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

Optimal. Leaf size=28 \[ -\frac{\sqrt{a \csc ^2(x)}}{a}-\frac{1}{\sqrt{a \csc ^2(x)}} \]

[Out]

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

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Rubi [A]  time = 0.0962848, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {3657, 4124, 43} \[ -\frac{\sqrt{a \csc ^2(x)}}{a}-\frac{1}{\sqrt{a \csc ^2(x)}} \]

Antiderivative was successfully verified.

[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 3657

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 4124

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]

Rule 43

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])

Rubi steps

\begin{align*} \int \frac{\cot ^3(x)}{\sqrt{a+a \cot ^2(x)}} \, dx &=\int \frac{\cot ^3(x)}{\sqrt{a \csc ^2(x)}} \, dx\\ &=-\left (\frac{1}{2} a \operatorname{Subst}\left (\int \frac{-1+x}{(a x)^{3/2}} \, dx,x,\csc ^2(x)\right )\right )\\ &=-\left (\frac{1}{2} a \operatorname{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]  time = 0.0236956, size = 19, normalized size = 0.68 \[ \frac{-\csc ^2(x)-1}{\sqrt{a \csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 - Csc[x]^2)/Sqrt[a*Csc[x]^2]

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Maple [A]  time = 0.036, size = 29, normalized size = 1. \begin{align*} -{\frac{1}{a}\sqrt{a+a \left ( \cot \left ( x \right ) \right ) ^{2}}}-{\frac{1}{\sqrt{a+a \left ( \cot \left ( x \right ) \right ) ^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.966181, size = 32, normalized size = 1.14 \begin{align*} -\frac{1}{\sqrt{\frac{a}{\sin \left (x\right )^{2}}}} - \frac{\sqrt{\frac{a}{\sin \left (x\right )^{2}}}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.52765, size = 73, normalized size = 2.61 \begin{align*} \frac{\sqrt{2} \sqrt{-\frac{a}{\cos \left (2 \, x\right ) - 1}}{\left (\cos \left (2 \, x\right ) - 3\right )}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{3}{\left (x \right )}}{\sqrt{a \left (\cot ^{2}{\left (x \right )} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.19753, size = 41, normalized size = 1.46 \begin{align*} -\sqrt{a}{\left (\frac{\sin \left (x\right )}{a \mathrm{sgn}\left (\sin \left (x\right )\right )} + \frac{1}{a \mathrm{sgn}\left (\sin \left (x\right )\right ) \sin \left (x\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a)*(sin(x)/(a*sgn(sin(x))) + 1/(a*sgn(sin(x))*sin(x)))