Optimal. Leaf size=51 \[ -\frac {3 \cos (a+b x) \sqrt [3]{\csc (a+b x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)}} \]
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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 = {3772, 2643} \[ -\frac {3 \cos (a+b x) \sqrt [3]{\csc (a+b x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3772
Rubi steps
\begin {align*} \int \csc ^{\frac {4}{3}}(a+b x) \, dx &=\sqrt [3]{\csc (a+b x)} \sqrt [3]{\sin (a+b x)} \int \frac {1}{\sin ^{\frac {4}{3}}(a+b x)} \, dx\\ &=-\frac {3 \cos (a+b x) \sqrt [3]{\csc (a+b x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 54, normalized size = 1.06 \[ \frac {\cos (a+b x) \sqrt [3]{\csc (a+b x)} \left (2 \sqrt [6]{\sin ^2(a+b x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};\cos ^2(a+b x)\right )-3\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\csc \left (b x + a\right )^{\frac {4}{3}}, x\right ) \]
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 \[ \int \csc \left (b x + a\right )^{\frac {4}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \csc ^{\frac {4}{3}}\left (b x +a \right )\, dx \]
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 \[ \int \csc \left (b x + a\right )^{\frac {4}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{4/3} \,d x \]
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 \[ \int \csc ^{\frac {4}{3}}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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