Optimal. Leaf size=213 \[ \frac {2 (c \cos (e+f x))^m (c \sec (e+f x))^m \text {Int}\left (\frac {(c \cos (e+f x))^{-m} \left (\frac {1}{2} c \left (a^2 A-2 a b B (1-m)+A b^2 (1-2 m)\right )-\frac {1}{2} b c (3-2 m) (A b-a B) \cos ^2(e+f x)-\frac {1}{2} a c (A b-a B) \cos (e+f x)\right )}{\sqrt {a+b \cos (e+f x)}},x\right )}{a c \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (e+f x) \cos (e+f x) (c \sec (e+f x))^m}{a f \left (a^2-b^2\right ) \sqrt {a+b \cos (e+f x)}} \]
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Rubi [A] time = 0.69, 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 = {} \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx \]
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}{(a+b \cos (e+f x))^{3/2}} \, 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))}{(a+b \cos (e+f x))^{3/2}} \, dx\\ &=\frac {2 b (A b-a B) \cos (e+f x) (c \sec (e+f x))^m \sin (e+f x)}{a \left (a^2-b^2\right ) f \sqrt {a+b \cos (e+f x)}}+\frac {\left (2 (c \cos (e+f x))^m (c \sec (e+f x))^m\right ) \int \frac {(c \cos (e+f x))^{-m} \left (\frac {1}{2} c \left (a^2 A+A b^2 (1-2 m)-2 a b B (1-m)\right )-\frac {1}{2} a (A b-a B) c \cos (e+f x)-\frac {1}{2} b (A b-a B) c (3-2 m) \cos ^2(e+f x)\right )}{\sqrt {a+b \cos (e+f x)}} \, dx}{a \left (a^2-b^2\right ) c}\\ \end {align*}
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Mathematica [A] time = 11.21, size = 0, normalized size = 0.00 \[ \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \cos \left (f x + e\right ) + A\right )} \sqrt {b \cos \left (f x + e\right ) + a} \left (c \sec \left (f x + e\right )\right )^{m}}{b^{2} \cos \left (f x + e\right )^{2} + 2 \, a b \cos \left (f x + e\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \sec \left (f x + e\right )\right )^{m}}{{\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {\left (A +B \cos \left (f x +e \right )\right ) \left (c \sec \left (f x +e \right )\right )^{m}}{\left (a +b \cos \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \sec \left (f x + e\right )\right )^{m}}{{\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )}{{\left (a+b\,\cos \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (c \sec {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right )}{\left (a + b \cos {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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