Optimal. Leaf size=209 \[ \frac{2 (c \cos (e+f x))^m (c \sec (e+f x))^m \text{Unintegrable}\left (\frac{(c \cos (e+f x))^{-m} \left (\frac{1}{2} c \cos (e+f x) \left (a (5-2 m) (a B+2 A b)+b^2 B (3-2 m)\right )+\frac{1}{2} b c \cos ^2(e+f x) (2 a B (3-m)+A b (5-2 m))+\frac{1}{2} a c \left (2 a A \left (\frac{5}{2}-m\right )+2 b B (1-m)\right )\right )}{\sqrt{a+b \cos (e+f x)}},x\right )}{c (5-2 m)}+\frac{2 b B \sin (e+f x) \cos (e+f x) \sqrt{a+b \cos (e+f x)} (c \sec (e+f x))^m}{f (5-2 m)} \]
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Rubi [A] time = 0.721117, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}
Mathematica [A] time = 57.0767, size = 0, normalized size = 0. \[ \int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.714, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\cos \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( A+B\cos \left ( fx+e \right ) \right ) \left ( c\sec \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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