Integrand size = 12, antiderivative size = 54 \[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\frac {5}{2} \arcsin (\cot (x))-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )+\frac {1}{2} \cot (x) \sqrt {1-\cot ^2(x)} \] Output:
5/2*arcsin(cot(x))-2*2^(1/2)*arctan(2^(1/2)*cot(x)/(1-cot(x)^2)^(1/2))+1/2 *cot(x)*(1-cot(x)^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(123\) vs. \(2(54)=108\).
Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.28 \[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\frac {1}{2} \left (1-\cot ^2(x)\right )^{3/2} \sec ^2(2 x) \left (\arctan \left (\frac {\cos (x)}{\sqrt {-\cos (2 x)}}\right ) \sqrt {-\cos (2 x)} \sin ^3(x)+4 \text {arctanh}\left (\frac {\cos (x)}{\sqrt {\cos (2 x)}}\right ) \sqrt {\cos (2 x)} \sin ^3(x)-4 \sqrt {2} \sqrt {\cos (2 x)} \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right ) \sin ^3(x)-\frac {1}{4} \sin (4 x)\right ) \] Input:
Integrate[(1 - Cot[x]^2)^(3/2),x]
Output:
((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
Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {3042, 4144, 318, 398, 223, 291, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (1-\cot ^2(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (1-\tan \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4144 |
\(\displaystyle -\int \frac {\left (1-\cot ^2(x)\right )^{3/2}}{\cot ^2(x)+1}d\cot (x)\) |
\(\Big \downarrow \) 318 |
\(\displaystyle \frac {1}{2} \cot (x) \sqrt {1-\cot ^2(x)}-\frac {1}{2} \int \frac {3-5 \cot ^2(x)}{\sqrt {1-\cot ^2(x)} \left (\cot ^2(x)+1\right )}d\cot (x)\) |
\(\Big \downarrow \) 398 |
\(\displaystyle \frac {1}{2} \left (5 \int \frac {1}{\sqrt {1-\cot ^2(x)}}d\cot (x)-8 \int \frac {1}{\sqrt {1-\cot ^2(x)} \left (\cot ^2(x)+1\right )}d\cot (x)\right )+\frac {1}{2} \sqrt {1-\cot ^2(x)} \cot (x)\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {1}{2} \left (5 \arcsin (\cot (x))-8 \int \frac {1}{\sqrt {1-\cot ^2(x)} \left (\cot ^2(x)+1\right )}d\cot (x)\right )+\frac {1}{2} \sqrt {1-\cot ^2(x)} \cot (x)\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {1}{2} \left (5 \arcsin (\cot (x))-8 \int \frac {1}{\frac {2 \cot ^2(x)}{1-\cot ^2(x)}+1}d\frac {\cot (x)}{\sqrt {1-\cot ^2(x)}}\right )+\frac {1}{2} \sqrt {1-\cot ^2(x)} \cot (x)\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{2} \left (5 \arcsin (\cot (x))-4 \sqrt {2} \arctan \left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )\right )+\frac {1}{2} \cot (x) \sqrt {1-\cot ^2(x)}\) |
Input:
Int[(1 - Cot[x]^2)^(3/2),x]
Output:
(5*ArcSin[Cot[x]] - 4*Sqrt[2]*ArcTan[(Sqrt[2]*Cot[x])/Sqrt[1 - Cot[x]^2]]) /2 + (Cot[x]*Sqrt[1 - Cot[x]^2])/2
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[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])
Time = 0.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\cot \left (x \right ) \sqrt {1-\cot \left (x \right )^{2}}}{2}+\frac {5 \arcsin \left (\cot \left (x \right )\right )}{2}+2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {1-\cot \left (x \right )^{2}}\, \cot \left (x \right )}{-1+\cot \left (x \right )^{2}}\right )\) | \(51\) |
default | \(\frac {\cot \left (x \right ) \sqrt {1-\cot \left (x \right )^{2}}}{2}+\frac {5 \arcsin \left (\cot \left (x \right )\right )}{2}+2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {1-\cot \left (x \right )^{2}}\, \cot \left (x \right )}{-1+\cot \left (x \right )^{2}}\right )\) | \(51\) |
Input:
int((1-cot(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
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))
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (42) = 84\).
Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.04 \[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\frac {4 \, \sqrt {2} \arctan \left (\frac {\sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1}\right ) \sin \left (2 \, x\right ) + \sqrt {2} \sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) + 1\right )} - 5 \, \arctan \left (\frac {\sqrt {2} \sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1}\right ) \sin \left (2 \, x\right )}{2 \, \sin \left (2 \, x\right )} \] Input:
integrate((1-cot(x)^2)^(3/2),x, algorithm="fricas")
Output:
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*arc tan(sqrt(2)*sqrt(cos(2*x)/(cos(2*x) - 1))*sin(2*x)/(cos(2*x) + 1))*sin(2*x ))/sin(2*x)
\[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\int \left (1 - \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((1-cot(x)**2)**(3/2),x)
Output:
Integral((1 - cot(x)**2)**(3/2), x)
\[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\int { {\left (-\cot \left (x\right )^{2} + 1\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((1-cot(x)^2)^(3/2),x, algorithm="maxima")
Output:
integrate((-cot(x)^2 + 1)^(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (42) = 84\).
Time = 0.22 (sec) , antiderivative size = 257, normalized size of antiderivative = 4.76 \[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\frac {1}{4} \, {\left (5 \, \pi \mathrm {sgn}\left (\cos \left (x\right )\right ) - 4 \, \sqrt {2} {\left (\pi \mathrm {sgn}\left (\cos \left (x\right )\right ) + 2 \, \arctan \left (-\frac {{\left (\frac {{\left (\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}\right )}^{2}}{\cos \left (x\right )^{2}} - 4\right )} \cos \left (x\right )}{4 \, {\left (\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}\right )}}\right )\right )} + \frac {4 \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}}{\cos \left (x\right )} - \frac {4 \, \cos \left (x\right )}{\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}}\right )}}{{\left (\frac {\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}}{\cos \left (x\right )} - \frac {4 \, \cos \left (x\right )}{\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}}\right )}^{2} + 8} + 10 \, \arctan \left (-\frac {\sqrt {2} {\left (\frac {{\left (\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}\right )}^{2}}{\cos \left (x\right )^{2}} - 4\right )} \cos \left (x\right )}{4 \, {\left (\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}\right )}}\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \] Input:
integrate((1-cot(x)^2)^(3/2),x, algorithm="giac")
Output:
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)))/(((s qrt(2)*sqrt(-2*cos(x)^2 + 1) - sqrt(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)*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))))*sgn(sin(x))
Time = 9.56 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.93 \[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\frac {5\,\mathrm {asin}\left (\mathrm {cot}\left (x\right )\right )}{2}+\frac {\mathrm {cot}\left (x\right )\,\sqrt {1-{\mathrm {cot}\left (x\right )}^2}}{2}-\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (-1+\mathrm {cot}\left (x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\sqrt {1-{\mathrm {cot}\left (x\right )}^2}\,1{}\mathrm {i}}{\mathrm {cot}\left (x\right )-\mathrm {i}}\right )\,1{}\mathrm {i}+\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (1+\mathrm {cot}\left (x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\sqrt {1-{\mathrm {cot}\left (x\right )}^2}\,1{}\mathrm {i}}{\mathrm {cot}\left (x\right )+1{}\mathrm {i}}\right )\,1{}\mathrm {i} \] Input:
int((1 - cot(x)^2)^(3/2),x)
Output:
(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)/(cot( x) + 1i))*1i
\[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\int \sqrt {-\cot \left (x \right )^{2}+1}d x -\left (\int \sqrt {-\cot \left (x \right )^{2}+1}\, \cot \left (x \right )^{2}d x \right ) \] Input:
int((1-cot(x)^2)^(3/2),x)
Output:
int(sqrt( - cot(x)**2 + 1),x) - int(sqrt( - cot(x)**2 + 1)*cot(x)**2,x)