3.441 \(\int (d \cos (e+f x))^n (a+a \sec (e+f x))^3 \, dx\)

Optimal. Leaf size=244 \[ -\frac{a^3 (7-4 n) \sin (e+f x) (d \cos (e+f x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n}{2},\frac{n+2}{2},\cos ^2(e+f x)\right )}{f (2-n) n \sqrt{\sin ^2(e+f x)}}-\frac{a^3 (1-4 n) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+1}{2},\frac{n+3}{2},\cos ^2(e+f x)\right )}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (5-2 n) \tan (e+f x) (d \cos (e+f x))^n}{f (1-n) (2-n)}+\frac{\tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right ) (d \cos (e+f x))^n}{f (2-n)} \]

[Out]

-((a^3*(7 - 4*n)*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(2
- n)*n*Sqrt[Sin[e + f*x]^2])) - (a^3*(1 - 4*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)
/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n)*(1 + n)*Sqrt[Sin[e + f*x]^2]) + (a^3*(5 - 2*n)*(d*Cos[
e + f*x])^n*Tan[e + f*x])/(f*(1 - n)*(2 - n)) + ((d*Cos[e + f*x])^n*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(f*
(2 - n))

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Rubi [A]  time = 0.399734, antiderivative size = 244, 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, 3814, 3997, 3787, 3772, 2643} \[ -\frac{a^3 (7-4 n) \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right )}{f (2-n) n \sqrt{\sin ^2(e+f x)}}-\frac{a^3 (1-4 n) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right )}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (5-2 n) \tan (e+f x) (d \cos (e+f x))^n}{f (1-n) (2-n)}+\frac{\tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right ) (d \cos (e+f x))^n}{f (2-n)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cos[e + f*x])^n*(a + a*Sec[e + f*x])^3,x]

[Out]

-((a^3*(7 - 4*n)*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(2
- n)*n*Sqrt[Sin[e + f*x]^2])) - (a^3*(1 - 4*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)
/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n)*(1 + n)*Sqrt[Sin[e + f*x]^2]) + (a^3*(5 - 2*n)*(d*Cos[
e + f*x])^n*Tan[e + f*x])/(f*(1 - n)*(2 - n)) + ((d*Cos[e + f*x])^n*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(f*
(2 - n))

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 3814

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(b^2*
Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n)/(f*(m + n - 1)), x] + Dist[b/(m + n - 1), Int[(a
+ b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n*(b*(m + 2*n - 1) + a*(3*m + 2*n - 4)*Csc[e + f*x]), x], x] /; Fr
eeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + n - 1, 0] && IntegerQ[2*m]

Rule 3997

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.
) + (A_)), x_Symbol] :> -Simp[(b*B*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(n + 1)), x] + Dist[1/(n + 1), Int[(d*C
sc[e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f
, A, B}, x] && NeQ[A*b - a*B, 0] &&  !LeQ[n, -1]

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 (d \cos (e+f x))^n (a+a \sec (e+f x))^3 \, dx &=\left ((d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{-n} (a+a \sec (e+f x))^3 \, dx\\ &=\frac{(d \cos (e+f x))^n \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{f (2-n)}+\frac{\left (a (d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{-n} (a+a \sec (e+f x)) (a (2-2 n)+a (5-2 n) \sec (e+f x)) \, dx}{2-n}\\ &=\frac{a^3 (5-2 n) (d \cos (e+f x))^n \tan (e+f x)}{f (1-n) (2-n)}+\frac{(d \cos (e+f x))^n \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{f (2-n)}+\frac{\left (a (d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{-n} \left (a^2 (1-4 n) (2-n)+a^2 (7-4 n) (1-n) \sec (e+f x)\right ) \, dx}{(1-n) (2-n)}\\ &=\frac{a^3 (5-2 n) (d \cos (e+f x))^n \tan (e+f x)}{f (1-n) (2-n)}+\frac{(d \cos (e+f x))^n \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{f (2-n)}+\frac{\left (a^3 (1-4 n) (d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{-n} \, dx}{1-n}+\frac{\left (a^3 (7-4 n) (d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{1-n} \, dx}{d (2-n)}\\ &=\frac{a^3 (5-2 n) (d \cos (e+f x))^n \tan (e+f x)}{f (1-n) (2-n)}+\frac{(d \cos (e+f x))^n \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{f (2-n)}+\frac{\left (a^3 (1-4 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}{1-n}+\frac{\left (a^3 (7-4 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}{d (2-n)}\\ &=-\frac{a^3 (7-4 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)}{f (2-n) n \sqrt{\sin ^2(e+f x)}}-\frac{a^3 (1-4 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)}{f (1-n) (1+n) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (5-2 n) (d \cos (e+f x))^n \tan (e+f x)}{f (1-n) (2-n)}+\frac{(d \cos (e+f x))^n \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{f (2-n)}\\ \end{align*}

Mathematica [F]  time = 2.51072, size = 0, normalized size = 0. \[ \int (d \cos (e+f x))^n (a+a \sec (e+f x))^3 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(d*Cos[e + f*x])^n*(a + a*Sec[e + f*x])^3,x]

[Out]

Integrate[(d*Cos[e + f*x])^n*(a + a*Sec[e + f*x])^3, x]

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Maple [F]  time = 2.663, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( fx+e \right ) \right ) ^{n} \left ( a+a\sec \left ( fx+e \right ) \right ) ^{3}\, 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))^3,x)

[Out]

int((d*cos(f*x+e))^n*(a+a*sec(f*x+e))^3,x)

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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 )}^{3} \left (d \cos \left (f x + e\right )\right )^{n}\,{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))^3,x, algorithm="maxima")

[Out]

integrate((a*sec(f*x + e) + a)^3*(d*cos(f*x + e))^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{3} \sec \left (f x + e\right )^{3} + 3 \, a^{3} \sec \left (f x + e\right )^{2} + 3 \, a^{3} \sec \left (f x + e\right ) + a^{3}\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.

[In]

integrate((d*cos(f*x+e))^n*(a+a*sec(f*x+e))^3,x, algorithm="fricas")

[Out]

integral((a^3*sec(f*x + e)^3 + 3*a^3*sec(f*x + e)^2 + 3*a^3*sec(f*x + e) + a^3)*(d*cos(f*x + e))^n, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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))**3,x)

[Out]

Timed out

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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 )}^{3} \left (d \cos \left (f x + e\right )\right )^{n}\,{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))^3,x, algorithm="giac")

[Out]

integrate((a*sec(f*x + e) + a)^3*(d*cos(f*x + e))^n, x)