\(\int \frac {1}{(a \cot ^2(x))^{3/2}} \, dx\) [28]

Optimal result

Integrand size = 10, antiderivative size = 39 \[ \int \frac {1}{\left (a \cot ^2(x)\right )^{3/2}} \, dx=\frac {\cot (x) \log (\cos (x))}{a \sqrt {a \cot ^2(x)}}+\frac {\tan (x)}{2 a \sqrt {a \cot ^2(x)}} \]

[Out]

cot(x)*ln(cos(x))/a/(a*cot(x)^2)^(1/2)+1/2*tan(x)/a/(a*cot(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, 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} \[ \int \frac {1}{\left (a \cot ^2(x)\right )^{3/2}} \, dx=\frac {\tan (x)}{2 a \sqrt {a \cot ^2(x)}}+\frac {\cot (x) \log (\cos (x))}{a \sqrt {a \cot ^2(x)}} \]

[In]

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

[Out]

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

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}& = \frac {\cot (x) \int \tan ^3(x) \, dx}{a \sqrt {a \cot ^2(x)}} \\ & = \frac {\tan (x)}{2 a \sqrt {a \cot ^2(x)}}-\frac {\cot (x) \int \tan (x) \, dx}{a \sqrt {a \cot ^2(x)}} \\ & = \frac {\cot (x) \log (\cos (x))}{a \sqrt {a \cot ^2(x)}}+\frac {\tan (x)}{2 a \sqrt {a \cot ^2(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\left (a \cot ^2(x)\right )^{3/2}} \, dx=\frac {2 \cot (x) \log (\cos (x))+\tan (x)}{2 a \sqrt {a \cot ^2(x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(-\frac {\cot \left (x \right ) \left (\ln \left (\cot \left (x \right )^{2}+1\right ) \cot \left (x \right )^{2}-2 \ln \left (\cot \left (x \right )\right ) \cot \left (x \right )^{2}-1\right )}{2 \left (a \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}\)	\(36\)
default	\(-\frac {\cot \left (x \right ) \left (\ln \left (\cot \left (x \right )^{2}+1\right ) \cot \left (x \right )^{2}-2 \ln \left (\cot \left (x \right )\right ) \cot \left (x \right )^{2}-1\right )}{2 \left (a \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}\)	\(36\)
risch	\(\frac {i {\mathrm e}^{4 i x} \ln \left ({\mathrm e}^{2 i x}+1\right )+{\mathrm e}^{4 i x} x +2 i {\mathrm e}^{2 i x} \ln \left ({\mathrm e}^{2 i x}+1\right )+2 i {\mathrm e}^{2 i x}+2 \,{\mathrm e}^{2 i x} x +i \ln \left ({\mathrm e}^{2 i x}+1\right )+x}{a \left ({\mathrm e}^{2 i x}+1\right ) \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}\)	\(114\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (33) = 66\).

Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.90 \[ \int \frac {1}{\left (a \cot ^2(x)\right )^{3/2}} \, dx=\frac {{\left ({\left (\cos \left (2 \, x\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (2 \, x\right ) + \frac {1}{2}\right ) \sin \left (2 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} \sqrt {-\frac {a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}}}{2 \, {\left (a^{2} \cos \left (2 \, x\right )^{2} + 2 \, a^{2} \cos \left (2 \, x\right ) + a^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{\left (a \cot ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral((a*cot(x)**2)**(-3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\left (a \cot ^2(x)\right )^{3/2}} \, dx=\frac {\tan \left (x\right )^{2}}{2 \, a^{\frac {3}{2}}} - \frac {\log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, a^{\frac {3}{2}}} \]

[In]

integrate(1/(a*cot(x)^2)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (a \cot ^2(x)\right )^{3/2}} \, dx=-\frac {\frac {\mathrm {sgn}\left (\sin \left (x\right )\right )}{\sqrt {a}} - \frac {2 \, \sqrt {a} \log \left ({\left | \cos \left (x\right ) \right |}\right ) + \frac {\sqrt {a}}{\cos \left (x\right )^{2}}}{a \mathrm {sgn}\left (\cos \left (x\right )\right ) \mathrm {sgn}\left (\sin \left (x\right )\right )}}{2 \, a} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \cot ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (a\,{\mathrm {cot}\left (x\right )}^2\right )}^{3/2}} \,d x \]

[In]

int(1/(a*cot(x)^2)^(3/2),x)

[Out]

int(1/(a*cot(x)^2)^(3/2), x)