3.3.16 \(\int \cos ^3(c+d x) (b \sec (c+d x))^n \, dx\) [216]

Optimal. Leaf size=75 \[ -\frac {b^4 \, _2F_1\left (\frac {1}{2},\frac {4-n}{2};\frac {6-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{-4+n} \sin (c+d x)}{d (4-n) \sqrt {\sin ^2(c+d x)}} \]

[Out]

-b^4*hypergeom([1/2, 2-1/2*n],[3-1/2*n],cos(d*x+c)^2)*(b*sec(d*x+c))^(-4+n)*sin(d*x+c)/d/(4-n)/(sin(d*x+c)^2)^
(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {16, 3857, 2722} \begin {gather*} -\frac {b^4 \sin (c+d x) (b \sec (c+d x))^{n-4} \, _2F_1\left (\frac {1}{2},\frac {4-n}{2};\frac {6-n}{2};\cos ^2(c+d x)\right )}{d (4-n) \sqrt {\sin ^2(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*(b*Sec[c + d*x])^n,x]

[Out]

-((b^4*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-4 + n)*Sin[c + d*x])/(d
*(4 - n)*Sqrt[Sin[c + d*x]^2]))

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rule 3857

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]

Rubi steps

\begin {align*} \int \cos ^3(c+d x) (b \sec (c+d x))^n \, dx &=b^3 \int (b \sec (c+d x))^{-3+n} \, dx\\ &=\left (b^3 \left (\frac {\cos (c+d x)}{b}\right )^n (b \sec (c+d x))^n\right ) \int \left (\frac {\cos (c+d x)}{b}\right )^{3-n} \, dx\\ &=-\frac {\cos ^4(c+d x) \, _2F_1\left (\frac {1}{2},\frac {4-n}{2};\frac {6-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (4-n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 73, normalized size = 0.97 \begin {gather*} \frac {\cos ^3(c+d x) \cot (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-3+n);\frac {1}{2} (-1+n);\sec ^2(c+d x)\right ) (b \sec (c+d x))^n \sqrt {-\tan ^2(c+d x)}}{d (-3+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*(b*Sec[c + d*x])^n,x]

[Out]

(Cos[c + d*x]^3*Cot[c + d*x]*Hypergeometric2F1[1/2, (-3 + n)/2, (-1 + n)/2, Sec[c + d*x]^2]*(b*Sec[c + d*x])^n
*Sqrt[-Tan[c + d*x]^2])/(d*(-3 + n))

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Maple [F]
time = 0.55, size = 0, normalized size = 0.00 \[\int \left (\cos ^{3}\left (d x +c \right )\right ) \left (b \sec \left (d x +c \right )\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(b*sec(d*x+c))^n,x)

[Out]

int(cos(d*x+c)^3*(b*sec(d*x+c))^n,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(b*sec(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c))^n*cos(d*x + c)^3, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(b*sec(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((b*sec(d*x + c))^n*cos(d*x + c)^3, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3*(b*sec(d*x+c))**n,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(b*sec(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c))^n*cos(d*x + c)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\cos \left (c+d\,x\right )}^3\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^3*(b/cos(c + d*x))^n,x)

[Out]

int(cos(c + d*x)^3*(b/cos(c + d*x))^n, x)

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