Optimal. Leaf size=51 \[ \frac {3 \cos (a+b x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)} \sqrt [3]{\csc (a+b x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3857, 2722}
\begin {gather*} \frac {3 \cos (a+b x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)} \sqrt [3]{\csc (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3857
Rubi steps
\begin {align*} \int \csc ^{\frac {2}{3}}(a+b x) \, dx &=\csc ^{\frac {2}{3}}(a+b x) \sin ^{\frac {2}{3}}(a+b x) \int \frac {1}{\sin ^{\frac {2}{3}}(a+b x)} \, dx\\ &=\frac {3 \cos (a+b x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)} \sqrt [3]{\csc (a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 51, normalized size = 1.00 \begin {gather*} -\frac {\cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\cos ^2(a+b x)\right )}{b \sqrt [3]{\csc (a+b x)} \sqrt [6]{\sin ^2(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \csc ^{\frac {2}{3}}\left (x b +a \right )\, dx\]
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 \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 \csc ^{\frac {2}{3}}{\left (a + b x \right )}\, 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 = -0.02 \begin {gather*} \int {\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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