Optimal. Leaf size=210 \[ \frac {2 (c \cos (e+f x))^m (c \sec (e+f x))^m \text {Int}\left (\frac {(c \cos (e+f x))^{-m} \left (\frac {1}{2} c \cos (e+f x) \left (a (5-2 m) (a B+2 A b)+b^2 B (3-2 m)\right )+\frac {1}{2} b c \cos ^2(e+f x) (2 a B (3-m)+A b (5-2 m))+\frac {1}{2} a c \left (2 a A \left (\frac {5}{2}-m\right )+2 b B (1-m)\right )\right )}{\sqrt {a+b \cos (e+f x)}},x\right )}{c (5-2 m)}+\frac {2 b B \sin (e+f x) \cos (e+f x) \sqrt {a+b \cos (e+f x)} (c \sec (e+f x))^m}{f (5-2 m)} \]
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Rubi [A] time = 0.72, 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 = {} \[ \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx &=\left ((c \cos (e+f x))^m (c \sec (e+f x))^m\right ) \int (c \cos (e+f x))^{-m} (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx\\ &=\frac {2 b B \cos (e+f x) \sqrt {a+b \cos (e+f x)} (c \sec (e+f x))^m \sin (e+f x)}{f (5-2 m)}+\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} a c \left (2 b B (1-m)+2 a A \left (\frac {5}{2}-m\right )\right )+\frac {1}{2} c \left (b^2 B (3-2 m)+a (2 A b+a B) (5-2 m)\right ) \cos (e+f x)+\frac {1}{2} b c (A b (5-2 m)+2 a B (3-m)) \cos ^2(e+f x)\right )}{\sqrt {a+b \cos (e+f x)}} \, dx}{c (5-2 m)}\\ \end {align*}
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Mathematica [A] time = 61.59, size = 0, normalized size = 0.00 \[ \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b \cos \left (f x + e\right )^{2} + A a + {\left (B a + A b\right )} \cos \left (f x + e\right )\right )} \sqrt {b \cos \left (f x + e\right ) + a} \left (c \sec \left (f x + e\right )\right )^{m}, x\right ) \]
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 \[ \int {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (c \sec \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 0, normalized size = 0.00 \[ \int \left (a +b \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \cos \left (f x +e \right )\right ) \left (c \sec \left (f x +e \right )\right )^{m}\, dx \]
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 \[ \int {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (c \sec \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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