Optimal. Leaf size=53 \[ \frac {3 \cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\sin ^2(a+b x)\right )}{7 b \sqrt {\cos ^2(a+b x)} \csc ^{\frac {7}{3}}(a+b x)} \]
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Rubi [A] time = 0.02, antiderivative size = 53, 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} \[ \frac {3 \cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\sin ^2(a+b x)\right )}{7 b \sqrt {\cos ^2(a+b x)} \csc ^{\frac {7}{3}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3772
Rubi steps
\begin {align*} \int \frac {1}{\csc ^{\frac {4}{3}}(a+b x)} \, dx &=\csc ^{\frac {2}{3}}(a+b x) \sin ^{\frac {2}{3}}(a+b x) \int \sin ^{\frac {4}{3}}(a+b x) \, dx\\ &=\frac {3 \cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\sin ^2(a+b x)\right )}{7 b \sqrt {\cos ^2(a+b x)} \csc ^{\frac {7}{3}}(a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 68, normalized size = 1.28 \[ -\frac {\cos (a+b x) \left (\, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\cos ^2(a+b x)\right )+3 \sqrt [6]{\sin ^2(a+b x)}\right )}{4 b \sqrt [6]{\sin ^2(a+b x)} \sqrt [3]{\csc (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\csc \left (b x + a\right )^{\frac {4}{3}}}, x\right ) \]
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 \[ \int \frac {1}{\csc \left (b x + a\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.61, size = 0, normalized size = 0.00 \[ \int \frac {1}{\csc \left (b x +a \right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {1}{\csc \left (b x + a\right )^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{4/3}} \,d x \]
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 \[ \int \frac {1}{\csc ^{\frac {4}{3}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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