Optimal. Leaf size=129 \[ \frac {13 \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{5\ 2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}}-\frac {3 \cos (c+d x)}{2 a d \sqrt [3]{a \sin (c+d x)+a}}-\frac {3 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}} \]
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Rubi [A] time = 0.14, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2758, 2751, 2652, 2651} \[ \frac {13 \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{5\ 2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}}-\frac {3 \cos (c+d x)}{2 a d \sqrt [3]{a \sin (c+d x)+a}}-\frac {3 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}} \]
Antiderivative was successfully verified.
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Rule 2651
Rule 2652
Rule 2751
Rule 2758
Rubi steps
\begin {align*} \int \frac {\sin ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx &=-\frac {3 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}+\frac {3 \int \frac {-\frac {4 a}{3}+\frac {5}{3} a \sin (c+d x)}{\sqrt [3]{a+a \sin (c+d x)}} \, dx}{5 a^2}\\ &=-\frac {3 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}-\frac {3 \cos (c+d x)}{2 a d \sqrt [3]{a+a \sin (c+d x)}}-\frac {13 \int \frac {1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx}{10 a}\\ &=-\frac {3 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}-\frac {3 \cos (c+d x)}{2 a d \sqrt [3]{a+a \sin (c+d x)}}-\frac {\left (13 \sqrt [3]{1+\sin (c+d x)}\right ) \int \frac {1}{\sqrt [3]{1+\sin (c+d x)}} \, dx}{10 a \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac {3 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}-\frac {3 \cos (c+d x)}{2 a d \sqrt [3]{a+a \sin (c+d x)}}+\frac {13 \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{5\ 2^{5/6} a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 108, normalized size = 0.84 \[ -\frac {3 \cos (c+d x) \left (13 \sqrt {2} (\sin (c+d x)+1) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2\left (\frac {1}{4} (2 c+2 d x+\pi )\right )\right )+\sqrt {1-\sin (c+d x)} (5 \sin (c+d x)+7)\right )}{10 d \sqrt {1-\sin (c+d x)} (a (\sin (c+d x)+1))^{4/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (\cos \left (d x + c\right )^{2} - 1\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}}}{a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}}, x\right ) \]
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 \[ \int \frac {\sin \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{2}\left (d x +c \right )}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {\sin \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (c+d\,x\right )}^2}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3}} \,d x \]
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 \[ \int \frac {\sin ^{2}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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