\(\int (b \cos (c+d x))^n \sec (c+d x) \, dx\) [248]

Optimal result

Integrand size = 17, antiderivative size = 60 \[ \int (b \cos (c+d x))^n \sec (c+d x) \, dx=-\frac {(b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d n \sqrt {\sin ^2(c+d x)}} \]

[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] (verified)

Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {16, 2722} \[ \int (b \cos (c+d x))^n \sec (c+d x) \, dx=-\frac {\sin (c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {n+2}{2},\cos ^2(c+d x)\right )}{d n \sqrt {\sin ^2(c+d x)}} \]

[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 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]

Rubi steps \begin{align*} \text {integral}& = b \int (b \cos (c+d x))^{-1+n} \, dx \\ & = -\frac {(b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\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] (verified)

Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.05 \[ \int (b \cos (c+d x))^n \sec (c+d x) \, dx=-\frac {b (b \cos (c+d x))^{-1+n} \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{d n} \]

[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))

Maple [F]

\[\int \left (\cos \left (d x +c \right ) b \right )^{n} \sec \left (d x +c \right )d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (b \cos (c+d x))^n \sec (c+d x) \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ) \,d x } \]

[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)

Sympy [F]

\[ \int (b \cos (c+d x))^n \sec (c+d x) \, dx=\int \left (b \cos {\left (c + d x \right )}\right )^{n} \sec {\left (c + d x \right )}\, dx \]

[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)

Maxima [F]

\[ \int (b \cos (c+d x))^n \sec (c+d x) \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ) \,d x } \]

[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)

Giac [F]

\[ \int (b \cos (c+d x))^n \sec (c+d x) \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ) \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (b \cos (c+d x))^n \sec (c+d x) \, dx=\int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^n}{\cos \left (c+d\,x\right )} \,d x \]

[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)