3.32 \(\int \frac {1}{(a \cot ^3(x))^{3/2}} \, dx\)

Optimal. Leaf size=212 \[ -\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \log \left (\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \log \left (\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2} a \sqrt {a \cot ^3(x)}} \]

[Out]

-2/3/a/(a*cot(x)^3)^(1/2)-1/2*arctan(-1+2^(1/2)*cot(x)^(1/2))*cot(x)^(3/2)/a*2^(1/2)/(a*cot(x)^3)^(1/2)-1/2*ar
ctan(1+2^(1/2)*cot(x)^(1/2))*cot(x)^(3/2)/a*2^(1/2)/(a*cot(x)^3)^(1/2)+1/4*cot(x)^(3/2)*ln(1+cot(x)-2^(1/2)*co
t(x)^(1/2))/a*2^(1/2)/(a*cot(x)^3)^(1/2)-1/4*cot(x)^(3/2)*ln(1+cot(x)+2^(1/2)*cot(x)^(1/2))/a*2^(1/2)/(a*cot(x
)^3)^(1/2)+2/7*tan(x)^2/a/(a*cot(x)^3)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3658, 3474, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \log \left (\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \log \left (\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2} a \sqrt {a \cot ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cot[x]^3)^(-3/2),x]

[Out]

-2/(3*a*Sqrt[a*Cot[x]^3]) + (ArcTan[1 - Sqrt[2]*Sqrt[Cot[x]]]*Cot[x]^(3/2))/(Sqrt[2]*a*Sqrt[a*Cot[x]^3]) - (Ar
cTan[1 + Sqrt[2]*Sqrt[Cot[x]]]*Cot[x]^(3/2))/(Sqrt[2]*a*Sqrt[a*Cot[x]^3]) + (Cot[x]^(3/2)*Log[1 - Sqrt[2]*Sqrt
[Cot[x]] + Cot[x]])/(2*Sqrt[2]*a*Sqrt[a*Cot[x]^3]) - (Cot[x]^(3/2)*Log[1 + Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*
Sqrt[2]*a*Sqrt[a*Cot[x]^3]) + (2*Tan[x]^2)/(7*a*Sqrt[a*Cot[x]^3])

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 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 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 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 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 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 3474

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

Rubi steps

\begin {align*} \int \frac {1}{\left (a \cot ^3(x)\right )^{3/2}} \, dx &=\frac {\cot ^{\frac {3}{2}}(x) \int \frac {1}{\cot ^{\frac {9}{2}}(x)} \, dx}{a \sqrt {a \cot ^3(x)}}\\ &=\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \int \frac {1}{\cot ^{\frac {5}{2}}(x)} \, dx}{a \sqrt {a \cot ^3(x)}}\\ &=-\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \int \frac {1}{\sqrt {\cot (x)}} \, dx}{a \sqrt {a \cot ^3(x)}}\\ &=-\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\cot (x)\right )}{a \sqrt {a \cot ^3(x)}}\\ &=-\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}-\frac {\left (2 \cot ^{\frac {3}{2}}(x)\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{a \sqrt {a \cot ^3(x)}}\\ &=-\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{a \sqrt {a \cot ^3(x)}}\\ &=-\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(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} a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(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} a \sqrt {a \cot ^3(x)}}\\ &=-\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} a \sqrt {a \cot ^3(x)}}\\ &=-\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \cot ^{\frac {3}{2}}(x)}{\sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \cot ^{\frac {3}{2}}(x)}{\sqrt {2} a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.02, size = 30, normalized size = 0.14 \[ \frac {2 \cot (x) \, _2F_1\left (-\frac {7}{4},1;-\frac {3}{4};-\cot ^2(x)\right )}{7 \left (a \cot ^3(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cot[x]^3)^(-3/2),x]

[Out]

(2*Cot[x]*Hypergeometric2F1[-7/4, 1, -3/4, -Cot[x]^2])/(7*(a*Cot[x]^3)^(3/2))

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \cot \relax (x)^{3}\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*cot(x)^3)^(-3/2), x)

________________________________________________________________________________________

maple [A]  time = 0.12, size = 185, normalized size = 0.87 \[ -\frac {\cot \relax (x ) \left (21 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (a \cot \relax (x )\right )^{\frac {7}{2}} \ln \left (-\frac {a \cot \relax (x )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \relax (x )}\, \sqrt {2}+\sqrt {a^{2}}}{\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \relax (x )}\, \sqrt {2}-a \cot \relax (x )-\sqrt {a^{2}}}\right )+42 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (a \cot \relax (x )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \relax (x )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+42 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (a \cot \relax (x )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \relax (x )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+56 \left (\cot ^{2}\relax (x )\right ) a^{4}-24 a^{4}\right )}{84 a^{4} \left (a \left (\cot ^{3}\relax (x )\right )\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/84*cot(x)/a^4*(21*(a^2)^(1/4)*2^(1/2)*(a*cot(x))^(7/2)*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)))+42*(a^2)^(1/4)*2^(1/2)*(a*cot(x))^(7/
2)*arctan((2^(1/2)*(a*cot(x))^(1/2)+(a^2)^(1/4))/(a^2)^(1/4))+42*(a^2)^(1/4)*2^(1/2)*(a*cot(x))^(7/2)*arctan((
2^(1/2)*(a*cot(x))^(1/2)-(a^2)^(1/4))/(a^2)^(1/4))+56*cot(x)^2*a^4-24*a^4)/(a*cot(x)^3)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 109, normalized size = 0.51 \[ \frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \relax (x)}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \relax (x)}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \relax (x)} + \tan \relax (x) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \relax (x)} + \tan \relax (x) + 1\right )}{4 \, a^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, \sqrt {a} \tan \relax (x)^{\frac {7}{2}} - 7 \, \sqrt {a} \tan \relax (x)^{\frac {3}{2}}\right )}}{21 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)^3)^(3/2),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*sqr
t(tan(x)))) - sqrt(2)*log(sqrt(2)*sqrt(tan(x)) + tan(x) + 1) + sqrt(2)*log(-sqrt(2)*sqrt(tan(x)) + tan(x) + 1)
)/a^(3/2) + 2/21*(3*sqrt(a)*tan(x)^(7/2) - 7*sqrt(a)*tan(x)^(3/2))/a^2

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a\,{\mathrm {cot}\relax (x)}^3\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a*cot(x)^3)^(3/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \cot ^{3}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*cot(x)**3)**(-3/2), x)

________________________________________________________________________________________