\(\int (a (b \cot (c+d x))^p)^n \, dx\) [38]

Optimal result

Integrand size = 14, antiderivative size = 62 \[ \int \left (a (b \cot (c+d x))^p\right )^n \, dx=-\frac {\cot (c+d x) \left (a (b \cot (c+d x))^p\right )^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),-\cot ^2(c+d x)\right )}{d (1+n p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3740, 3557, 371} \[ \int \left (a (b \cot (c+d x))^p\right )^n \, dx=-\frac {\cot (c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (n p+1),\frac {1}{2} (n p+3),-\cot ^2(c+d x)\right ) \left (a (b \cot (c+d x))^p\right )^n}{d (n p+1)} \]

[In]

Int[(a*(b*Cot[c + d*x])^p)^n,x]

[Out]

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

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]

Rule 3740

Int[(u_.)*((b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Tan[e + f*x
])^n)^FracPart[p]/(c*Tan[e + f*x])^(n*FracPart[p])), Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x], x] /; Fre
eQ[{b, c, e, f, n, p}, x] &&  !IntegerQ[p] &&  !IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x]
)^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \left (a (b \cot (c+d x))^p\right )^n \, dx=-\frac {\cot (c+d x) \left (a (b \cot (c+d x))^p\right )^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),-\cot ^2(c+d x)\right )}{d (1+n p)} \]

[In]

Integrate[(a*(b*Cot[c + d*x])^p)^n,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(((b*cot(d*x + c))^p*a)^n, x)

Sympy [F]

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

[In]

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

[Out]

Integral((a*(b*cot(c + d*x))**p)**n, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a*(b*cot(c + d*x))^p)^n,x)

[Out]

int((a*(b*cot(c + d*x))^p)^n, x)