3.1.38 \(\int (1-\cot ^2(x))^{3/2} \, dx\) [38]

Optimal. Leaf size=54 \[ \frac {5}{2} \text {ArcSin}(\cot (x))-2 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )+\frac {1}{2} \cot (x) \sqrt {1-\cot ^2(x)} \]

[Out]

5/2*arcsin(cot(x))-2*arctan(cot(x)*2^(1/2)/(1-cot(x)^2)^(1/2))*2^(1/2)+1/2*cot(x)*(1-cot(x)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3742, 427, 537, 222, 385, 209} \begin {gather*} \frac {5}{2} \text {ArcSin}(\cot (x))-2 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )+\frac {1}{2} \cot (x) \sqrt {1-\cot ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - Cot[x]^2)^(3/2),x]

[Out]

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

Rule 209

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

Rule 222

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

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*} \int \left (1-\cot ^2(x)\right )^{3/2} \, dx &=-\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 {5}{2} \sin ^{-1}(\cot (x))+\frac {1}{2} \cot (x) \sqrt {1-\cot ^2(x)}-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} \sin ^{-1}(\cot (x))-2 \sqrt {2} \tan ^{-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 [B] Leaf count is larger than twice the leaf count of optimal. \(123\) vs. \(2(54)=108\).
time = 0.44, size = 123, normalized size = 2.28 \begin {gather*} \frac {1}{2} \left (1-\cot ^2(x)\right )^{3/2} \sec ^2(2 x) \left (\text {ArcTan}\left (\frac {\cos (x)}{\sqrt {-\cos (2 x)}}\right ) \sqrt {-\cos (2 x)} \sin ^3(x)+4 \tanh ^{-1}\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 ) \end {gather*}

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]/S
qrt[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]
time = 0.20, size = 51, normalized size = 0.94

method	result	size
derivativedivides	\(\frac {\cot \left (x \right ) \sqrt {1-\left (\cot ^{2}\left (x \right )\right )}}{2}+\frac {5 \arcsin \left (\cot \left (x \right )\right )}{2}+2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {1-\left (\cot ^{2}\left (x \right )\right )}\, \cot \left (x \right )}{-1+\cot ^{2}\left (x \right )}\right )\)	\(51\)
default	\(\frac {\cot \left (x \right ) \sqrt {1-\left (\cot ^{2}\left (x \right )\right )}}{2}+\frac {5 \arcsin \left (\cot \left (x \right )\right )}{2}+2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {1-\left (\cot ^{2}\left (x \right )\right )}\, \cot \left (x \right )}{-1+\cot ^{2}\left (x \right )}\right )\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*cot(x)*(1-cot(x)^2)^(1/2)+5/2*arcsin(cot(x))+2*2^(1/2)*arctan(2^(1/2)*(1-cot(x)^2)^(1/2)/(-1+cot(x)^2)*cot
(x))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (42) = 84\).
time = 2.40, size = 110, normalized size = 2.04 \begin {gather*} \frac {4 \, \sqrt {2} \arctan \left (\frac {\sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1}\right ) \sin \left (2 \, x\right ) + \sqrt {2} \sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) + 1\right )} - 5 \, \arctan \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}\right ) \sin \left (2 \, x\right )}{2 \, \sin \left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (1 - \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((1 - cot(x)**2)**(3/2), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (42) = 84\).
time = 0.48, size = 257, normalized size = 4.76 \begin {gather*} \frac {1}{4} \, {\left (5 \, \pi \mathrm {sgn}\left (\cos \left (x\right )\right ) - 4 \, \sqrt {2} {\left (\pi \mathrm {sgn}\left (\cos \left (x\right )\right ) + 2 \, \arctan \left (-\frac {{\left (\frac {{\left (\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}\right )}^{2}}{\cos \left (x\right )^{2}} - 4\right )} \cos \left (x\right )}{4 \, {\left (\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}\right )}}\right )\right )} + \frac {4 \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}}{\cos \left (x\right )} - \frac {4 \, \cos \left (x\right )}{\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}}\right )}}{{\left (\frac {\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}}{\cos \left (x\right )} - \frac {4 \, \cos \left (x\right )}{\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}}\right )}^{2} + 8} + 10 \, \arctan \left (-\frac {\sqrt {2} {\left (\frac {{\left (\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}\right )}^{2}}{\cos \left (x\right )^{2}} - 4\right )} \cos \left (x\right )}{4 \, {\left (\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}\right )}}\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(5*pi*sgn(cos(x)) - 4*sqrt(2)*(pi*sgn(cos(x)) + 2*arctan(-1/4*((sqrt(2)*sqrt(-2*cos(x)^2 + 1) - sqrt(2))^2
/cos(x)^2 - 4)*cos(x)/(sqrt(2)*sqrt(-2*cos(x)^2 + 1) - sqrt(2)))) + 4*sqrt(2)*((sqrt(2)*sqrt(-2*cos(x)^2 + 1)
- sqrt(2))/cos(x) - 4*cos(x)/(sqrt(2)*sqrt(-2*cos(x)^2 + 1) - sqrt(2)))/(((sqrt(2)*sqrt(-2*cos(x)^2 + 1) - sqr
t(2))/cos(x) - 4*cos(x)/(sqrt(2)*sqrt(-2*cos(x)^2 + 1) - sqrt(2)))^2 + 8) + 10*arctan(-1/4*sqrt(2)*((sqrt(2)*s
qrt(-2*cos(x)^2 + 1) - sqrt(2))^2/cos(x)^2 - 4)*cos(x)/(sqrt(2)*sqrt(-2*cos(x)^2 + 1) - sqrt(2))))*sgn(sin(x))

________________________________________________________________________________________

Mupad [B]
time = 0.84, size = 104, normalized size = 1.93 \begin {gather*} \frac {5\,\mathrm {asin}\left (\mathrm {cot}\left (x\right )\right )}{2}+\frac {\mathrm {cot}\left (x\right )\,\sqrt {1-{\mathrm {cot}\left (x\right )}^2}}{2}-\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (-1+\mathrm {cot}\left (x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\sqrt {1-{\mathrm {cot}\left (x\right )}^2}\,1{}\mathrm {i}}{\mathrm {cot}\left (x\right )-\mathrm {i}}\right )\,1{}\mathrm {i}+\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (1+\mathrm {cot}\left (x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\sqrt {1-{\mathrm {cot}\left (x\right )}^2}\,1{}\mathrm {i}}{\mathrm {cot}\left (x\right )+1{}\mathrm {i}}\right )\,1{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - cot(x)^2)^(3/2),x)

[Out]

(5*asin(cot(x)))/2 + (cot(x)*(1 - cot(x)^2)^(1/2))/2 - 2^(1/2)*log(((2^(1/2)*(cot(x)*1i - 1)*1i)/2 - (1 - cot(
x)^2)^(1/2)*1i)/(cot(x) - 1i))*1i + 2^(1/2)*log(((2^(1/2)*(cot(x)*1i + 1)*1i)/2 + (1 - cot(x)^2)^(1/2)*1i)/(co
t(x) + 1i))*1i

________________________________________________________________________________________