\(\int (b \cot (c+d x))^n \, dx\) [24]

Optimal result

Integrand size = 10, antiderivative size = 51 \[ \int (b \cot (c+d x))^n \, dx=-\frac {(b \cot (c+d x))^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\cot ^2(c+d x)\right )}{b d (1+n)} \]

[Out]

-(b*cot(d*x+c))^(1+n)*hypergeom([1, 1/2+1/2*n],[3/2+1/2*n],-cot(d*x+c)^2)/b/d/(1+n)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3557, 371} \[ \int (b \cot (c+d x))^n \, dx=-\frac {(b \cot (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(c+d x)\right )}{b d (n+1)} \]

[In]

Int[(b*Cot[c + d*x])^n,x]

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = -\frac {b \text {Subst}\left (\int \frac {x^n}{b^2+x^2} \, dx,x,b \cot (c+d x)\right )}{d} \\ & = -\frac {(b \cot (c+d x))^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\cot ^2(c+d x)\right )}{b d (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.02 \[ \int (b \cot (c+d x))^n \, dx=-\frac {\cot (c+d x) (b \cot (c+d x))^n \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\cot ^2(c+d x)\right )}{d (1+n)} \]

[In]

Integrate[(b*Cot[c + d*x])^n,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*cot(d*x + c))^n, x)

Sympy [F]

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

[In]

integrate((b*cot(d*x+c))**n,x)

[Out]

Integral((b*cot(c + d*x))**n, x)

Maxima [F]

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

[In]

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

[Out]

integrate((b*cot(d*x + c))^n, x)

Giac [F]

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

[In]

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

[Out]

integrate((b*cot(d*x + c))^n, x)

Mupad [F(-1)]

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

[In]

int((b*cot(c + d*x))^n,x)

[Out]

int((b*cot(c + d*x))^n, x)