Optimal. Leaf size=28 \[ \text {Int}\left (\frac {\cos ^{\frac {2}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}},x\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\cos ^{\frac {2}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {2}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx &=\int \frac {\cos ^{\frac {2}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx\\ \end {align*}
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Mathematica [A]
time = 14.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{\frac {2}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\cos ^{\frac {2}{3}}\left (d x +c \right )}{\sqrt {a +b \cos \left (d x +c \right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{\frac {2}{3}}{\left (c + d x \right )}}{\sqrt {a + b \cos {\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^{2/3}}{\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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