Optimal. Leaf size=156 \[ \frac{a \sin (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{n p}{2},\frac{1}{2} (2-n p),\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n}{f n p \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) \cos (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (1-n p),\frac{1}{2} (3-n p),\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n}{f (1-n p) \sqrt{\sin ^2(e+f x)}} \]
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Rubi [A] time = 0.146012, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3948, 3787, 3772, 2643} \[ \frac{a \sin (e+f x) \, _2F_1\left (\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n}{f n p \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n}{f (1-n p) \sqrt{\sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3948
Rule 3787
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \left (c (d \sec (e+f x))^p\right )^n (a+a \sec (e+f x)) \, 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)) \, dx\\ &=\left (a (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+\frac{\left (a (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}\\ &=\left (a \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+\frac{\left (a \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{a \, _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 \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 (1-n p) \sqrt{\sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.201515, size = 124, normalized size = 0.79 \[ \frac{a \sqrt{-\tan ^2(e+f x)} \csc (e+f x) \left (c (d \sec (e+f x))^p\right )^n \left (n p \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (n p+1),\frac{1}{2} (n p+3),\sec ^2(e+f x)\right )+(n p+1) \cos (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n p}{2},\frac{n p}{2}+1,\sec ^2(e+f x)\right )\right )}{f n p (n p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.148, 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 ) \, 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 )} \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 \sec \left (f x + e\right ) + a\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 \left (\int \left (c \left (d \sec{\left (e + f x \right )}\right )^{p}\right )^{n}\, dx + \int \left (c \left (d \sec{\left (e + f x \right )}\right )^{p}\right )^{n} \sec{\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 )} \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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