\(\int \frac {\cot ^3(x)}{\sqrt {a+a \cot ^2(x)}} \, dx\) [14]

Optimal result

Integrand size = 17, antiderivative size = 28 \[ \int \frac {\cot ^3(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=-\frac {1}{\sqrt {a \csc ^2(x)}}-\frac {\sqrt {a \csc ^2(x)}}{a} \]

[Out]

-1/(a*csc(x)^2)^(1/2)-(a*csc(x)^2)^(1/2)/a

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3738, 4209, 45} \[ \int \frac {\cot ^3(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=-\frac {\sqrt {a \csc ^2(x)}}{a}-\frac {1}{\sqrt {a \csc ^2(x)}} \]

[In]

Int[Cot[x]^3/Sqrt[a + a*Cot[x]^2],x]

[Out]

-(1/Sqrt[a*Csc[x]^2]) - Sqrt[a*Csc[x]^2]/a

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 3738

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 4209

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Dist[b/(2*f), Subst[In
t[(-1 + x)^((m - 1)/2)*(b*x)^(p - 1), x], x, Sec[e + f*x]^2], x] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p] &&
IntegerQ[(m - 1)/2]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {\cot ^3(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {-1-\csc ^2(x)}{\sqrt {a \csc ^2(x)}} \]

[In]

Integrate[Cot[x]^3/Sqrt[a + a*Cot[x]^2],x]

[Out]

(-1 - Csc[x]^2)/Sqrt[a*Csc[x]^2]

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(-\frac {\sqrt {a +a \cot \left (x \right )^{2}}}{a}-\frac {1}{\sqrt {a +a \cot \left (x \right )^{2}}}\)	\(29\)
default	\(-\frac {\sqrt {a +a \cot \left (x \right )^{2}}}{a}-\frac {1}{\sqrt {a +a \cot \left (x \right )^{2}}}\)	\(29\)
risch	\(-\frac {{\mathrm e}^{4 i x}-6 \,{\mathrm e}^{2 i x}+1}{2 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )^{2}}\)	\(45\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\cot ^3(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\sqrt {2} \sqrt {-\frac {a}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 3\right )}}{2 \, a} \]

[In]

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

[Out]

1/2*sqrt(2)*sqrt(-a/(cos(2*x) - 1))*(cos(2*x) - 3)/a

Sympy [F]

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

[In]

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

[Out]

Integral(cot(x)**3/sqrt(a*(cot(x)**2 + 1)), x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\cot ^3(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=-\frac {1}{\sqrt {\frac {a}{\sin \left (x\right )^{2}}}} - \frac {\sqrt {\frac {a}{\sin \left (x\right )^{2}}}}{a} \]

[In]

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

[Out]

-1/sqrt(a/sin(x)^2) - sqrt(a/sin(x)^2)/a

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^3(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=-\frac {\sqrt {a} \sin \left (x\right ) + \frac {\sqrt {a}}{\sin \left (x\right )}}{a \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

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

[Out]

-(sqrt(a)*sin(x) + sqrt(a)/sin(x))/(a*sgn(sin(x)))

Mupad [B] (verification not implemented)

Time = 13.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^3(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=-\frac {{\sin \left (x\right )}^2+1}{\sqrt {a}\,\sqrt {{\sin \left (x\right )}^2}} \]

[In]

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

[Out]

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