3.29 \(\int (a \cot ^3(x))^{3/2} \, dx\)

Optimal. Leaf size=200 \[ -\frac{2}{7} a \cot ^2(x) \sqrt{a \cot ^3(x)}+\frac{2}{3} a \sqrt{a \cot ^3(x)}-\frac{a \sqrt{a \cot ^3(x)} \log \left (\cot (x)-\sqrt{2} \sqrt{\cot (x)}+1\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{a \sqrt{a \cot ^3(x)} \log \left (\cot (x)+\sqrt{2} \sqrt{\cot (x)}+1\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{a \sqrt{a \cot ^3(x)} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (x)}\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}-\frac{a \sqrt{a \cot ^3(x)} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (x)}+1\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)} \]

[Out]

(2*a*Sqrt[a*Cot[x]^3])/3 + (a*ArcTan[1 - Sqrt[2]*Sqrt[Cot[x]]]*Sqrt[a*Cot[x]^3])/(Sqrt[2]*Cot[x]^(3/2)) - (a*A
rcTan[1 + Sqrt[2]*Sqrt[Cot[x]]]*Sqrt[a*Cot[x]^3])/(Sqrt[2]*Cot[x]^(3/2)) - (2*a*Cot[x]^2*Sqrt[a*Cot[x]^3])/7 -
(a*Sqrt[a*Cot[x]^3]*Log[1 - Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*Cot[x]^(3/2)) + (a*Sqrt[a*Cot[x]^3]*Lo
g[1 + Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*Cot[x]^(3/2))

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Rubi [A]  time = 0.097225, antiderivative size = 200, 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, 3473, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{2}{7} a \cot ^2(x) \sqrt{a \cot ^3(x)}+\frac{2}{3} a \sqrt{a \cot ^3(x)}-\frac{a \sqrt{a \cot ^3(x)} \log \left (\cot (x)-\sqrt{2} \sqrt{\cot (x)}+1\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{a \sqrt{a \cot ^3(x)} \log \left (\cot (x)+\sqrt{2} \sqrt{\cot (x)}+1\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{a \sqrt{a \cot ^3(x)} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (x)}\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}-\frac{a \sqrt{a \cot ^3(x)} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (x)}+1\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*Sqrt[a*Cot[x]^3])/3 + (a*ArcTan[1 - Sqrt[2]*Sqrt[Cot[x]]]*Sqrt[a*Cot[x]^3])/(Sqrt[2]*Cot[x]^(3/2)) - (a*A
rcTan[1 + Sqrt[2]*Sqrt[Cot[x]]]*Sqrt[a*Cot[x]^3])/(Sqrt[2]*Cot[x]^(3/2)) - (2*a*Cot[x]^2*Sqrt[a*Cot[x]^3])/7 -
(a*Sqrt[a*Cot[x]^3]*Log[1 - Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*Cot[x]^(3/2)) + (a*Sqrt[a*Cot[x]^3]*Lo
g[1 + Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*Cot[x]^(3/2))

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 297

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

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])

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]

Rubi steps

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

Mathematica [C]  time = 0.0547907, size = 39, normalized size = 0.2 \[ -\frac{2}{21} a \sqrt{a \cot ^3(x)} \left (7 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(x)\right )+3 \cot ^2(x)-7\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*Sqrt[a*Cot[x]^3]*(-7 + 3*Cot[x]^2 + 7*Hypergeometric2F1[3/4, 1, 7/4, -Cot[x]^2]))/21

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Maple [A]  time = 0.062, size = 189, normalized size = 0.9 \begin{align*} -{\frac{1}{84\, \left ( \cot \left ( x \right ) \right ) ^{3}{a}^{2}} \left ( a \left ( \cot \left ( x \right ) \right ) ^{3} \right ) ^{{\frac{3}{2}}} \left ( 24\, \left ( a\cot \left ( x \right ) \right ) ^{7/2}\sqrt [4]{{a}^{2}}+21\,{a}^{4}\sqrt{2}\ln \left ( -{\frac{\sqrt [4]{{a}^{2}}\sqrt{a\cot \left ( x \right ) }\sqrt{2}-a\cot \left ( x \right ) -\sqrt{{a}^{2}}}{a\cot \left ( x \right ) +\sqrt [4]{{a}^{2}}\sqrt{a\cot \left ( x \right ) }\sqrt{2}+\sqrt{{a}^{2}}}} \right ) +42\,{a}^{4}\sqrt{2}\arctan \left ({\frac{\sqrt{2}\sqrt{a\cot \left ( x \right ) }+\sqrt [4]{{a}^{2}}}{\sqrt [4]{{a}^{2}}}} \right ) +42\,{a}^{4}\sqrt{2}\arctan \left ({\frac{\sqrt{2}\sqrt{a\cot \left ( x \right ) }-\sqrt [4]{{a}^{2}}}{\sqrt [4]{{a}^{2}}}} \right ) -56\, \left ( a\cot \left ( x \right ) \right ) ^{3/2}{a}^{2}\sqrt [4]{{a}^{2}} \right ) \left ( a\cot \left ( x \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.56199, size = 153, normalized size = 0.76 \begin{align*} \frac{1}{4} \,{\left (2 \, \sqrt{2} \sqrt{a} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (x\right )}\right )}\right ) + 2 \, \sqrt{2} \sqrt{a} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (x\right )}\right )}\right ) + \sqrt{2} \sqrt{a} \log \left (\sqrt{2} \sqrt{\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) - \sqrt{2} \sqrt{a} \log \left (-\sqrt{2} \sqrt{\tan \left (x\right )} + \tan \left (x\right ) + 1\right )\right )} a + \frac{2 \, a^{\frac{3}{2}}}{3 \, \tan \left (x\right )^{\frac{3}{2}}} - \frac{2 \, a^{\frac{3}{2}}}{7 \, \tan \left (x\right )^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*sqrt(2)*sqrt(a)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(x)))) + 2*sqrt(2)*sqrt(a)*arctan(-1/2*sqrt(2)*
(sqrt(2) - 2*sqrt(tan(x)))) + sqrt(2)*sqrt(a)*log(sqrt(2)*sqrt(tan(x)) + tan(x) + 1) - sqrt(2)*sqrt(a)*log(-sq
rt(2)*sqrt(tan(x)) + tan(x) + 1))*a + 2/3*a^(3/2)/tan(x)^(3/2) - 2/7*a^(3/2)/tan(x)^(7/2)

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

[Out]

integrate((a*cot(x)^3)^(3/2), x)