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

Optimal. Leaf size=55 \[ -\frac{\left (a+b \sin ^n(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sin ^n(c+d x)}{a}+1\right )}{a d n (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0765276, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3230, 266, 65} \[ -\frac{\left (a+b \sin ^n(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sin ^n(c+d x)}{a}+1\right )}{a d n (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]*(a + b*Sin[c + d*x]^n)^p,x]

[Out]

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

Rule 3230

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + b*(c*ff*x)^n)^p)/(1 - ff^2*x^2)^(
(m + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

\begin{align*} \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^n\right )^p}{x} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^p}{x} \, dx,x,\sin ^n(c+d x)\right )}{d n}\\ &=-\frac{\, _2F_1\left (1,1+p;2+p;1+\frac{b \sin ^n(c+d x)}{a}\right ) \left (a+b \sin ^n(c+d x)\right )^{1+p}}{a d n (1+p)}\\ \end{align*}

Mathematica [A]  time = 0.0575393, size = 55, normalized size = 1. \[ -\frac{\left (a+b \sin ^n(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sin ^n(c+d x)}{a}+1\right )}{a d n (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]*(a + b*Sin[c + d*x]^n)^p,x]

[Out]

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

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Maple [F]  time = 0.624, size = 0, normalized size = 0. \begin{align*} \int \cot \left ( dx+c \right ) \left ( a+b \left ( \sin \left ( dx+c \right ) \right ) ^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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