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

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

[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])

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Rubi [A]  time = 0.096023, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., 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 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 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 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 \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 = 0.0142202, size = 30, normalized size = 0.14 \[ \frac{2 \cot (x) \text{Hypergeometric2F1}\left (-\frac{7}{4},1,-\frac{3}{4},-\cot ^2(x)\right )}{7 \left (a \cot ^3(x)\right )^{3/2}} \]

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))

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Maple [A]  time = 0.051, size = 185, normalized size = 0.9 \begin{align*} -{\frac{\cot \left ( x \right ) }{84\,{a}^{4}} \left ( 21\,\sqrt [4]{{a}^{2}}\sqrt{2} \left ( a\cot \left ( x \right ) \right ) ^{7/2}\ln \left ( -{\frac{a\cot \left ( x \right ) +\sqrt [4]{{a}^{2}}\sqrt{a\cot \left ( x \right ) }\sqrt{2}+\sqrt{{a}^{2}}}{\sqrt [4]{{a}^{2}}\sqrt{a\cot \left ( x \right ) }\sqrt{2}-a\cot \left ( x \right ) -\sqrt{{a}^{2}}}} \right ) +42\,\sqrt [4]{{a}^{2}}\sqrt{2} \left ( a\cot \left ( x \right ) \right ) ^{7/2}\arctan \left ({\frac{\sqrt{2}\sqrt{a\cot \left ( x \right ) }+\sqrt [4]{{a}^{2}}}{\sqrt [4]{{a}^{2}}}} \right ) +42\,\sqrt [4]{{a}^{2}}\sqrt{2} \left ( a\cot \left ( x \right ) \right ) ^{7/2}\arctan \left ({\frac{\sqrt{2}\sqrt{a\cot \left ( x \right ) }-\sqrt [4]{{a}^{2}}}{\sqrt [4]{{a}^{2}}}} \right ) +56\, \left ( \cot \left ( x \right ) \right ) ^{2}{a}^{4}-24\,{a}^{4} \right ) \left ( a \left ( \cot \left ( x \right ) \right ) ^{3} \right ) ^{-{\frac{3}{2}}}} \end{align*}

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)

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Maxima [A]  time = 1.57804, size = 147, normalized size = 0.69 \begin{align*} \frac{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 )}{4 \, a^{\frac{3}{2}}} + \frac{2 \,{\left (3 \, \sqrt{a} \tan \left (x\right )^{\frac{7}{2}} - 7 \, \sqrt{a} \tan \left (x\right )^{\frac{3}{2}}\right )}}{21 \, a^{2}} \end{align*}

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

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

[Out]

Timed out

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

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)

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

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)