Integrand size = 12, antiderivative size = 35 \[ \int \left (-1-\cot ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \arctan \left (\frac {\cot (x)}{\sqrt {-\csc ^2(x)}}\right )+\frac {1}{2} \cot (x) \sqrt {-\csc ^2(x)} \] Output:
-1/2*arctan(cot(x)/(-csc(x)^2)^(1/2))+1/2*cot(x)*(-csc(x)^2)^(1/2)
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37 \[ \int \left (-1-\cot ^2(x)\right )^{3/2} \, dx=-\frac {\csc \left (\frac {x}{2}\right ) \left (\cot (x) \csc (x)+\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sec \left (\frac {x}{2}\right )}{4 \sqrt {-\csc ^2(x)}} \] Input:
Integrate[(-1 - Cot[x]^2)^(3/2),x]
Output:
-1/4*(Csc[x/2]*(Cot[x]*Csc[x] + Log[Cos[x/2]] - Log[Sin[x/2]])*Sec[x/2])/S qrt[-Csc[x]^2]
Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {3042, 4140, 3042, 4610, 211, 224, 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.
\(\displaystyle \int \left (-\cot ^2(x)-1\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (-\tan \left (x+\frac {\pi }{2}\right )^2-1\right )^{3/2}dx\) |
\(\Big \downarrow \) 4140 |
\(\displaystyle \int \left (-\csc ^2(x)\right )^{3/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (-\sec \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4610 |
\(\displaystyle \int \sqrt {-\cot ^2(x)-1}d\cot (x)\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {1}{2} \cot (x) \sqrt {-\cot ^2(x)-1}-\frac {1}{2} \int \frac {1}{\sqrt {-\cot ^2(x)-1}}d\cot (x)\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{2} \cot (x) \sqrt {-\cot ^2(x)-1}-\frac {1}{2} \int \frac {1}{\frac {\cot ^2(x)}{-\cot ^2(x)-1}+1}d\frac {\cot (x)}{\sqrt {-\cot ^2(x)-1}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{2} \cot (x) \sqrt {-\cot ^2(x)-1}-\frac {1}{2} \arctan \left (\frac {\cot (x)}{\sqrt {-\cot ^2(x)-1}}\right )\) |
Input:
Int[(-1 - Cot[x]^2)^(3/2),x]
Output:
-1/2*ArcTan[Cot[x]/Sqrt[-1 - Cot[x]^2]] + (Cot[x]*Sqrt[-1 - Cot[x]^2])/2
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(a*sec[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a, b]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFac tors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p]
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {\cot \left (x \right ) \sqrt {-1-\cot \left (x \right )^{2}}}{2}-\frac {\arctan \left (\frac {\cot \left (x \right )}{\sqrt {-1-\cot \left (x \right )^{2}}}\right )}{2}\) | \(32\) |
default | \(\frac {\cot \left (x \right ) \sqrt {-1-\cot \left (x \right )^{2}}}{2}-\frac {\arctan \left (\frac {\cot \left (x \right )}{\sqrt {-1-\cot \left (x \right )^{2}}}\right )}{2}\) | \(32\) |
risch | \(\frac {i \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )}{{\mathrm e}^{2 i x}-1}-\sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )+\sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )\) | \(95\) |
Input:
int((-1-cot(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2*cot(x)*(-1-cot(x)^2)^(1/2)-1/2*arctan(cot(x)/(-1-cot(x)^2)^(1/2))
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.09 \[ \int \left (-1-\cot ^2(x)\right )^{3/2} \, dx=\frac {{\left (-i \, e^{\left (4 i \, x\right )} + 2 i \, e^{\left (2 i \, x\right )} - i\right )} \log \left (e^{\left (i \, x\right )} + 1\right ) + {\left (i \, e^{\left (4 i \, x\right )} - 2 i \, e^{\left (2 i \, x\right )} + i\right )} \log \left (e^{\left (i \, x\right )} - 1\right ) + 2 i \, e^{\left (3 i \, x\right )} + 2 i \, e^{\left (i \, x\right )}}{2 \, {\left (e^{\left (4 i \, x\right )} - 2 \, e^{\left (2 i \, x\right )} + 1\right )}} \] Input:
integrate((-1-cot(x)^2)^(3/2),x, algorithm="fricas")
Output:
1/2*((-I*e^(4*I*x) + 2*I*e^(2*I*x) - I)*log(e^(I*x) + 1) + (I*e^(4*I*x) - 2*I*e^(2*I*x) + I)*log(e^(I*x) - 1) + 2*I*e^(3*I*x) + 2*I*e^(I*x))/(e^(4*I *x) - 2*e^(2*I*x) + 1)
\[ \int \left (-1-\cot ^2(x)\right )^{3/2} \, dx=\int \left (- \cot ^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \] Input:
integrate((-1-cot(x)**2)**(3/2),x)
Output:
Integral((-cot(x)**2 - 1)**(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (27) = 54\).
Time = 0.15 (sec) , antiderivative size = 284, normalized size of antiderivative = 8.11 \[ \int \left (-1-\cot ^2(x)\right )^{3/2} \, dx=\frac {{\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \, {\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \, {\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + 4 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \cos \left (x\right ) \sin \left (2 \, x\right ) - 4 \, \cos \left (2 \, x\right ) \sin \left (x\right ) + 2 \, \sin \left (x\right )}{2 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \] Input:
integrate((-1-cot(x)^2)^(3/2),x, algorithm="maxima")
Output:
1/2*((2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)*arctan2(sin(x), co s(x) + 1) - (2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin (4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)*arctan2(sin (x), cos(x) - 1) + 2*(sin(3*x) + sin(x))*cos(4*x) - 2*(cos(3*x) + cos(x))* sin(4*x) - 2*(2*cos(2*x) - 1)*sin(3*x) + 4*cos(3*x)*sin(2*x) + 4*cos(x)*si n(2*x) - 4*cos(2*x)*sin(x) + 2*sin(x))/(2*(2*cos(2*x) - 1)*cos(4*x) - cos( 4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \left (-1-\cot ^2(x)\right )^{3/2} \, dx=-\frac {1}{4} \, {\left (\frac {2 i \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} - i \, \log \left (\cos \left (x\right ) + 1\right ) + i \, \log \left (-\cos \left (x\right ) + 1\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \] Input:
integrate((-1-cot(x)^2)^(3/2),x, algorithm="giac")
Output:
-1/4*(2*I*cos(x)/(cos(x)^2 - 1) - I*log(cos(x) + 1) + I*log(-cos(x) + 1))* sgn(sin(x))
Time = 8.63 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \left (-1-\cot ^2(x)\right )^{3/2} \, dx=\frac {\mathrm {cot}\left (x\right )\,\sqrt {-{\mathrm {cot}\left (x\right )}^2-1}}{2}-\frac {\mathrm {atan}\left (\frac {\mathrm {cot}\left (x\right )}{\sqrt {-{\mathrm {cot}\left (x\right )}^2-1}}\right )}{2} \] Input:
int((- cot(x)^2 - 1)^(3/2),x)
Output:
(cot(x)*(- cot(x)^2 - 1)^(1/2))/2 - atan(cot(x)/(- cot(x)^2 - 1)^(1/2))/2
\[ \int \left (-1-\cot ^2(x)\right )^{3/2} \, dx=-i \left (\int \sqrt {\cot \left (x \right )^{2}+1}d x +\int \sqrt {\cot \left (x \right )^{2}+1}\, \cot \left (x \right )^{2}d x \right ) \] Input:
int((-1-cot(x)^2)^(3/2),x)
Output:
- i*(int(sqrt(cot(x)**2 + 1),x) + int(sqrt(cot(x)**2 + 1)*cot(x)**2,x))