Optimal. Leaf size=99 \[ \frac {3 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}-\frac {4 \sqrt [6]{2} \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{5 a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}} \]
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Rubi [A]
time = 0.06, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2829, 2731,
2730} \begin {gather*} \frac {3 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}-\frac {4 \sqrt [6]{2} \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 a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2730
Rule 2731
Rule 2829
Rubi steps
\begin {align*} \int \frac {\sin (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 {4 \int \frac {1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx}{5 a}\\ &=\frac {3 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}+\frac {\left (4 \sqrt [3]{1+\sin (c+d x)}\right ) \int \frac {1}{\sqrt [3]{1+\sin (c+d x)}} \, dx}{5 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 {4 \sqrt [6]{2} \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 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 = 0.19, size = 130, normalized size = 1.31 \begin {gather*} \frac {3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sqrt {2-2 \sin (c+d x)}+8 \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right ) (1+\sin (c+d x))\right )}{5 d \sqrt {2-2 \sin (c+d x)} (a (1+\sin (c+d x)))^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (d x +c \right )}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sin {\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {4}{3}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sin \left (c+d\,x\right )}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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