Optimal. Leaf size=205 \[ -\frac{a^2 (2 n p+1) \sin (e+f x) \cos (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (1-n p),\frac{1}{2} (3-n p),\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n}{f \left (1-n^2 p^2\right ) \sqrt{\sin ^2(e+f x)}}+\frac{2 a^2 \sin (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{n p}{2},\frac{1}{2} (2-n p),\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n}{f n p \sqrt{\sin ^2(e+f x)}}+\frac{a^2 \tan (e+f x) \left (c (d \sec (e+f x))^p\right )^n}{f (n p+1)} \]
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Rubi [A] time = 0.248729, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {3948, 3788, 3772, 2643, 4046} \[ -\frac{a^2 (2 n p+1) \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n}{f \left (1-n^2 p^2\right ) \sqrt{\sin ^2(e+f x)}}+\frac{2 a^2 \sin (e+f x) \, _2F_1\left (\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n}{f n p \sqrt{\sin ^2(e+f x)}}+\frac{a^2 \tan (e+f x) \left (c (d \sec (e+f x))^p\right )^n}{f (n p+1)} \]
Antiderivative was successfully verified.
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Rule 3948
Rule 3788
Rule 3772
Rule 2643
Rule 4046
Rubi steps
\begin{align*} \int \left (c (d \sec (e+f x))^p\right )^n (a+a \sec (e+f x))^2 \, dx &=\left ((d \sec (e+f x))^{-n p} \left (c (d \sec (e+f x))^p\right )^n\right ) \int (d \sec (e+f x))^{n p} (a+a \sec (e+f x))^2 \, dx\\ &=\left ((d \sec (e+f x))^{-n p} \left (c (d \sec (e+f x))^p\right )^n\right ) \int (d \sec (e+f x))^{n p} \left (a^2+a^2 \sec ^2(e+f x)\right ) \, dx+\frac{\left (2 a^2 (d \sec (e+f x))^{-n p} \left (c (d \sec (e+f x))^p\right )^n\right ) \int (d \sec (e+f x))^{1+n p} \, dx}{d}\\ &=\frac{a^2 \left (c (d \sec (e+f x))^p\right )^n \tan (e+f x)}{f (1+n p)}+\frac{\left (2 a^2 \left (\frac{\cos (e+f x)}{d}\right )^{n p} \left (c (d \sec (e+f x))^p\right )^n\right ) \int \left (\frac{\cos (e+f x)}{d}\right )^{-1-n p} \, dx}{d}+\frac{\left (a^2 (1+2 n p) (d \sec (e+f x))^{-n p} \left (c (d \sec (e+f x))^p\right )^n\right ) \int (d \sec (e+f x))^{n p} \, dx}{1+n p}\\ &=\frac{2 a^2 \, _2F_1\left (\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n \sin (e+f x)}{f n p \sqrt{\sin ^2(e+f x)}}+\frac{a^2 \left (c (d \sec (e+f x))^p\right )^n \tan (e+f x)}{f (1+n p)}+\frac{\left (a^2 (1+2 n p) \left (\frac{\cos (e+f x)}{d}\right )^{n p} \left (c (d \sec (e+f x))^p\right )^n\right ) \int \left (\frac{\cos (e+f x)}{d}\right )^{-n p} \, dx}{1+n p}\\ &=\frac{2 a^2 \, _2F_1\left (\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n \sin (e+f x)}{f n p \sqrt{\sin ^2(e+f x)}}-\frac{a^2 (1+2 n p) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n \sin (e+f x)}{f \left (1-n^2 p^2\right ) \sqrt{\sin ^2(e+f x)}}+\frac{a^2 \left (c (d \sec (e+f x))^p\right )^n \tan (e+f x)}{f (1+n p)}\\ \end{align*}
Mathematica [F] time = 1.00003, size = 0, normalized size = 0. \[ \int \left (c (d \sec (e+f x))^p\right )^n (a+a \sec (e+f x))^2 \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.161, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( d\sec \left ( fx+e \right ) \right ) ^{p} \right ) ^{n} \left ( a+a\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 (a \sec \left (f x + e\right ) + a\right )}^{2} \left (\left (d \sec \left (f x + e\right )\right )^{p} c\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^{2} \sec \left (f x + e\right )^{2} + 2 \, a^{2} \sec \left (f x + e\right ) + a^{2}\right )} \left (\left (d \sec \left (f x + e\right )\right )^{p} c\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^{2} \left (\int \left (c \left (d \sec{\left (e + f x \right )}\right )^{p}\right )^{n}\, dx + \int 2 \left (c \left (d \sec{\left (e + f x \right )}\right )^{p}\right )^{n} \sec{\left (e + f x \right )}\, dx + \int \left (c \left (d \sec{\left (e + f x \right )}\right )^{p}\right )^{n} \sec ^{2}{\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 )}^{2} \left (\left (d \sec \left (f x + e\right )\right )^{p} c\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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