3.37 \(\int (b \cot ^p(c+d x))^n \, dx\)

Optimal. Leaf size=60 \[ -\frac{\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \text{Hypergeometric2F1}\left (1,\frac{1}{2} (n p+1),\frac{1}{2} (n p+3),-\cot ^2(c+d x)\right )}{d (n p+1)} \]

[Out]

-((Cot[c + d*x]*(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)))

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Rubi [A]  time = 0.0397252, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3659, 3476, 364} \[ -\frac{\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \, _2F_1\left (1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\cot ^2(c+d x)\right )}{d (n p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(b*Cot[c + d*x]^p)^n,x]

[Out]

-((Cot[c + d*x]*(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 3659

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]])

Rule 3476

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 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
 (ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \left (b \cot ^p(c+d x)\right )^n \, dx &=\left (\cot ^{-n p}(c+d x) \left (b \cot ^p(c+d x)\right )^n\right ) \int \cot ^{n p}(c+d x) \, dx\\ &=-\frac{\left (\cot ^{-n p}(c+d x) \left (b \cot ^p(c+d x)\right )^n\right ) \operatorname{Subst}\left (\int \frac{x^{n p}}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \, _2F_1\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]  time = 0.0516629, size = 58, normalized size = 0.97 \[ -\frac{\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \text{Hypergeometric2F1}\left (1,\frac{1}{2} (n p+1),\frac{1}{2} (n p+3),-\cot ^2(c+d x)\right )}{d n p+d} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Cot[c + d*x]^p)^n,x]

[Out]

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

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Maple [F]  time = 4.602, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ( \cot \left ( dx+c \right ) \right ) ^{p} \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cot \left (d x + c\right )^{p}\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cot(d*x + c)^p)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \cot \left (d x + c\right )^{p}\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*cot(d*x + c)^p)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cot ^{p}{\left (c + d x \right )}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b*cot(c + d*x)**p)**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cot \left (d x + c\right )^{p}\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cot(d*x + c)^p)^n, x)