3.21 \(\int \frac{1}{(c \cot (a+b x))^{2/3}} \, dx\)

Optimal. Leaf size=225 \[ \frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{2/3}}-\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{2/3}}+\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b c^{2/3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt{3}\right )}{2 b c^{2/3}} \]

[Out]

-(ArcTan[(c*Cot[a + b*x])^(1/3)/c^(1/3)]/(b*c^(2/3))) + ArcTan[Sqrt[3] - (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)]/(
2*b*c^(2/3)) - ArcTan[Sqrt[3] + (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)]/(2*b*c^(2/3)) + (Sqrt[3]*Log[c^(2/3) - Sqr
t[3]*c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b*c^(2/3)) - (Sqrt[3]*Log[c^(2/3) + Sqrt[3]*
c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b*c^(2/3))

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Rubi [A]  time = 0.30955, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {3476, 329, 209, 634, 618, 204, 628, 203} \[ \frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{2/3}}-\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{2/3}}+\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b c^{2/3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt{3}\right )}{2 b c^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Cot[a + b*x])^(-2/3),x]

[Out]

-(ArcTan[(c*Cot[a + b*x])^(1/3)/c^(1/3)]/(b*c^(2/3))) + ArcTan[Sqrt[3] - (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)]/(
2*b*c^(2/3)) - ArcTan[Sqrt[3] + (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)]/(2*b*c^(2/3)) + (Sqrt[3]*Log[c^(2/3) - Sqr
t[3]*c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b*c^(2/3)) - (Sqrt[3]*Log[c^(2/3) + Sqrt[3]*
c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b*c^(2/3))

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 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 209

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u, v}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x] +
Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 +
s^2*x^2), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && PosQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{(c \cot (a+b x))^{2/3}} \, dx &=-\frac{c \operatorname{Subst}\left (\int \frac{1}{x^{2/3} \left (c^2+x^2\right )} \, dx,x,c \cot (a+b x)\right )}{b}\\ &=-\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{c^2+x^6} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt [3]{c}-\frac{\sqrt{3} x}{2}}{c^{2/3}-\sqrt{3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt [3]{c}+\frac{\sqrt{3} x}{2}}{c^{2/3}+\sqrt{3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{c^{2/3}+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b \sqrt [3]{c}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{2/3}}+\frac{\sqrt{3} \operatorname{Subst}\left (\int \frac{-\sqrt{3} \sqrt [3]{c}+2 x}{c^{2/3}-\sqrt{3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c^{2/3}}-\frac{\sqrt{3} \operatorname{Subst}\left (\int \frac{\sqrt{3} \sqrt [3]{c}+2 x}{c^{2/3}+\sqrt{3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{c^{2/3}-\sqrt{3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b \sqrt [3]{c}}-\frac{\operatorname{Subst}\left (\int \frac{1}{c^{2/3}+\sqrt{3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b \sqrt [3]{c}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{2/3}}+\frac{\sqrt{3} \log \left (c^{2/3}-\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{2/3}}-\frac{\sqrt{3} \log \left (c^{2/3}+\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt{3} \sqrt [3]{c}}\right )}{2 \sqrt{3} b c^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt{3} \sqrt [3]{c}}\right )}{2 \sqrt{3} b c^{2/3}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{2/3}}+\frac{\tan ^{-1}\left (\frac{1}{3} \left (3 \sqrt{3}-\frac{6 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )\right )}{2 b c^{2/3}}-\frac{\tan ^{-1}\left (\frac{1}{3} \left (3 \sqrt{3}+\frac{6 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )\right )}{2 b c^{2/3}}+\frac{\sqrt{3} \log \left (c^{2/3}-\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{2/3}}-\frac{\sqrt{3} \log \left (c^{2/3}+\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0260848, size = 38, normalized size = 0.17 \[ -\frac{3 \sqrt [3]{c \cot (a+b x)} \text{Hypergeometric2F1}\left (\frac{1}{6},1,\frac{7}{6},-\cot ^2(a+b x)\right )}{b c} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Cot[a + b*x])^(-2/3),x]

[Out]

(-3*(c*Cot[a + b*x])^(1/3)*Hypergeometric2F1[1/6, 1, 7/6, -Cot[a + b*x]^2])/(b*c)

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Maple [A]  time = 0.059, size = 209, normalized size = 0.9 \begin{align*}{\frac{\sqrt{3}}{4\,bc}\sqrt [6]{{c}^{2}}\ln \left ( \left ( c\cot \left ( bx+a \right ) \right ) ^{{\frac{2}{3}}}-\sqrt{3}\sqrt [6]{{c}^{2}}\sqrt [3]{c\cot \left ( bx+a \right ) }+\sqrt [3]{{c}^{2}} \right ) }-{\frac{1}{2\,bc}\sqrt [6]{{c}^{2}}\arctan \left ( 2\,{\frac{\sqrt [3]{c\cot \left ( bx+a \right ) }}{\sqrt [6]{{c}^{2}}}}-\sqrt{3} \right ) }-{\frac{1}{bc}\sqrt [6]{{c}^{2}}\arctan \left ({\sqrt [3]{c\cot \left ( bx+a \right ) }{\frac{1}{\sqrt [6]{{c}^{2}}}}} \right ) }-{\frac{\sqrt{3}}{4\,bc}\sqrt [6]{{c}^{2}}\ln \left ( \left ( c\cot \left ( bx+a \right ) \right ) ^{{\frac{2}{3}}}+\sqrt{3}\sqrt [6]{{c}^{2}}\sqrt [3]{c\cot \left ( bx+a \right ) }+\sqrt [3]{{c}^{2}} \right ) }-{\frac{1}{2\,bc}\sqrt [6]{{c}^{2}}\arctan \left ( 2\,{\frac{\sqrt [3]{c\cot \left ( bx+a \right ) }}{\sqrt [6]{{c}^{2}}}}+\sqrt{3} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*cot(b*x+a))^(2/3),x)

[Out]

1/4/b/c*3^(1/2)*(c^2)^(1/6)*ln((c*cot(b*x+a))^(2/3)-3^(1/2)*(c^2)^(1/6)*(c*cot(b*x+a))^(1/3)+(c^2)^(1/3))-1/2/
b/c*(c^2)^(1/6)*arctan(2*(c*cot(b*x+a))^(1/3)/(c^2)^(1/6)-3^(1/2))-1/b/c*(c^2)^(1/6)*arctan((c*cot(b*x+a))^(1/
3)/(c^2)^(1/6))-1/4/b/c*3^(1/2)*(c^2)^(1/6)*ln((c*cot(b*x+a))^(2/3)+3^(1/2)*(c^2)^(1/6)*(c*cot(b*x+a))^(1/3)+(
c^2)^(1/3))-1/2/b/c*(c^2)^(1/6)*arctan(2*(c*cot(b*x+a))^(1/3)/(c^2)^(1/6)+3^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*cot(b*x+a))^(2/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*cot(b*x+a))^(2/3),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \cot{\left (a + b x \right )}\right )^{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*cot(b*x+a))**(2/3),x)

[Out]

Integral((c*cot(a + b*x))**(-2/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \cot \left (b x + a\right )\right )^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*cot(b*x+a))^(2/3),x, algorithm="giac")

[Out]

integrate((c*cot(b*x + a))^(-2/3), x)