Optimal. Leaf size=132 \[ -\frac{a \sin (e+f x) (d \cos (e+f x))^{n+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+1}{2},\frac{n+3}{2},\cos ^2(e+f x)\right )}{d f (n+1) \sqrt{\sin ^2(e+f x)}}-\frac{b \sin (e+f x) (d \cos (e+f x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n}{2},\frac{n+2}{2},\cos ^2(e+f x)\right )}{f n \sqrt{\sin ^2(e+f x)}} \]
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Rubi [A] time = 0.114997, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4225, 16, 2748, 2643} \[ -\frac{a \sin (e+f x) (d \cos (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1) \sqrt{\sin ^2(e+f x)}}-\frac{b \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right )}{f n \sqrt{\sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 4225
Rule 16
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int (d \cos (e+f x))^n (a+b \sec (e+f x)) \, dx &=\int (d \cos (e+f x))^n (b+a \cos (e+f x)) \sec (e+f x) \, dx\\ &=d \int (d \cos (e+f x))^{-1+n} (b+a \cos (e+f x)) \, dx\\ &=a \int (d \cos (e+f x))^n \, dx+(b d) \int (d \cos (e+f x))^{-1+n} \, dx\\ &=-\frac{b (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)}{f n \sqrt{\sin ^2(e+f x)}}-\frac{a (d \cos (e+f x))^{1+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)}{d f (1+n) \sqrt{\sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.11885, size = 106, normalized size = 0.8 \[ -\frac{\sqrt{\sin ^2(e+f x)} \csc (e+f x) (d \cos (e+f x))^n \left (a n \cos (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+1}{2},\frac{n+3}{2},\cos ^2(e+f x)\right )+b (n+1) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n}{2},\frac{n+2}{2},\cos ^2(e+f x)\right )\right )}{f n (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.576, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( fx+e \right ) \right ) ^{n} \left ( a+b\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 (b \sec \left (f x + e\right ) + a\right )} \left (d \cos \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 (b \sec \left (f x + e\right ) + a\right )} \left (d \cos \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos{\left (e + f x \right )}\right )^{n} \left (a + b \sec{\left (e + f x \right )}\right )\, dx \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 (b \sec \left (f x + e\right ) + a\right )} \left (d \cos \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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