Optimal. Leaf size=61 \[ (c \cos (e+f x))^m (c \sec (e+f x))^m \text {Int}\left (\frac {(A+B \cos (e+f x)) (c \cos (e+f x))^{-m}}{\sqrt {a+b \cos (e+f x)}},x\right ) \]
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Rubi [A] time = 0.24, 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 = {} \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx &=\left ((c \cos (e+f x))^m (c \sec (e+f x))^m\right ) \int \frac {(c \cos (e+f x))^{-m} (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx\\ \end {align*}
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Mathematica [A] time = 8.87, size = 0, normalized size = 0.00 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt {a+b \cos (e+f x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \sec \left (f x + e\right )\right )^{m}}{\sqrt {b \cos \left (f x + e\right ) + a}}, x\right ) \]
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 \[ \int \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \sec \left (f x + e\right )\right )^{m}}{\sqrt {b \cos \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {\left (A +B \cos \left (f x +e \right )\right ) \left (c \sec \left (f x +e \right )\right )^{m}}{\sqrt {a +b \cos \left (f x +e \right )}}\, dx \]
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 \[ \int \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \sec \left (f x + e\right )\right )^{m}}{\sqrt {b \cos \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )}{\sqrt {a+b\,\cos \left (e+f\,x\right )}} \,d x \]
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 \[ \int \frac {\left (c \sec {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right )}{\sqrt {a + b \cos {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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