Optimal. Leaf size=130 \[ \frac{2 a^2 (4 n+1) \tan (e+f x) (-\sec (e+f x))^{-n} (d \sec (e+f x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},1-n,\frac{3}{2},\sec (e+f x)+1\right )}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \tan (e+f x) (d \sec (e+f x))^n}{f (2 n+1) \sqrt{a-a \sec (e+f x)}} \]
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Rubi [A] time = 0.179818, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3814, 21, 3806, 67, 65} \[ \frac{2 a^2 (4 n+1) \tan (e+f x) (-\sec (e+f x))^{-n} (d \sec (e+f x))^n \, _2F_1\left (\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right )}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \tan (e+f x) (d \sec (e+f x))^n}{f (2 n+1) \sqrt{a-a \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3814
Rule 21
Rule 3806
Rule 67
Rule 65
Rubi steps
\begin{align*} \int (d \sec (e+f x))^n (a-a \sec (e+f x))^{3/2} \, dx &=\frac{2 a^2 (d \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt{a-a \sec (e+f x)}}-\frac{(2 a) \int \frac{(d \sec (e+f x))^n \left (-a \left (\frac{1}{2}+2 n\right )+a \left (\frac{1}{2}+2 n\right ) \sec (e+f x)\right )}{\sqrt{a-a \sec (e+f x)}} \, dx}{1+2 n}\\ &=\frac{2 a^2 (d \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt{a-a \sec (e+f x)}}+\frac{(a (1+4 n)) \int (d \sec (e+f x))^n \sqrt{a-a \sec (e+f x)} \, dx}{1+2 n}\\ &=\frac{2 a^2 (d \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt{a-a \sec (e+f x)}}-\frac{\left (a^3 d (1+4 n) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(d x)^{-1+n}}{\sqrt{a+a x}} \, dx,x,\sec (e+f x)\right )}{f (1+2 n) \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{2 a^2 (d \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt{a-a \sec (e+f x)}}+\frac{\left (a^3 (1+4 n) (-\sec (e+f x))^{-n} (d \sec (e+f x))^n \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(-x)^{-1+n}}{\sqrt{a+a x}} \, dx,x,\sec (e+f x)\right )}{f (1+2 n) \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{2 a^2 (d \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 (1+4 n) \, _2F_1\left (\frac{1}{2},1-n;\frac{3}{2};1+\sec (e+f x)\right ) (-\sec (e+f x))^{-n} (d \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt{a-a \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 1.14138, size = 346, normalized size = 2.66 \[ -\frac{2^{n-\frac{3}{2}} e^{-\frac{1}{2} i (2 n+1) (e+f x)} \left (\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^{n+\frac{1}{2}} \csc ^3\left (\frac{1}{2} (e+f x)\right ) (a-a \sec (e+f x))^{3/2} \sec ^{-n-\frac{3}{2}}(e+f x) \left (\left (n^3+6 n^2+11 n+6\right ) e^{i n (e+f x)} \text{Hypergeometric2F1}\left (1,\frac{1}{2} (-n-1),\frac{n+2}{2},-e^{2 i (e+f x)}\right )+3 n \left (n^2+4 n+3\right ) e^{i (n+2) (e+f x)} \text{Hypergeometric2F1}\left (1,\frac{1-n}{2},\frac{n+4}{2},-e^{2 i (e+f x)}\right )-n (n+2) \left ((n+1) e^{i (n+3) (e+f x)} \text{Hypergeometric2F1}\left (1,1-\frac{n}{2},\frac{n+5}{2},-e^{2 i (e+f x)}\right )+3 (n+3) e^{i (n+1) (e+f x)} \text{Hypergeometric2F1}\left (1,-\frac{n}{2},\frac{n+3}{2},-e^{2 i (e+f x)}\right )\right )\right ) (d \sec (e+f x))^n}{f n (n+1) (n+2) (n+3)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.176, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{n} \left ( a-a\sec \left ( fx+e \right ) \right ) ^{{\frac{3}{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 (-a \sec \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \left (d \sec \left (f x + e\right )\right )^{n}\,{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 (a \sec \left (f x + e\right ) - a\right )} \sqrt{-a \sec \left (f x + e\right ) + a} \left (d \sec \left (f x + e\right )\right )^{n}, 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 (-a \sec \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \left (d \sec \left (f x + e\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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