### 3.136 $$\int \frac{x^2}{(a^2-x^2)^{3/2}} \, dx$$

Optimal. Leaf size=34 $\frac{x}{\sqrt{a^2-x^2}}-\tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right )$

[Out]

x/Sqrt[a^2 - x^2] - ArcTan[x/Sqrt[a^2 - x^2]]

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Rubi [A]  time = 0.0063195, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.176, Rules used = {288, 217, 203} $\frac{x}{\sqrt{a^2-x^2}}-\tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[x^2/(a^2 - x^2)^(3/2),x]

[Out]

x/Sqrt[a^2 - x^2] - ArcTan[x/Sqrt[a^2 - x^2]]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0301574, size = 39, normalized size = 1.15 $\frac{x-a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )}{\sqrt{a^2-x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^2/(a^2 - x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 31, normalized size = 0.9 \begin{align*} -\arctan \left ({x{\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}}} \right ) +{x{\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a^2-x^2)^(3/2),x)

[Out]

-arctan(x/(a^2-x^2)^(1/2))+x/(a^2-x^2)^(1/2)

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Maxima [A]  time = 1.40408, size = 32, normalized size = 0.94 \begin{align*} \frac{x}{\sqrt{a^{2} - x^{2}}} - \arcsin \left (\frac{x}{\sqrt{a^{2}}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2-x^2)^(3/2),x, algorithm="maxima")

[Out]

x/sqrt(a^2 - x^2) - arcsin(x/sqrt(a^2))

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Fricas [A]  time = 2.22168, size = 111, normalized size = 3.26 \begin{align*} \frac{2 \,{\left (a^{2} - x^{2}\right )} \arctan \left (-\frac{a - \sqrt{a^{2} - x^{2}}}{x}\right ) + \sqrt{a^{2} - x^{2}} x}{a^{2} - x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2-x^2)^(3/2),x, algorithm="fricas")

[Out]

(2*(a^2 - x^2)*arctan(-(a - sqrt(a^2 - x^2))/x) + sqrt(a^2 - x^2)*x)/(a^2 - x^2)

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Sympy [A]  time = 1.52178, size = 51, normalized size = 1.5 \begin{align*} \begin{cases} i \operatorname{acosh}{\left (\frac{x}{a} \right )} - \frac{i x}{a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \frac{\left |{x^{2}}\right |}{\left |{a^{2}}\right |} > 1 \\- \operatorname{asin}{\left (\frac{x}{a} \right )} + \frac{x}{a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a**2-x**2)**(3/2),x)

[Out]

Piecewise((I*acosh(x/a) - I*x/(a*sqrt(-1 + x**2/a**2)), Abs(x**2)/Abs(a**2) > 1), (-asin(x/a) + x/(a*sqrt(1 -
x**2/a**2)), True))

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Giac [A]  time = 1.07619, size = 32, normalized size = 0.94 \begin{align*} -\arcsin \left (\frac{x}{a}\right ) \mathrm{sgn}\left (a\right ) + \frac{x}{\sqrt{a^{2} - x^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2-x^2)^(3/2),x, algorithm="giac")

[Out]

-arcsin(x/a)*sgn(a) + x/sqrt(a^2 - x^2)