\(\int (-1+\cot ^2(x))^{3/2} \, dx\) [41]

Optimal result

Integrand size = 10, antiderivative size = 61 \[ \int \left (-1+\cot ^2(x)\right )^{3/2} \, dx=\frac {5}{2} \text {arctanh}\left (\frac {\cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )-\frac {1}{2} \cot (x) \sqrt {-1+\cot ^2(x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3742, 427, 537, 223, 212, 385} \[ \int \left (-1+\cot ^2(x)\right )^{3/2} \, dx=\frac {5}{2} \text {arctanh}\left (\frac {\cot (x)}{\sqrt {\cot ^2(x)-1}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {\cot ^2(x)-1}}\right )-\frac {1}{2} \cot (x) \sqrt {\cot ^2(x)-1} \]

[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 212

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

Rule 223

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 385

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]

Rule 427

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 537

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 3742

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])

Rubi steps \begin{align*} \text {integral}& = -\text {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} \text {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} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,\cot (x)\right )-4 \text {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} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )-4 \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\cot (x)}{\sqrt {-1+\cot ^2(x)}}\right ) \\ & = \frac {5}{2} \text {arctanh}\left (\frac {\cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )-2 \sqrt {2} \text {arctanh}\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] (verified)

Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.98 \[ \int \left (-1+\cot ^2(x)\right )^{3/2} \, dx=\frac {1}{2} \left (-1+\cot ^2(x)\right )^{3/2} \sec ^2(2 x) \left (\arctan \left (\frac {\cos (x)}{\sqrt {-\cos (2 x)}}\right ) \sqrt {-\cos (2 x)} \sin ^3(x)+4 \text {arctanh}\left (\frac {\cos (x)}{\sqrt {\cos (2 x)}}\right ) \sqrt {\cos (2 x)} \sin ^3(x)-4 \sqrt {2} \sqrt {\cos (2 x)} \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right ) \sin ^3(x)-\frac {1}{4} \sin (4 x)\right ) \]

[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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {5 \ln \left (\cot \left (x \right )+\sqrt {-1+\cot \left (x \right )^{2}}\right )}{2}-\frac {\cot \left (x \right ) \sqrt {-1+\cot \left (x \right )^{2}}}{2}-2 \,\operatorname {arctanh}\left (\frac {\cot \left (x \right ) \sqrt {2}}{\sqrt {-1+\cot \left (x \right )^{2}}}\right ) \sqrt {2}\)	\(48\)
default	\(\frac {5 \ln \left (\cot \left (x \right )+\sqrt {-1+\cot \left (x \right )^{2}}\right )}{2}-\frac {\cot \left (x \right ) \sqrt {-1+\cot \left (x \right )^{2}}}{2}-2 \,\operatorname {arctanh}\left (\frac {\cot \left (x \right ) \sqrt {2}}{\sqrt {-1+\cot \left (x \right )^{2}}}\right ) \sqrt {2}\)	\(48\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (47) = 94\).

Time = 0.28 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.79 \[ \int \left (-1+\cot ^2(x)\right )^{3/2} \, dx=\frac {4 \, \sqrt {2} \log \left (2 \, \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - 2 \, \cos \left (2 \, x\right ) - 1\right ) \sin \left (2 \, x\right ) - 2 \, \sqrt {2} \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) + 1\right )} + 5 \, \log \left (\frac {\sqrt {2} \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) + 1}{\cos \left (2 \, x\right ) + 1}\right ) \sin \left (2 \, x\right ) - 5 \, \log \left (\frac {\sqrt {2} \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 1}{\cos \left (2 \, x\right ) + 1}\right ) \sin \left (2 \, x\right )}{4 \, \sin \left (2 \, x\right )} \]

[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)

Sympy [F]

\[ \int \left (-1+\cot ^2(x)\right )^{3/2} \, dx=\int \left (\cot ^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

Integral((cot(x)**2 - 1)**(3/2), x)

Maxima [F]

\[ \int \left (-1+\cot ^2(x)\right )^{3/2} \, dx=\int { {\left (\cot \left (x\right )^{2} - 1\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

integrate((cot(x)^2 - 1)^(3/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (47) = 94\).

Time = 0.52 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.93 \[ \int \left (-1+\cot ^2(x)\right )^{3/2} \, dx=\frac {1}{4} \, {\left (4 \, \sqrt {2} \log \left ({\left (\sqrt {2} \cos \left (x\right ) - \sqrt {2 \, \cos \left (x\right )^{2} - 1}\right )}^{2}\right ) - \frac {4 \, \sqrt {2} {\left (3 \, {\left (\sqrt {2} \cos \left (x\right ) - \sqrt {2 \, \cos \left (x\right )^{2} - 1}\right )}^{2} - 1\right )}}{{\left (\sqrt {2} \cos \left (x\right ) - \sqrt {2 \, \cos \left (x\right )^{2} - 1}\right )}^{4} - 6 \, {\left (\sqrt {2} \cos \left (x\right ) - \sqrt {2 \, \cos \left (x\right )^{2} - 1}\right )}^{2} + 1} + 5 \, \log \left (\frac {{\left | 2 \, {\left (\sqrt {2} \cos \left (x\right ) - \sqrt {2 \, \cos \left (x\right )^{2} - 1}\right )}^{2} - 4 \, \sqrt {2} - 6 \right |}}{{\left | 2 \, {\left (\sqrt {2} \cos \left (x\right ) - \sqrt {2 \, \cos \left (x\right )^{2} - 1}\right )}^{2} + 4 \, \sqrt {2} - 6 \right |}}\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \]

[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))

Mupad [F(-1)]

Timed out. \[ \int \left (-1+\cot ^2(x)\right )^{3/2} \, dx=\int {\left ({\mathrm {cot}\left (x\right )}^2-1\right )}^{3/2} \,d x \]

[In]

int((cot(x)^2 - 1)^(3/2),x)

[Out]

int((cot(x)^2 - 1)^(3/2), x)