Optimal. Leaf size=230 \[ -\frac{a^3 (4 n+1) \sin (e+f x) \sec ^{n-1}(e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-n}{2},\frac{3-n}{2},\cos ^2(e+f x)\right )}{f \left (1-n^2\right ) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (4 n+7) \sin (e+f x) \sec ^n(e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{n}{2},\frac{2-n}{2},\cos ^2(e+f x)\right )}{f n (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (2 n+5) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (n+1) (n+2)}+\frac{\sin (e+f x) \left (a^3 \sec (e+f x)+a^3\right ) \sec ^{n+1}(e+f x)}{f (n+2)} \]
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Rubi [A] time = 0.276106, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3814, 3997, 3787, 3772, 2643} \[ -\frac{a^3 (4 n+1) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right )}{f \left (1-n^2\right ) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (4 n+7) \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left (\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right )}{f n (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (2 n+5) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (n+1) (n+2)}+\frac{\sin (e+f x) \left (a^3 \sec (e+f x)+a^3\right ) \sec ^{n+1}(e+f x)}{f (n+2)} \]
Antiderivative was successfully verified.
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Rule 3814
Rule 3997
Rule 3787
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx &=\frac{\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}+\frac{a \int \sec ^n(e+f x) (a+a \sec (e+f x)) (a (2+2 n)+a (5+2 n) \sec (e+f x)) \, dx}{2+n}\\ &=\frac{a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac{\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}+\frac{a \int \sec ^n(e+f x) \left (a^2 (2+n) (1+4 n)+a^2 (1+n) (7+4 n) \sec (e+f x)\right ) \, dx}{2+3 n+n^2}\\ &=\frac{a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac{\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}+\frac{\left (a^3 (1+4 n)\right ) \int \sec ^n(e+f x) \, dx}{1+n}+\frac{\left (a^3 (7+4 n)\right ) \int \sec ^{1+n}(e+f x) \, dx}{2+n}\\ &=\frac{a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac{\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}+\frac{\left (a^3 (1+4 n) \cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-n}(e+f x) \, dx}{1+n}+\frac{\left (a^3 (7+4 n) \cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-1-n}(e+f x) \, dx}{2+n}\\ &=\frac{a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac{\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}-\frac{a^3 (1+4 n) \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) \sin (e+f x)}{f \left (1-n^2\right ) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (7+4 n) \, _2F_1\left (\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right ) \sec ^n(e+f x) \sin (e+f x)}{f n (2+n) \sqrt{\sin ^2(e+f x)}}\\ \end{align*}
Mathematica [F] time = 2.0934, size = 0, normalized size = 0. \[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 2.337, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( fx+e \right ) \right ) ^{n} \left ( a+a\sec \left ( fx+e \right ) \right ) ^{3}\, 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 )}^{3} \sec \left (f x + e\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^{3} \sec \left (f x + e\right )^{3} + 3 \, a^{3} \sec \left (f x + e\right )^{2} + 3 \, a^{3} \sec \left (f x + e\right ) + a^{3}\right )} \sec \left (f x + e\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int 3 \sec{\left (e + f x \right )} \sec ^{n}{\left (e + f x \right )}\, dx + \int 3 \sec ^{2}{\left (e + f x \right )} \sec ^{n}{\left (e + f x \right )}\, dx + \int \sec ^{3}{\left (e + f x \right )} \sec ^{n}{\left (e + f x \right )}\, dx + \int \sec ^{n}{\left (e + f x \right )}\, dx\right ) \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 )}^{3} \sec \left (f x + e\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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