Optimal. Leaf size=120 \[ -\frac {96 \sqrt {a \sin (c+d x)+a}}{d^4}+\frac {48 x \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a}}{d^3}+\frac {12 x^2 \sqrt {a \sin (c+d x)+a}}{d^2}-\frac {2 x^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a}}{d} \]
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Rubi [A] time = 0.14, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3319, 3296, 2638} \[ \frac {12 x^2 \sqrt {a \sin (c+d x)+a}}{d^2}-\frac {96 \sqrt {a \sin (c+d x)+a}}{d^4}+\frac {48 x \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a}}{d^3}-\frac {2 x^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a}}{d} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3319
Rubi steps
\begin {align*} \int x^3 \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^3 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx\\ &=-\frac {2 x^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}}{d}+\frac {\left (6 \csc \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}\right ) \int x^2 \cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{d}\\ &=\frac {12 x^2 \sqrt {a+a \sin (c+d x)}}{d^2}-\frac {2 x^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}}{d}-\frac {\left (24 \csc \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}\right ) \int x \cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{d^2}\\ &=\frac {12 x^2 \sqrt {a+a \sin (c+d x)}}{d^2}+\frac {48 x \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}}{d^3}-\frac {2 x^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}}{d}-\frac {\left (48 \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^3}\\ &=-\frac {96 \sqrt {a+a \sin (c+d x)}}{d^4}+\frac {12 x^2 \sqrt {a+a \sin (c+d x)}}{d^2}+\frac {48 x \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}}{d^3}-\frac {2 x^3 \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 = 0.34, size = 108, normalized size = 0.90 \[ -\frac {2 \sqrt {a (\sin (c+d x)+1)} \left (\left (-d^3 x^3-6 d^2 x^2+24 d x+48\right ) \sin \left (\frac {1}{2} (c+d x)\right )+\left (d^3 x^3-6 d^2 x^2-24 d x+48\right ) \cos \left (\frac {1}{2} (c+d x)\right )\right )}{d^4 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
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 = 1.09, size = 116, normalized size = 0.97 \[ 2 \, \sqrt {2} \sqrt {a} {\left (\frac {6 \, {\left (d^{2} x^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 8 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d^{4}} + \frac {{\left (d^{3} x^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 24 \, d x \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 145, normalized size = 1.21 \[ -\frac {i \sqrt {2}\, \sqrt {-a \left (-2-2 \sin \left (d x +c \right )\right )}\, \left (-i x^{3} d^{3}+d^{3} x^{3} {\mathrm e}^{i \left (d x +c \right )}+6 i d^{2} x^{2} {\mathrm e}^{i \left (d x +c \right )}-6 d^{2} x^{2}+24 i d x -24 d x \,{\mathrm e}^{i \left (d x +c \right )}-48 i {\mathrm e}^{i \left (d x +c \right )}+48\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1+2 i {\mathrm e}^{i \left (d x +c \right )}\right ) d^{4}} \]
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 \[ \int \sqrt {a \sin \left (d x + c\right ) + a} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 82, normalized size = 0.68 \[ -\frac {2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (48\,\sin \left (c+d\,x\right )-6\,d^2\,x^2+d^3\,x^3\,\cos \left (c+d\,x\right )-6\,d^2\,x^2\,\sin \left (c+d\,x\right )-24\,d\,x\,\cos \left (c+d\,x\right )+48\right )}{d^4\,\left (\sin \left (c+d\,x\right )+1\right )} \]
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 \[ \int x^{3} \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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