3.30 \(\int \sqrt{a \cot ^3(x)} \, dx\)

Optimal. Leaf size=176 \[ -\frac{\sqrt{a \cot ^3(x)} \log \left (\cot (x)-\sqrt{2} \sqrt{\cot (x)}+1\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \log \left (\cot (x)+\sqrt{2} \sqrt{\cot (x)}+1\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}-\frac{\sqrt{a \cot ^3(x)} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (x)}\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (x)}+1\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}-2 \tan (x) \sqrt{a \cot ^3(x)} \]

[Out]

-((ArcTan[1 - Sqrt[2]*Sqrt[Cot[x]]]*Sqrt[a*Cot[x]^3])/(Sqrt[2]*Cot[x]^(3/2))) + (ArcTan[1 + Sqrt[2]*Sqrt[Cot[x
]]]*Sqrt[a*Cot[x]^3])/(Sqrt[2]*Cot[x]^(3/2)) - (Sqrt[a*Cot[x]^3]*Log[1 - Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sq
rt[2]*Cot[x]^(3/2)) + (Sqrt[a*Cot[x]^3]*Log[1 + Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*Cot[x]^(3/2)) - 2*S
qrt[a*Cot[x]^3]*Tan[x]

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Rubi [A]  time = 0.0872821, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {3658, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt{a \cot ^3(x)} \log \left (\cot (x)-\sqrt{2} \sqrt{\cot (x)}+1\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \log \left (\cot (x)+\sqrt{2} \sqrt{\cot (x)}+1\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}-\frac{\sqrt{a \cot ^3(x)} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (x)}\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (x)}+1\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}-2 \tan (x) \sqrt{a \cot ^3(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Cot[x]^3],x]

[Out]

-((ArcTan[1 - Sqrt[2]*Sqrt[Cot[x]]]*Sqrt[a*Cot[x]^3])/(Sqrt[2]*Cot[x]^(3/2))) + (ArcTan[1 + Sqrt[2]*Sqrt[Cot[x
]]]*Sqrt[a*Cot[x]^3])/(Sqrt[2]*Cot[x]^(3/2)) - (Sqrt[a*Cot[x]^3]*Log[1 - Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sq
rt[2]*Cot[x]^(3/2)) + (Sqrt[a*Cot[x]^3]*Log[1 + Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*Cot[x]^(3/2)) - 2*S
qrt[a*Cot[x]^3]*Tan[x]

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, 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 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

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 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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])

Rubi steps

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

Mathematica [A]  time = 0.104914, size = 122, normalized size = 0.69 \[ -\frac{\sqrt{a \cot ^3(x)} \left (8 \sqrt{\cot (x)}+\sqrt{2} \log \left (\cot (x)-\sqrt{2} \sqrt{\cot (x)}+1\right )-\sqrt{2} \log \left (\cot (x)+\sqrt{2} \sqrt{\cot (x)}+1\right )+2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (x)}+1\right )\right )}{4 \cot ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Cot[x]^3],x]

[Out]

-(Sqrt[a*Cot[x]^3]*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[x]]] +
8*Sqrt[Cot[x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[x]] + Cot[x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[x]] + Cot[x
]]))/(4*Cot[x]^(3/2))

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Maple [A]  time = 0.079, size = 165, normalized size = 0.9 \begin{align*}{\frac{1}{4\,\cot \left ( x \right ) }\sqrt{a \left ( \cot \left ( x \right ) \right ) ^{3}} \left ( \sqrt [4]{{a}^{2}}\sqrt{2}\ln \left ( -{ \left ( a\cot \left ( x \right ) +\sqrt [4]{{a}^{2}}\sqrt{a\cot \left ( x \right ) }\sqrt{2}+\sqrt{{a}^{2}} \right ) \left ( \sqrt [4]{{a}^{2}}\sqrt{a\cot \left ( x \right ) }\sqrt{2}-a\cot \left ( x \right ) -\sqrt{{a}^{2}} \right ) ^{-1}} \right ) +2\,\sqrt [4]{{a}^{2}}\sqrt{2}\arctan \left ({\frac{\sqrt{2}\sqrt{a\cot \left ( x \right ) }+\sqrt [4]{{a}^{2}}}{\sqrt [4]{{a}^{2}}}} \right ) +2\,\sqrt [4]{{a}^{2}}\sqrt{2}\arctan \left ({\frac{\sqrt{2}\sqrt{a\cot \left ( x \right ) }-\sqrt [4]{{a}^{2}}}{\sqrt [4]{{a}^{2}}}} \right ) -8\,\sqrt{a\cot \left ( x \right ) } \right ){\frac{1}{\sqrt{a\cot \left ( x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(a*cot(x)^3)^(1/2)*((a^2)^(1/4)*2^(1/2)*ln(-(a*cot(x)+(a^2)^(1/4)*(a*cot(x))^(1/2)*2^(1/2)+(a^2)^(1/2))/((
a^2)^(1/4)*(a*cot(x))^(1/2)*2^(1/2)-a*cot(x)-(a^2)^(1/2)))+2*(a^2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(a*cot(x))^(1
/2)+(a^2)^(1/4))/(a^2)^(1/4))+2*(a^2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(a*cot(x))^(1/2)-(a^2)^(1/4))/(a^2)^(1/4))
-8*(a*cot(x))^(1/2))/cot(x)/(a*cot(x))^(1/2)

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Maxima [A]  time = 1.62105, size = 127, normalized size = 0.72 \begin{align*} -\frac{1}{4} \,{\left (2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (x\right )}\right )}\right ) + 2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (x\right )}\right )}\right ) - \sqrt{2} \log \left (\sqrt{2} \sqrt{\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) + \sqrt{2} \log \left (-\sqrt{2} \sqrt{\tan \left (x\right )} + \tan \left (x\right ) + 1\right )\right )} \sqrt{a} - \frac{2 \, \sqrt{a}}{\sqrt{\tan \left (x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(x)))) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sq
rt(tan(x)))) - sqrt(2)*log(sqrt(2)*sqrt(tan(x)) + tan(x) + 1) + sqrt(2)*log(-sqrt(2)*sqrt(tan(x)) + tan(x) + 1
))*sqrt(a) - 2*sqrt(a)/sqrt(tan(x))

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Fricas [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((a*cot(x)^3)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*cot(x)**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*cot(x)^3), x)