right ) + a} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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