Optimal. Leaf size=82 \[ -\frac {3 \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (5+3 m),\frac {1}{6} (11+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+3 m) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {20, 2722}
\begin {gather*} -\frac {3 \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+5);\frac {1}{6} (3 m+11);\cos ^2(c+d x)\right )}{d (3 m+5) \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 2722
Rubi steps
\begin {align*} \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx &=\frac {(b \cos (c+d x))^{2/3} \int \cos ^{\frac {2}{3}+m}(c+d x) \, dx}{\cos ^{\frac {2}{3}}(c+d x)}\\ &=-\frac {3 \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac {1}{2},\frac {1}{6} (5+3 m);\frac {1}{6} (11+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+3 m) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 82, normalized size = 1.00 \begin {gather*} -\frac {\cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \csc (c+d x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (\frac {5}{3}+m\right ),\frac {1}{2} \left (\frac {11}{3}+m\right ),\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{d \left (\frac {5}{3}+m\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (\cos ^{m}\left (d x +c \right )\right ) \left (b \cos \left (d x +c \right )\right )^{\frac {2}{3}}\, dx\]
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 \left (b \cos {\left (c + d x \right )}\right )^{\frac {2}{3}} \cos ^{m}{\left (c + d 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 = -0.01 \begin {gather*} \int {\cos \left (c+d\,x\right )}^m\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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