Optimal. Leaf size=215 \[ \frac{2 (n+2) \sin (e+f x) (d \cos (e+f x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n}{2},\frac{n+2}{2},\cos ^2(e+f x)\right )}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{(2 n+3) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+1}{2},\frac{n+3}{2},\cos ^2(e+f x)\right )}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{2 (n+2) \tan (e+f x) (d \cos (e+f x))^n}{3 a^2 f (\sec (e+f x)+1)}-\frac{\tan (e+f x) (d \cos (e+f x))^n}{3 f (a \sec (e+f x)+a)^2} \]
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Rubi [A] time = 0.411779, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {4264, 3817, 4020, 3787, 3772, 2643} \[ \frac{2 (n+2) \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right )}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{(2 n+3) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right )}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{2 (n+2) \tan (e+f x) (d \cos (e+f x))^n}{3 a^2 f (\sec (e+f x)+1)}-\frac{\tan (e+f x) (d \cos (e+f x))^n}{3 f (a \sec (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3817
Rule 4020
Rule 3787
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \frac{(d \cos (e+f x))^n}{(a+a \sec (e+f x))^2} \, dx &=\left ((d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int \frac{(d \sec (e+f x))^{-n}}{(a+a \sec (e+f x))^2} \, dx\\ &=-\frac{(d \cos (e+f x))^n \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac{\left ((d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int \frac{(d \sec (e+f x))^{-n} (a (-3-n)-a (-1-n) \sec (e+f x))}{a+a \sec (e+f x)} \, dx}{3 a^2}\\ &=-\frac{2 (2+n) (d \cos (e+f x))^n \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))}-\frac{(d \cos (e+f x))^n \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac{\left ((d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{-n} \left (-a^2 (1+n) (3+2 n)+2 a^2 n (2+n) \sec (e+f x)\right ) \, dx}{3 a^4}\\ &=-\frac{2 (2+n) (d \cos (e+f x))^n \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))}-\frac{(d \cos (e+f x))^n \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac{\left (2 n (2+n) (d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{1-n} \, dx}{3 a^2 d}+\frac{\left ((1+n) (3+2 n) (d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{-n} \, dx}{3 a^2}\\ &=-\frac{2 (2+n) (d \cos (e+f x))^n \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))}-\frac{(d \cos (e+f x))^n \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac{\left (2 n (2+n) \left (\frac{\cos (e+f x)}{d}\right )^{-n} (d \cos (e+f x))^n\right ) \int \left (\frac{\cos (e+f x)}{d}\right )^{-1+n} \, dx}{3 a^2 d}+\frac{\left ((1+n) (3+2 n) \left (\frac{\cos (e+f x)}{d}\right )^{-n} (d \cos (e+f x))^n\right ) \int \left (\frac{\cos (e+f x)}{d}\right )^n \, dx}{3 a^2}\\ &=\frac{2 (2+n) (d \cos (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{2+n}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{(3+2 n) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{1+n}{2};\frac{3+n}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{2 (2+n) (d \cos (e+f x))^n \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))}-\frac{(d \cos (e+f x))^n \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}\\ \end{align*}
Mathematica [F] time = 1.67316, size = 0, normalized size = 0. \[ \int \frac{(d \cos (e+f x))^n}{(a+a \sec (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.299, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d\cos \left ( fx+e \right ) \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 \frac{\left (d \cos \left (f x + e\right )\right )^{n}}{{\left (a \sec \left (f x + e\right ) + a\right )}^{2}}\,{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 (\frac{\left (d \cos \left (f x + e\right )\right )^{n}}{a^{2} \sec \left (f x + e\right )^{2} + 2 \, a^{2} \sec \left (f x + e\right ) + a^{2}}, 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*} \frac{\int \frac{\left (d \cos{\left (e + f x \right )}\right )^{n}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx}{a^{2}} \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 \frac{\left (d \cos \left (f x + e\right )\right )^{n}}{{\left (a \sec \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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