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

Optimal. Leaf size=62 \[ -\frac{\cot (c+d x) \text{Hypergeometric2F1}\left (1,\frac{1}{2} (n p+1),\frac{1}{2} (n p+3),-\cot ^2(c+d x)\right ) \left (a (b \cot (c+d x))^p\right )^n}{d (n p+1)} \]

[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)))

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Rubi [A]  time = 0.0449681, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3659, 3476, 364} \[ -\frac{\cot (c+d x) \, _2F_1\left (1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\cot ^2(c+d x)\right ) \left (a (b \cot (c+d x))^p\right )^n}{d (n p+1)} \]

Antiderivative was successfully verified.

[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 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 (a (b \cot (c+d x))^p\right )^n \, dx &=\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 ) \operatorname{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 \, _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 = 0.0473146, size = 60, normalized size = 0.97 \[ -\frac{\cot (c+d x) \text{Hypergeometric2F1}\left (1,\frac{1}{2} (n p+1),\frac{1}{2} (n p+3),-\cot ^2(c+d x)\right ) \left (a (b \cot (c+d x))^p\right )^n}{d n p+d} \]

Antiderivative was successfully verified.

[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 +
d*n*p))

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Maple [F]  time = 5.854, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ( b\cot \left ( dx+c \right ) \right ) ^{p} \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*(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 (\left (b \cot \left (d x + c\right )\right )^{p} a\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((b*cot(d*x + c))^p*a)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*(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 (\left (b \cot \left (d x + c\right )\right )^{p} a\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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