Optimal. Leaf size=58 \[ \frac {3 \cos (e+f x) \, _2F_1\left (-\frac {1}{2},\frac {1}{3};\frac {4}{3};\sin ^2(e+f x)\right ) (b \sin (e+f x))^{2/3}}{2 b f \sqrt {\cos ^2(e+f x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2657}
\begin {gather*} \frac {3 \cos (e+f x) (b \sin (e+f x))^{2/3} \, _2F_1\left (-\frac {1}{2},\frac {1}{3};\frac {4}{3};\sin ^2(e+f x)\right )}{2 b f \sqrt {\cos ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rubi steps
\begin {align*} \int \frac {\cos ^2(e+f x)}{\sqrt [3]{b \sin (e+f x)}} \, dx &=\frac {3 \cos (e+f x) \, _2F_1\left (-\frac {1}{2},\frac {1}{3};\frac {4}{3};\sin ^2(e+f x)\right ) (b \sin (e+f x))^{2/3}}{2 b f \sqrt {\cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 55, normalized size = 0.95 \begin {gather*} \frac {3 \sqrt {\cos ^2(e+f x)} \, _2F_1\left (-\frac {1}{2},\frac {1}{3};\frac {4}{3};\sin ^2(e+f x)\right ) \tan (e+f x)}{2 f \sqrt [3]{b \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\cos ^{2}\left (f x +e \right )}{\left (b \sin \left (f x +e \right )\right )^{\frac {1}{3}}}\, dx\]
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 {\cos ^{2}{\left (e + f x \right )}}{\sqrt [3]{b \sin {\left (e + f 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 \frac {{\cos \left (e+f\,x\right )}^2}{{\left (b\,\sin \left (e+f\,x\right )\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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