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 = 0.07, antiderivative size = 58, normalized size of antiderivative = 1.00, 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 2638
Rule 3296
Rule 3319
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*}
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Mathematica [A] time = 0.20, 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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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.83, size = 69, normalized size = 1.19 \[ 2 \, \sqrt {2} {\left (\frac {x \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} + \frac {2 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d^{2}}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 93, normalized size = 1.60 \[ -\frac {i \sqrt {2}\, \sqrt {-a \left (-2-2 \sin \left (d x +c \right )\right )}\, \left (-i d x +d x \,{\mathrm e}^{i \left (d x +c \right )}+2 i {\mathrm e}^{i \left (d x +c \right )}-2\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1+2 i {\mathrm e}^{i \left (d x +c \right )}\right ) d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sin \left (d x + c\right ) + a} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 47, normalized size = 0.81 \[ \frac {2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (2\,\sin \left (c+d\,x\right )-d\,x\,\cos \left (c+d\,x\right )+2\right )}{d^2\,\left (\sin \left (c+d\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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