\(\int (a \cot ^2(x))^{3/2} \, dx\) [25]

Optimal result

Integrand size = 10, antiderivative size = 36 \[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} a \cot (x) \sqrt {a \cot ^2(x)}-a \sqrt {a \cot ^2(x)} \log (\sin (x)) \tan (x) \]

[Out]

-1/2*a*cot(x)*(a*cot(x)^2)^(1/2)-a*ln(sin(x))*(a*cot(x)^2)^(1/2)*tan(x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3739, 3554, 3556} \[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} a \cot (x) \sqrt {a \cot ^2(x)}-a \tan (x) \sqrt {a \cot ^2(x)} \log (\sin (x)) \]

[In]

Int[(a*Cot[x]^2)^(3/2),x]

[Out]

-1/2*(a*Cot[x]*Sqrt[a*Cot[x]^2]) - a*Sqrt[a*Cot[x]^2]*Log[Sin[x]]*Tan[x]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3739

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Tan[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
]])

Rubi steps \begin{align*} \text {integral}& = \left (a \sqrt {a \cot ^2(x)} \tan (x)\right ) \int \cot ^3(x) \, dx \\ & = -\frac {1}{2} a \cot (x) \sqrt {a \cot ^2(x)}-\left (a \sqrt {a \cot ^2(x)} \tan (x)\right ) \int \cot (x) \, dx \\ & = -\frac {1}{2} a \cot (x) \sqrt {a \cot ^2(x)}-a \sqrt {a \cot ^2(x)} \log (\sin (x)) \tan (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} a \sqrt {a \cot ^2(x)} \left (\cot ^2(x)+2 (\log (\cos (x))+\log (\tan (x)))\right ) \tan (x) \]

[In]

Integrate[(a*Cot[x]^2)^(3/2),x]

[Out]

-1/2*(a*Sqrt[a*Cot[x]^2]*(Cot[x]^2 + 2*(Log[Cos[x]] + Log[Tan[x]]))*Tan[x])

Maple [A] (verified)

Time = 0.10 (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	\(\frac {a \sqrt {-\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left (i \ln \left ({\mathrm e}^{2 i x}-1\right ) {\mathrm e}^{4 i x}-2 i \ln \left ({\mathrm e}^{2 i x}-1\right ) {\mathrm e}^{2 i x}+{\mathrm e}^{4 i x} x +i \ln \left ({\mathrm e}^{2 i x}-1\right )-2 i {\mathrm e}^{2 i x}-2 \,{\mathrm e}^{2 i x} x +x \right )}{\left ({\mathrm e}^{2 i x}+1\right ) \left ({\mathrm e}^{2 i x}-1\right )}\)	\(112\)

[In]

int((a*cot(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*(a*cot(x)^2)^(3/2)*(-cot(x)^2+ln(cot(x)^2+1))/cot(x)^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.44 \[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=\frac {{\left ({\left (a \cos \left (2 \, x\right ) - a\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, x\right ) + \frac {1}{2}\right ) - 2 \, a\right )} \sqrt {-\frac {a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}}}{2 \, \sin \left (2 \, x\right )} \]

[In]

integrate((a*cot(x)^2)^(3/2),x, algorithm="fricas")

[Out]

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)

Sympy [F]

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

[In]

integrate((a*cot(x)**2)**(3/2),x)

[Out]

Integral((a*cot(x)**2)**(3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=\frac {1}{2} \, a^{\frac {3}{2}} \log \left (\tan \left (x\right )^{2} + 1\right ) - a^{\frac {3}{2}} \log \left (\tan \left (x\right )\right ) - \frac {a^{\frac {3}{2}}}{2 \, \tan \left (x\right )^{2}} \]

[In]

integrate((a*cot(x)^2)^(3/2),x, algorithm="maxima")

[Out]

1/2*a^(3/2)*log(tan(x)^2 + 1) - a^(3/2)*log(tan(x)) - 1/2*a^(3/2)/tan(x)^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \left (a \cot ^2(x)\right )^{3/2} \, dx=\frac {1}{2} \, a^{\frac {3}{2}} {\left (\frac {1}{\cos \left (x\right )^{2} - 1} - \log \left (-\cos \left (x\right )^{2} + 1\right )\right )} \mathrm {sgn}\left (\cos \left (x\right )\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) \]

[In]

integrate((a*cot(x)^2)^(3/2),x, algorithm="giac")

[Out]

1/2*a^(3/2)*(1/(cos(x)^2 - 1) - log(-cos(x)^2 + 1))*sgn(cos(x))*sgn(sin(x))

Mupad [F(-1)]

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

[In]

int((a*cot(x)^2)^(3/2),x)

[Out]

int((a*cot(x)^2)^(3/2), x)