3.282 \(\int \frac{\cot ^2(x)}{(a+a \tan ^2(x))^{3/2}} \, dx\)

Optimal. Leaf size=60 \[ -\frac{\csc (x) \sec (x)}{a \sqrt{a \sec ^2(x)}}-\frac{2 \tan (x)}{a \sqrt{a \sec ^2(x)}}+\frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}} \]

[Out]

-((Csc[x]*Sec[x])/(a*Sqrt[a*Sec[x]^2])) - (2*Tan[x])/(a*Sqrt[a*Sec[x]^2]) + (Sin[x]^2*Tan[x])/(3*a*Sqrt[a*Sec[
x]^2])

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Rubi [A]  time = 0.114722, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3657, 4125, 2590, 270} \[ -\frac{\csc (x) \sec (x)}{a \sqrt{a \sec ^2(x)}}-\frac{2 \tan (x)}{a \sqrt{a \sec ^2(x)}}+\frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Csc[x]*Sec[x])/(a*Sqrt[a*Sec[x]^2])) - (2*Tan[x])/(a*Sqrt[a*Sec[x]^2]) + (Sin[x]^2*Tan[x])/(3*a*Sqrt[a*Sec[
x]^2])

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 4125

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

Rule 2590

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\cot ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx &=\int \frac{\cot ^2(x)}{\left (a \sec ^2(x)\right )^{3/2}} \, dx\\ &=\frac{\sec (x) \int \cos ^3(x) \cot ^2(x) \, dx}{a \sqrt{a \sec ^2(x)}}\\ &=-\frac{\sec (x) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^2} \, dx,x,-\sin (x)\right )}{a \sqrt{a \sec ^2(x)}}\\ &=-\frac{\sec (x) \operatorname{Subst}\left (\int \left (-2+\frac{1}{x^2}+x^2\right ) \, dx,x,-\sin (x)\right )}{a \sqrt{a \sec ^2(x)}}\\ &=-\frac{\csc (x) \sec (x)}{a \sqrt{a \sec ^2(x)}}-\frac{2 \tan (x)}{a \sqrt{a \sec ^2(x)}}+\frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.056845, size = 31, normalized size = 0.52 \[ \frac{\sec ^3(x) \left (\sin ^3(x)-6 \sin (x)-3 \csc (x)\right )}{3 \left (a \sec ^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[x]^3*(-3*Csc[x] - 6*Sin[x] + Sin[x]^3))/(3*(a*Sec[x]^2)^(3/2))

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Maple [A]  time = 0.071, size = 31, normalized size = 0.5 \begin{align*}{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}+4\, \left ( \cos \left ( x \right ) \right ) ^{2}-8}{3\,\sin \left ( x \right ) \left ( \cos \left ( x \right ) \right ) ^{3}} \left ({\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(cos(x)^4+4*cos(x)^2-8)/sin(x)/cos(x)^3/(a/cos(x)^2)^(3/2)

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Maxima [B]  time = 1.88868, size = 304, normalized size = 5.07 \begin{align*} \frac{{\left ({\left (\sin \left (5 \, x\right ) - \sin \left (3 \, x\right )\right )} \cos \left (8 \, x\right ) + 20 \,{\left (\sin \left (5 \, x\right ) - \sin \left (3 \, x\right )\right )} \cos \left (6 \, x\right ) + 10 \,{\left (9 \, \sin \left (4 \, x\right ) - 2 \, \sin \left (2 \, x\right )\right )} \cos \left (5 \, x\right ) -{\left (\cos \left (5 \, x\right ) - \cos \left (3 \, x\right )\right )} \sin \left (8 \, x\right ) - 20 \,{\left (\cos \left (5 \, x\right ) - \cos \left (3 \, x\right )\right )} \sin \left (6 \, x\right ) -{\left (90 \, \cos \left (4 \, x\right ) - 20 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (5 \, x\right ) - 90 \, \cos \left (3 \, x\right ) \sin \left (4 \, x\right ) -{\left (20 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (3 \, x\right ) + 90 \, \cos \left (4 \, x\right ) \sin \left (3 \, x\right ) + 20 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right )\right )} \sqrt{a}}{24 \,{\left (a^{2} \cos \left (5 \, x\right )^{2} - 2 \, a^{2} \cos \left (5 \, x\right ) \cos \left (3 \, x\right ) + a^{2} \cos \left (3 \, x\right )^{2} + a^{2} \sin \left (5 \, x\right )^{2} - 2 \, a^{2} \sin \left (5 \, x\right ) \sin \left (3 \, x\right ) + a^{2} \sin \left (3 \, x\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((sin(5*x) - sin(3*x))*cos(8*x) + 20*(sin(5*x) - sin(3*x))*cos(6*x) + 10*(9*sin(4*x) - 2*sin(2*x))*cos(5*
x) - (cos(5*x) - cos(3*x))*sin(8*x) - 20*(cos(5*x) - cos(3*x))*sin(6*x) - (90*cos(4*x) - 20*cos(2*x) - 1)*sin(
5*x) - 90*cos(3*x)*sin(4*x) - (20*cos(2*x) + 1)*sin(3*x) + 90*cos(4*x)*sin(3*x) + 20*cos(3*x)*sin(2*x))*sqrt(a
)/(a^2*cos(5*x)^2 - 2*a^2*cos(5*x)*cos(3*x) + a^2*cos(3*x)^2 + a^2*sin(5*x)^2 - 2*a^2*sin(5*x)*sin(3*x) + a^2*
sin(3*x)^2)

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Fricas [A]  time = 1.59882, size = 139, normalized size = 2.32 \begin{align*} -\frac{{\left (8 \, \tan \left (x\right )^{4} + 12 \, \tan \left (x\right )^{2} + 3\right )} \sqrt{a \tan \left (x\right )^{2} + a}}{3 \,{\left (a^{2} \tan \left (x\right )^{5} + 2 \, a^{2} \tan \left (x\right )^{3} + a^{2} \tan \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(8*tan(x)^4 + 12*tan(x)^2 + 3)*sqrt(a*tan(x)^2 + a)/(a^2*tan(x)^5 + 2*a^2*tan(x)^3 + a^2*tan(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1077, size = 74, normalized size = 1.23 \begin{align*} -\frac{{\left (5 \, \tan \left (x\right )^{2} + 6\right )} \tan \left (x\right )}{3 \,{\left (a \tan \left (x\right )^{2} + a\right )}^{\frac{3}{2}}} + \frac{2}{{\left ({\left (\sqrt{a} \tan \left (x\right ) - \sqrt{a \tan \left (x\right )^{2} + a}\right )}^{2} - a\right )} \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(5*tan(x)^2 + 6)*tan(x)/(a*tan(x)^2 + a)^(3/2) + 2/(((sqrt(a)*tan(x) - sqrt(a*tan(x)^2 + a))^2 - a)*sqrt(
a))