Optimal. Leaf size=92 \[ \frac{a 2^{m+\frac{1}{2}} \tan (e+f x) (c-c \sec (e+f x))^2 (\sec (e+f x)+1)^{\frac{1}{2}-m} (a \sec (e+f x)+a)^{m-1} \text{Hypergeometric2F1}\left (\frac{5}{2},\frac{1}{2}-m,\frac{7}{2},\frac{1}{2} (1-\sec (e+f x))\right )}{5 f} \]
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Rubi [A] time = 0.108356, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {3961, 70, 69} \[ \frac{a 2^{m+\frac{1}{2}} \tan (e+f x) (c-c \sec (e+f x))^2 (\sec (e+f x)+1)^{\frac{1}{2}-m} (a \sec (e+f x)+a)^{m-1} \, _2F_1\left (\frac{5}{2},\frac{1}{2}-m;\frac{7}{2};\frac{1}{2} (1-\sec (e+f x))\right )}{5 f} \]
Antiderivative was successfully verified.
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Rule 3961
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^2 \, dx &=-\frac{(a c \tan (e+f x)) \operatorname{Subst}\left (\int (a+a x)^{-\frac{1}{2}+m} (c-c x)^{3/2} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=-\frac{\left (2^{-\frac{1}{2}+m} a c (a+a \sec (e+f x))^{-1+m} \left (\frac{a+a \sec (e+f x)}{a}\right )^{\frac{1}{2}-m} \tan (e+f x)\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}+\frac{x}{2}\right )^{-\frac{1}{2}+m} (c-c x)^{3/2} \, dx,x,\sec (e+f x)\right )}{f \sqrt{c-c \sec (e+f x)}}\\ &=\frac{2^{\frac{1}{2}+m} a \, _2F_1\left (\frac{5}{2},\frac{1}{2}-m;\frac{7}{2};\frac{1}{2} (1-\sec (e+f x))\right ) (1+\sec (e+f x))^{\frac{1}{2}-m} (a+a \sec (e+f x))^{-1+m} (c-c \sec (e+f x))^2 \tan (e+f x)}{5 f}\\ \end{align*}
Mathematica [A] time = 0.24753, size = 89, normalized size = 0.97 \[ \frac{c^2 2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)-1)^2 (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a (\sec (e+f x)+1))^m \text{Hypergeometric2F1}\left (\frac{5}{2},\frac{1}{2}-m,\frac{7}{2},\frac{1}{2} (1-\sec (e+f x))\right )}{5 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.433, size = 0, normalized size = 0. \begin{align*} \int \sec \left ( fx+e \right ) \left ( a+a\sec \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sec \left ( fx+e \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c \sec \left (f x + e\right ) - c\right )}^{2}{\left (a \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c^{2} \sec \left (f x + e\right )^{3} - 2 \, c^{2} \sec \left (f x + e\right )^{2} + c^{2} \sec \left (f x + e\right )\right )}{\left (a \sec \left (f x + e\right ) + a\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c \sec \left (f x + e\right ) - c\right )}^{2}{\left (a \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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