3.2.36 \(\int \frac {x^2}{(a^2-x^2)^{3/2}} \, dx\) [136]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 = {294, 223, 209} \begin {gather*} \frac {x}{\sqrt {a^2-x^2}}-\text {ArcTan}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \end {gather*}

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 209

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

Rule 223

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 294

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]

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}}-\text {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*}

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Mathematica [A]
time = 0.06, size = 34, normalized size = 1.00 \begin {gather*} \frac {x}{\sqrt {a^2-x^2}}-\tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a^2 - x^2)^(3/2),x]

[Out]

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

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Maple [A]
time = 0.04, size = 31, normalized size = 0.91

method	result	size
default	\(-\arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+\frac {x}{\sqrt {a^{2}-x^{2}}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a^2-x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.17, size = 22, normalized size = 0.65 \begin {gather*} \frac {x}{\sqrt {a^{2} - x^{2}}} - \arcsin \left (\frac {x}{a}\right ) \end {gather*}

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/a)

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Fricas [A]
time = 0.47, size = 58, normalized size = 1.71 \begin {gather*} \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 {gather*}

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 [C] Result contains complex when optimal does not.
time = 0.76, size = 49, normalized size = 1.44 \begin {gather*} \begin {cases} i \operatorname {acosh}{\left (\frac {x}{a} \right )} - \frac {i x}{a \sqrt {-1 + \frac {x^{2}}{a^{2}}}} & \text {for}\: \left |{\frac {x^{2}}{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 {gather*}

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/a**2) > 1), (-asin(x/a) + x/(a*sqrt(1 - x**2/
a**2)), True))

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Giac [A]
time = 0.79, size = 24, normalized size = 0.71 \begin {gather*} -\arcsin \left (\frac {x}{a}\right ) \mathrm {sgn}\left (a\right ) + \frac {x}{\sqrt {a^{2} - x^{2}}} \end {gather*}

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)

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Mupad [B]
time = 0.22, size = 34, normalized size = 1.00 \begin {gather*} \frac {x}{\sqrt {a^2-x^2}}+\ln \left (\sqrt {a^2-x^2}+x\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a^2 - x^2)^(3/2),x)

[Out]

log(x*1i + (a^2 - x^2)^(1/2))*1i + x/(a^2 - x^2)^(1/2)

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