Integrand size = 10, antiderivative size = 36 \[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} a \cot (x) \sqrt {a \cot ^2(x)}-a \sqrt {a \cot ^2(x)} \log (\sin (x)) \tan (x) \] Output:
-1/2*a*cot(x)*(a*cot(x)^2)^(1/2)-a*(a*cot(x)^2)^(1/2)*ln(sin(x))*tan(x)
Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} a \sqrt {a \cot ^2(x)} \left (\csc ^2(x)+2 \log (\sin (x))\right ) \tan (x) \] Input:
Integrate[(a*Cot[x]^2)^(3/2),x]
Output:
-1/2*(a*Sqrt[a*Cot[x]^2]*(Csc[x]^2 + 2*Log[Sin[x]])*Tan[x])
Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {3042, 4141, 3042, 25, 3954, 25, 3042, 25, 3956}
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.
\(\displaystyle \int \left (a \cot ^2(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \tan \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4141 |
\(\displaystyle a \tan (x) \sqrt {a \cot ^2(x)} \int \cot ^3(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \tan (x) \sqrt {a \cot ^2(x)} \int -\tan \left (x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a \tan (x) \sqrt {a \cot ^2(x)} \int \tan \left (x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle -a \tan (x) \sqrt {a \cot ^2(x)} \left (\frac {\cot ^2(x)}{2}-\int -\cot (x)dx\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a \tan (x) \sqrt {a \cot ^2(x)} \left (\int \cot (x)dx+\frac {\cot ^2(x)}{2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -a \tan (x) \sqrt {a \cot ^2(x)} \left (\int -\tan \left (x+\frac {\pi }{2}\right )dx+\frac {\cot ^2(x)}{2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a \tan (x) \sqrt {a \cot ^2(x)} \left (\frac {\cot ^2(x)}{2}-\int \tan \left (x+\frac {\pi }{2}\right )dx\right )\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -a \tan (x) \sqrt {a \cot ^2(x)} \left (\frac {\cot ^2(x)}{2}+\log (\sin (x))\right )\) |
Input:
Int[(a*Cot[x]^2)^(3/2),x]
Output:
-(a*Sqrt[a*Cot[x]^2]*(Cot[x]^2/2 + Log[Sin[x]])*Tan[x])
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {\left (a \cot \left (x \right )^{2}\right )^{\frac {3}{2}} \left (-\cot \left (x \right )^{2}+\ln \left (\cot \left (x \right )^{2}+1\right )\right )}{2 \cot \left (x \right )^{3}}\) | \(29\) |
default | \(\frac {\left (a \cot \left (x \right )^{2}\right )^{\frac {3}{2}} \left (-\cot \left (x \right )^{2}+\ln \left (\cot \left (x \right )^{2}+1\right )\right )}{2 \cot \left (x \right )^{3}}\) | \(29\) |
risch | \(\frac {a \sqrt {-\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left (i {\mathrm e}^{4 i x} \ln \left ({\mathrm e}^{2 i x}-1\right )+{\mathrm e}^{4 i x} x -2 i {\mathrm e}^{2 i x} \ln \left ({\mathrm e}^{2 i x}-1\right )-2 i {\mathrm e}^{2 i x}-2 \,{\mathrm e}^{2 i x} x +i \ln \left ({\mathrm e}^{2 i x}-1\right )+x \right )}{\left ({\mathrm e}^{2 i x}+1\right ) \left ({\mathrm e}^{2 i x}-1\right )}\) | \(112\) |
Input:
int((a*cot(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2*(a*cot(x)^2)^(3/2)*(-cot(x)^2+ln(cot(x)^2+1))/cot(x)^3
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.44 \[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=\frac {{\left ({\left (a \cos \left (2 \, x\right ) - a\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, x\right ) + \frac {1}{2}\right ) - 2 \, a\right )} \sqrt {-\frac {a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}}}{2 \, \sin \left (2 \, x\right )} \] Input:
integrate((a*cot(x)^2)^(3/2),x, algorithm="fricas")
Output:
1/2*((a*cos(2*x) - a)*log(-1/2*cos(2*x) + 1/2) - 2*a)*sqrt(-(a*cos(2*x) + a)/(cos(2*x) - 1))/sin(2*x)
\[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=\int \left (a \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((a*cot(x)**2)**(3/2),x)
Output:
Integral((a*cot(x)**2)**(3/2), x)
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=\frac {1}{2} \, a^{\frac {3}{2}} \log \left (\tan \left (x\right )^{2} + 1\right ) - a^{\frac {3}{2}} \log \left (\tan \left (x\right )\right ) - \frac {a^{\frac {3}{2}}}{2 \, \tan \left (x\right )^{2}} \] Input:
integrate((a*cot(x)^2)^(3/2),x, algorithm="maxima")
Output:
1/2*a^(3/2)*log(tan(x)^2 + 1) - a^(3/2)*log(tan(x)) - 1/2*a^(3/2)/tan(x)^2
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=\frac {1}{2} \, a^{\frac {3}{2}} {\left (\frac {1}{\cos \left (x\right )^{2} - 1} - \log \left (-\cos \left (x\right )^{2} + 1\right )\right )} \mathrm {sgn}\left (\cos \left (x\right )\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) \] Input:
integrate((a*cot(x)^2)^(3/2),x, algorithm="giac")
Output:
1/2*a^(3/2)*(1/(cos(x)^2 - 1) - log(-cos(x)^2 + 1))*sgn(cos(x))*sgn(sin(x) )
Timed out. \[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=\int {\left (a\,{\mathrm {cot}\left (x\right )}^2\right )}^{3/2} \,d x \] Input:
int((a*cot(x)^2)^(3/2),x)
Output:
int((a*cot(x)^2)^(3/2), x)
Time = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=\frac {\sqrt {a}\, a \left (4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2}+1\right ) \sin \left (x \right )^{2}-4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2}+\sin \left (x \right )^{2}-2\right )}{4 \sin \left (x \right )^{2}} \] Input:
int((a*cot(x)^2)^(3/2),x)
Output:
(sqrt(a)*a*(4*log(tan(x/2)**2 + 1)*sin(x)**2 - 4*log(tan(x/2))*sin(x)**2 + sin(x)**2 - 2))/(4*sin(x)**2)