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

Optimal. Leaf size=23 \[ \frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.101071, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3657, 4125, 2564, 30} \[ \frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0179405, size = 18, normalized size = 0.78 \[ \frac{\tan ^3(x)}{3 \left (a \sec ^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Tan[x]^3/(3*(a*Sec[x]^2)^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 56, normalized size = 2.4 \begin{align*}{\frac{\tan \left ( x \right ) }{a}{\frac{1}{\sqrt{a+a \left ( \tan \left ( x \right ) \right ) ^{2}}}}}-a \left ({\frac{\tan \left ( x \right ) }{3\,a} \left ( a+a \left ( \tan \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,\tan \left ( x \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt{a+a \left ( \tan \left ( x \right ) \right ) ^{2}}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.92509, size = 19, normalized size = 0.83 \begin{align*} -\frac{\sin \left (3 \, x\right ) - 3 \, \sin \left (x\right )}{12 \, a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(sin(3*x) - 3*sin(x))/a^(3/2)

________________________________________________________________________________________

Fricas [B]  time = 1.34167, size = 99, normalized size = 4.3 \begin{align*} \frac{\sqrt{a \tan \left (x\right )^{2} + a} \tan \left (x\right )^{3}}{3 \,{\left (a^{2} \tan \left (x\right )^{4} + 2 \, a^{2} \tan \left (x\right )^{2} + a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(a*tan(x)^2 + a)*tan(x)^3/(a^2*tan(x)^4 + 2*a^2*tan(x)^2 + a^2)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.11288, size = 22, normalized size = 0.96 \begin{align*} \frac{\tan \left (x\right )^{3}}{3 \,{\left (a \tan \left (x\right )^{2} + a\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*tan(x)^3/(a*tan(x)^2 + a)^(3/2)