3.1.30 \(\int \sqrt {a \cot ^3(x)} \, dx\) [30]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3739, 3554, 3557, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {\sqrt {a \cot ^3(x)} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \cot ^3(x)} \text {ArcTan}\left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-2 \tan (x) \sqrt {a \cot ^3(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)} \log \left (\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)} \end {gather*}

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 210

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

Rule 217

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 335

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 631

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 642

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 1176

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 1179

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 3554

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 3557

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 3739

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
]])

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)} \text {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 ) \text {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)} \text {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)} \text {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)} \text {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)} \text {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)} \text {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)} \text {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)} \text {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)} \text {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*}

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Mathematica [A]
time = 0.10, size = 122, normalized size = 0.69 \begin {gather*} -\frac {\sqrt {a \cot ^3(x)} \left (2 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (x)}\right )-2 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (x)}\right )+8 \sqrt {\cot (x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )\right )}{4 \cot ^{\frac {3}{2}}(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(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]] + C
ot[x]]))/Cot[x]^(3/2)

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Maple [A]
time = 0.19, size = 162, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {\sqrt {a \left (\cot ^{3}\left (x \right )\right )}\, \left (\left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}{a \cot \left (x \right )-\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}\right )+2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )-8 \sqrt {a \cot \left (x \right )}\right )}{4 \cot \left (x \right ) \sqrt {a \cot \left (x \right )}}\)	\(162\)
default	\(\frac {\sqrt {a \left (\cot ^{3}\left (x \right )\right )}\, \left (\left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}{a \cot \left (x \right )-\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}\right )+2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )-8 \sqrt {a \cot \left (x \right )}\right )}{4 \cot \left (x \right ) \sqrt {a \cot \left (x \right )}}\)	\(162\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*
cot(x)-(a^2)^(1/4)*(a*cot(x))^(1/2)*2^(1/2)+(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 = 0.51, size = 94, normalized size = 0.53 \begin {gather*} -\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 {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \cot ^{3}{\left (x \right )}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {a\,{\mathrm {cot}\left (x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*cot(x)^3)^(1/2), x)

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