Optimal. Leaf size=69 \[ -\frac {\cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {11}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2652, 2651} \[ -\frac {\cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {11}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2651
Rule 2652
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (c+d x))^{4/3}} \, dx &=\frac {\sqrt [3]{1+\sin (c+d x)} \int \frac {1}{(1+\sin (c+d x))^{4/3}} \, dx}{a \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac {\cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {11}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{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 = 0.21, size = 130, normalized size = 1.88 \[ -\frac {3 \left (\sqrt {2-2 \sin (c+d x)}-2 (\sin (c+d x)+1) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2\left (\frac {1}{4} (2 c+2 d x+\pi )\right )\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{5 d \sqrt {2-2 \sin (c+d x)} (a (\sin (c+d x)+1))^{4/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}}}{a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3}} \,d x \]
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 \[ \int \frac {1}{\left (a \sin {\left (c + d x \right )} + a\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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