Optimal. Leaf size=56 \[ \frac {3 c \cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\sin ^2(a+b x)\right )}{7 b \sqrt {\cos ^2(a+b x)} (c \csc (a+b x))^{7/3}} \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3772, 2643} \[ \frac {3 c \cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\sin ^2(a+b x)\right )}{7 b \sqrt {\cos ^2(a+b x)} (c \csc (a+b x))^{7/3}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3772
Rubi steps
\begin {align*} \int \frac {1}{(c \csc (a+b x))^{4/3}} \, dx &=(c \csc (a+b x))^{2/3} \left (\frac {\sin (a+b x)}{c}\right )^{2/3} \int \left (\frac {\sin (a+b x)}{c}\right )^{4/3} \, dx\\ &=\frac {3 \cos (a+b x) (c \csc (a+b x))^{2/3} \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\sin ^2(a+b x)\right ) \sin ^3(a+b x)}{7 b c^2 \sqrt {\cos ^2(a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 76, normalized size = 1.36 \[ -\frac {\csc ^2(a+b x) \left (2 \sin ^2(a+b x)^{5/6} \cot (a+b x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\cos ^2(a+b x)\right )+3 \sin (2 (a+b x))\right )}{8 b (c \csc (a+b x))^{4/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (c \csc \left (b x + a\right )\right )^{\frac {2}{3}}}{c^{2} \csc \left (b x + a\right )^{2}}, x\right ) \]
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 \[ \int \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.52, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \csc \left (b x +a \right )\right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{4/3}} \,d x \]
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 \[ \int \frac {1}{\left (c \csc {\left (a + b x \right )}\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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