Optimal. Leaf size=293 \[ -\frac {16 \text {Li}_3\left (-e^{\frac {1}{4} i (2 c+2 d x+\pi )}\right ) \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d^3 \sqrt {a \sin (c+d x)+a}}+\frac {16 \text {Li}_3\left (e^{\frac {1}{4} i (2 c+2 d x+\pi )}\right ) \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d^3 \sqrt {a \sin (c+d x)+a}}+\frac {8 i x \text {Li}_2\left (-e^{\frac {1}{4} i (2 c+2 d x+\pi )}\right ) \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d^2 \sqrt {a \sin (c+d x)+a}}-\frac {8 i x \text {Li}_2\left (e^{\frac {1}{4} i (2 c+2 d x+\pi )}\right ) \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d^2 \sqrt {a \sin (c+d x)+a}}-\frac {4 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \tanh ^{-1}\left (e^{\frac {1}{4} i (2 c+2 d x+\pi )}\right )}{d \sqrt {a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.18, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3319, 4183, 2531, 2282, 6589} \[ \frac {8 i x \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \text {PolyLog}\left (2,-e^{\frac {1}{4} i (2 c+2 d x+\pi )}\right )}{d^2 \sqrt {a \sin (c+d x)+a}}-\frac {8 i x \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \text {PolyLog}\left (2,e^{\frac {1}{4} i (2 c+2 d x+\pi )}\right )}{d^2 \sqrt {a \sin (c+d x)+a}}-\frac {16 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \text {PolyLog}\left (3,-e^{\frac {1}{4} i (2 c+2 d x+\pi )}\right )}{d^3 \sqrt {a \sin (c+d x)+a}}+\frac {16 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \text {PolyLog}\left (3,e^{\frac {1}{4} i (2 c+2 d x+\pi )}\right )}{d^3 \sqrt {a \sin (c+d x)+a}}-\frac {4 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \tanh ^{-1}\left (e^{\frac {1}{4} i (2 c+2 d x+\pi )}\right )}{d \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 3319
Rule 4183
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a+a \sin (c+d x)}} \, dx &=\frac {\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \int x^2 \csc \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{\sqrt {a+a \sin (c+d x)}}\\ &=-\frac {4 x^2 \tanh ^{-1}\left (e^{\frac {1}{4} i (2 c+\pi +2 d x)}\right ) \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{d \sqrt {a+a \sin (c+d x)}}-\frac {\left (4 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int x \log \left (1-e^{i \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}\right ) \, dx}{d \sqrt {a+a \sin (c+d x)}}+\frac {\left (4 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int x \log \left (1+e^{i \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}\right ) \, dx}{d \sqrt {a+a \sin (c+d x)}}\\ &=-\frac {4 x^2 \tanh ^{-1}\left (e^{\frac {1}{4} i (2 c+\pi +2 d x)}\right ) \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{d \sqrt {a+a \sin (c+d x)}}+\frac {8 i x \text {Li}_2\left (-e^{\frac {1}{4} i (2 c+\pi +2 d x)}\right ) \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{d^2 \sqrt {a+a \sin (c+d x)}}-\frac {8 i x \text {Li}_2\left (e^{\frac {1}{4} i (2 c+\pi +2 d x)}\right ) \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{d^2 \sqrt {a+a \sin (c+d x)}}-\frac {\left (8 i \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int \text {Li}_2\left (-e^{i \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}\right ) \, dx}{d^2 \sqrt {a+a \sin (c+d x)}}+\frac {\left (8 i \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int \text {Li}_2\left (e^{i \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}\right ) \, dx}{d^2 \sqrt {a+a \sin (c+d x)}}\\ &=-\frac {4 x^2 \tanh ^{-1}\left (e^{\frac {1}{4} i (2 c+\pi +2 d x)}\right ) \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{d \sqrt {a+a \sin (c+d x)}}+\frac {8 i x \text {Li}_2\left (-e^{\frac {1}{4} i (2 c+\pi +2 d x)}\right ) \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{d^2 \sqrt {a+a \sin (c+d x)}}-\frac {8 i x \text {Li}_2\left (e^{\frac {1}{4} i (2 c+\pi +2 d x)}\right ) \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{d^2 \sqrt {a+a \sin (c+d x)}}-\frac {\left (16 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}\right )}{d^3 \sqrt {a+a \sin (c+d x)}}+\frac {\left (16 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}\right )}{d^3 \sqrt {a+a \sin (c+d x)}}\\ &=-\frac {4 x^2 \tanh ^{-1}\left (e^{\frac {1}{4} i (2 c+\pi +2 d x)}\right ) \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{d \sqrt {a+a \sin (c+d x)}}+\frac {8 i x \text {Li}_2\left (-e^{\frac {1}{4} i (2 c+\pi +2 d x)}\right ) \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{d^2 \sqrt {a+a \sin (c+d x)}}-\frac {8 i x \text {Li}_2\left (e^{\frac {1}{4} i (2 c+\pi +2 d x)}\right ) \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{d^2 \sqrt {a+a \sin (c+d x)}}-\frac {16 \text {Li}_3\left (-e^{\frac {1}{4} i (2 c+\pi +2 d x)}\right ) \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{d^3 \sqrt {a+a \sin (c+d x)}}+\frac {16 \text {Li}_3\left (e^{\frac {1}{4} i (2 c+\pi +2 d x)}\right ) \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{d^3 \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.62, size = 245, normalized size = 0.84 \[ \frac {\sqrt [4]{-1} \sqrt {2} e^{-\frac {1}{2} i (c+d x)} \left (e^{i (c+d x)}+i\right ) \left (4 d x \text {Li}_2\left (-\sqrt [4]{-1} e^{\frac {1}{2} i (c+d x)}\right )-i \left (d^2 x^2 \log \left (1-\sqrt [4]{-1} e^{\frac {1}{2} i (c+d x)}\right )-d^2 x^2 \log \left (1+\sqrt [4]{-1} e^{\frac {1}{2} i (c+d x)}\right )-4 i d x \text {Li}_2\left (\sqrt [4]{-1} e^{\frac {1}{2} i (c+d x)}\right )-8 \text {Li}_3\left (-\sqrt [4]{-1} e^{\frac {1}{2} i (c+d x)}\right )+8 \text {Li}_3\left (\sqrt [4]{-1} e^{\frac {1}{2} i (c+d x)}\right )\right )\right )}{d^3 \sqrt {-i a e^{-i (c+d x)} \left (e^{i (c+d x)}+i\right )^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\sqrt {a \sin \left (d x + c\right ) + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {a \sin \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {a +a \sin \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {x^{2}}{\sqrt {a \sin \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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