Optimal. Leaf size=28 \[ \text {Int}\left (\frac {1}{\cos ^{\frac {2}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}},x\right ) \]
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Rubi [A] time = 0.05, 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 = {} \[ \int \frac {1}{\cos ^{\frac {2}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\cos ^{\frac {2}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx &=\int \frac {1}{\cos ^{\frac {2}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.53, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cos ^{\frac {2}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {1}{3}}}{b \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cos \left (d x +c \right )^{\frac {2}{3}} \sqrt {a +b \cos \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {1}{\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {2}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^{2/3}\,\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a + b \cos {\left (c + d x \right )}} \cos ^{\frac {2}{3}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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