3.444 \(\int \frac{(d \cos (e+f x))^n}{a+a \sec (e+f x)} \, dx\)

Optimal. Leaf size=178 \[ -\frac{\sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+1}{2},\frac{n+3}{2},\cos ^2(e+f x)\right )}{a f \sqrt{\sin ^2(e+f x)}}+\frac{(n+1) \sin (e+f x) \cos ^2(e+f x) (d \cos (e+f x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+2}{2},\frac{n+4}{2},\cos ^2(e+f x)\right )}{a f (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{\sin (e+f x) (d \cos (e+f x))^n}{f (a \sec (e+f x)+a)} \]

[Out]

((d*Cos[e + f*x])^n*Sin[e + f*x])/(f*(a + a*Sec[e + f*x])) - (Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F
1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(a*f*Sqrt[Sin[e + f*x]^2]) + ((1 + n)*Cos[e + f*x]^
2*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(a*f*(2 + n)*S
qrt[Sin[e + f*x]^2])

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Rubi [A]  time = 0.244499, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4264, 3820, 3787, 3772, 2643} \[ -\frac{\sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right )}{a f \sqrt{\sin ^2(e+f x)}}+\frac{(n+1) \sin (e+f x) \cos ^2(e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(e+f x)\right )}{a f (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{\sin (e+f x) (d \cos (e+f x))^n}{f (a \sec (e+f x)+a)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cos[e + f*x])^n/(a + a*Sec[e + f*x]),x]

[Out]

((d*Cos[e + f*x])^n*Sin[e + f*x])/(f*(a + a*Sec[e + f*x])) - (Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F
1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(a*f*Sqrt[Sin[e + f*x]^2]) + ((1 + n)*Cos[e + f*x]^
2*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(a*f*(2 + n)*S
qrt[Sin[e + f*x]^2])

Rule 4264

Int[(u_)*((c_.)*sin[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Csc[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSecantIntegrandQ[
u, x]

Rule 3820

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[(b*d*Cot[e
+ f*x]*(d*Csc[e + f*x])^(n - 1))/(a*f*(a + b*Csc[e + f*x])), x] + Dist[(d*(n - 1))/(a*b), Int[(d*Csc[e + f*x]
)^(n - 1)*(a - b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3772

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x])^(n - 1)*((Sin[c + d*x]/b)^(n - 1)
*Int[1/(Sin[c + d*x]/b)^n, x]), x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int \frac{(d \cos (e+f x))^n}{a+a \sec (e+f x)} \, 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)} \, dx\\ &=\frac{(d \cos (e+f x))^n \sin (e+f x)}{f (a+a \sec (e+f x))}-\frac{\left (d (1+n) (d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{-1-n} (a-a \sec (e+f x)) \, dx}{a^2}\\ &=\frac{(d \cos (e+f x))^n \sin (e+f x)}{f (a+a \sec (e+f x))}+\frac{\left ((1+n) (d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{-n} \, dx}{a}-\frac{\left (d (1+n) (d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{-1-n} \, dx}{a}\\ &=\frac{(d \cos (e+f x))^n \sin (e+f x)}{f (a+a \sec (e+f x))}+\frac{\left ((1+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}{a}-\frac{\left (d (1+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}{a}\\ &=\frac{(d \cos (e+f x))^n \sin (e+f x)}{f (a+a \sec (e+f x))}-\frac{\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)}{a f \sqrt{\sin ^2(e+f x)}}+\frac{(1+n) \cos ^2(e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{2+n}{2};\frac{4+n}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{a f (2+n) \sqrt{\sin ^2(e+f x)}}\\ \end{align*}

Mathematica [F]  time = 0.969104, size = 0, normalized size = 0. \[ \int \frac{(d \cos (e+f x))^n}{a+a \sec (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(d*Cos[e + f*x])^n/(a + a*Sec[e + f*x]),x]

[Out]

Integrate[(d*Cos[e + f*x])^n/(a + a*Sec[e + f*x]), x]

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Maple [F]  time = 0.848, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d\cos \left ( fx+e \right ) \right ) ^{n}}{a+a\sec \left ( fx+e \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(f*x+e))^n/(a+a*sec(f*x+e)),x)

[Out]

int((d*cos(f*x+e))^n/(a+a*sec(f*x+e)),x)

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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}}{a \sec \left (f x + e\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(f*x+e))^n/(a+a*sec(f*x+e)),x, algorithm="maxima")

[Out]

integrate((d*cos(f*x + e))^n/(a*sec(f*x + e) + a), x)

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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 \sec \left (f x + e\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(f*x+e))^n/(a+a*sec(f*x+e)),x, algorithm="fricas")

[Out]

integral((d*cos(f*x + e))^n/(a*sec(f*x + e) + a), x)

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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{\left (e + f x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(f*x+e))**n/(a+a*sec(f*x+e)),x)

[Out]

Integral((d*cos(e + f*x))**n/(sec(e + f*x) + 1), x)/a

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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}}{a \sec \left (f x + e\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(f*x+e))^n/(a+a*sec(f*x+e)),x, algorithm="giac")

[Out]

integrate((d*cos(f*x + e))^n/(a*sec(f*x + e) + a), x)