3.41 \(\int (-1+\cot ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=61 \[ -\frac{1}{2} \cot (x) \sqrt{\cot ^2(x)-1}+\frac{5}{2} \tanh ^{-1}\left (\frac{\cot (x)}{\sqrt{\cot ^2(x)-1}}\right )-2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \cot (x)}{\sqrt{\cot ^2(x)-1}}\right ) \]

[Out]

(5*ArcTanh[Cot[x]/Sqrt[-1 + Cot[x]^2]])/2 - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*Cot[x])/Sqrt[-1 + Cot[x]^2]] - (Cot[x]*
Sqrt[-1 + Cot[x]^2])/2

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Rubi [A]  time = 0.0425351, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3661, 416, 523, 217, 206, 377} \[ -\frac{1}{2} \cot (x) \sqrt{\cot ^2(x)-1}+\frac{5}{2} \tanh ^{-1}\left (\frac{\cot (x)}{\sqrt{\cot ^2(x)-1}}\right )-2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \cot (x)}{\sqrt{\cot ^2(x)-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + Cot[x]^2)^(3/2),x]

[Out]

(5*ArcTanh[Cot[x]/Sqrt[-1 + Cot[x]^2]])/2 - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*Cot[x])/Sqrt[-1 + Cot[x]^2]] - (Cot[x]*
Sqrt[-1 + Cot[x]^2])/2

Rule 3661

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[(c*ff)/f, Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

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

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rubi steps

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

Mathematica [A]  time = 0.123553, size = 121, normalized size = 1.98 \[ \frac{1}{2} \left (\cot ^2(x)-1\right )^{3/2} \sec ^2(2 x) \left (-\frac{1}{4} \sin (4 x)-4 \sqrt{2} \sin ^3(x) \sqrt{\cos (2 x)} \log \left (\sqrt{2} \cos (x)+\sqrt{\cos (2 x)}\right )+\sin ^3(x) \sqrt{-\cos (2 x)} \tan ^{-1}\left (\frac{\cos (x)}{\sqrt{-\cos (2 x)}}\right )+4 \sin ^3(x) \sqrt{\cos (2 x)} \tanh ^{-1}\left (\frac{\cos (x)}{\sqrt{\cos (2 x)}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Cot[x]^2)^(3/2),x]

[Out]

((-1 + Cot[x]^2)^(3/2)*Sec[2*x]^2*(ArcTan[Cos[x]/Sqrt[-Cos[2*x]]]*Sqrt[-Cos[2*x]]*Sin[x]^3 + 4*ArcTanh[Cos[x]/
Sqrt[Cos[2*x]]]*Sqrt[Cos[2*x]]*Sin[x]^3 - 4*Sqrt[2]*Sqrt[Cos[2*x]]*Log[Sqrt[2]*Cos[x] + Sqrt[Cos[2*x]]]*Sin[x]
^3 - Sin[4*x]/4))/2

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Maple [A]  time = 0.03, size = 48, normalized size = 0.8 \begin{align*} -{\frac{\cot \left ( x \right ) }{2}\sqrt{-1+ \left ( \cot \left ( x \right ) \right ) ^{2}}}+{\frac{5}{2}\ln \left ( \cot \left ( x \right ) +\sqrt{-1+ \left ( \cot \left ( x \right ) \right ) ^{2}} \right ) }-2\,{\it Artanh} \left ({\frac{\cot \left ( x \right ) \sqrt{2}}{\sqrt{-1+ \left ( \cot \left ( x \right ) \right ) ^{2}}}} \right ) \sqrt{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+cot(x)^2)^(3/2),x)

[Out]

-1/2*cot(x)*(-1+cot(x)^2)^(1/2)+5/2*ln(cot(x)+(-1+cot(x)^2)^(1/2))-2*arctanh(cot(x)*2^(1/2)/(-1+cot(x)^2)^(1/2
))*2^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+cot(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((cot(x)^2 - 1)^(3/2), x)

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Fricas [B]  time = 1.92736, size = 477, normalized size = 7.82 \begin{align*} \frac{4 \, \sqrt{2} \log \left (2 \, \sqrt{-\frac{\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - 2 \, \cos \left (2 \, x\right ) - 1\right ) \sin \left (2 \, x\right ) - 2 \, \sqrt{2} \sqrt{-\frac{\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}}{\left (\cos \left (2 \, x\right ) + 1\right )} + 5 \, \log \left (\frac{\sqrt{2} \sqrt{-\frac{\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) + 1}{\cos \left (2 \, x\right ) + 1}\right ) \sin \left (2 \, x\right ) - 5 \, \log \left (\frac{\sqrt{2} \sqrt{-\frac{\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 1}{\cos \left (2 \, x\right ) + 1}\right ) \sin \left (2 \, x\right )}{4 \, \sin \left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+cot(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/4*(4*sqrt(2)*log(2*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) - 2*cos(2*x) - 1)*sin(2*x) - 2*sqrt(2)*sqrt(-cos(
2*x)/(cos(2*x) - 1))*(cos(2*x) + 1) + 5*log((sqrt(2)*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) + cos(2*x) + 1)/(
cos(2*x) + 1))*sin(2*x) - 5*log((sqrt(2)*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) - cos(2*x) - 1)/(cos(2*x) + 1
))*sin(2*x))/sin(2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+cot(x)**2)**(3/2),x)

[Out]

Integral((cot(x)**2 - 1)**(3/2), x)

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Giac [B]  time = 2.48973, size = 242, normalized size = 3.97 \begin{align*} \frac{1}{4} \,{\left (4 \, \sqrt{2} \log \left ({\left (\sqrt{2} \cos \left (x\right ) - \sqrt{2 \, \cos \left (x\right )^{2} - 1}\right )}^{2}\right ) - \frac{4 \, \sqrt{2}{\left (3 \,{\left (\sqrt{2} \cos \left (x\right ) - \sqrt{2 \, \cos \left (x\right )^{2} - 1}\right )}^{2} - 1\right )}}{{\left (\sqrt{2} \cos \left (x\right ) - \sqrt{2 \, \cos \left (x\right )^{2} - 1}\right )}^{4} - 6 \,{\left (\sqrt{2} \cos \left (x\right ) - \sqrt{2 \, \cos \left (x\right )^{2} - 1}\right )}^{2} + 1} + 5 \, \log \left (\frac{{\left | 2 \,{\left (\sqrt{2} \cos \left (x\right ) - \sqrt{2 \, \cos \left (x\right )^{2} - 1}\right )}^{2} - 4 \, \sqrt{2} - 6 \right |}}{{\left | 2 \,{\left (\sqrt{2} \cos \left (x\right ) - \sqrt{2 \, \cos \left (x\right )^{2} - 1}\right )}^{2} + 4 \, \sqrt{2} - 6 \right |}}\right )\right )} \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+cot(x)^2)^(3/2),x, algorithm="giac")

[Out]

1/4*(4*sqrt(2)*log((sqrt(2)*cos(x) - sqrt(2*cos(x)^2 - 1))^2) - 4*sqrt(2)*(3*(sqrt(2)*cos(x) - sqrt(2*cos(x)^2
- 1))^2 - 1)/((sqrt(2)*cos(x) - sqrt(2*cos(x)^2 - 1))^4 - 6*(sqrt(2)*cos(x) - sqrt(2*cos(x)^2 - 1))^2 + 1) +
5*log(abs(2*(sqrt(2)*cos(x) - sqrt(2*cos(x)^2 - 1))^2 - 4*sqrt(2) - 6)/abs(2*(sqrt(2)*cos(x) - sqrt(2*cos(x)^2
- 1))^2 + 4*sqrt(2) - 6)))*sgn(sin(x))