3.248 \(\int (b \cos (c+d x))^n \sec (c+d x) \, dx\)

Optimal. Leaf size=60 \[ -\frac {\sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {n+2}{2};\cos ^2(c+d x)\right )}{d n \sqrt {\sin ^2(c+d x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {16, 2643} \[ -\frac {\sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {n+2}{2};\cos ^2(c+d x)\right )}{d n \sqrt {\sin ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*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 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 (b \cos (c+d x))^n \sec (c+d x) \, dx &=b \int (b \cos (c+d x))^{-1+n} \, dx\\ &=-\frac {(b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {2+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d n \sqrt {\sin ^2(c+d x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 63, normalized size = 1.05 \[ -\frac {b \sqrt {\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {n+2}{2};\cos ^2(c+d x)\right )}{d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.48, size = 0, normalized size = 0.00 \[ \int \left (b \cos \left (d x +c \right )\right )^{n} \sec \left (d x +c \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^n}{\cos \left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________