3.14 \(\int \frac{x^2}{(a^2+x^2) (b^2+x^2)} \, dx\)

Optimal. Leaf size=40 \[ \frac{a \tan ^{-1}\left (\frac{x}{a}\right )}{a^2-b^2}-\frac{b \tan ^{-1}\left (\frac{x}{b}\right )}{a^2-b^2} \]

[Out]

(a*ArcTan[x/a])/(a^2 - b^2) - (b*ArcTan[x/b])/(a^2 - b^2)

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Rubi [A]  time = 0.019227, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {481, 203} \[ \frac{a \tan ^{-1}\left (\frac{x}{a}\right )}{a^2-b^2}-\frac{b \tan ^{-1}\left (\frac{x}{b}\right )}{a^2-b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((a^2 + x^2)*(b^2 + x^2)),x]

[Out]

(a*ArcTan[x/a])/(a^2 - b^2) - (b*ArcTan[x/b])/(a^2 - b^2)

Rule 481

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> -Dist[(a*e^n)/(b*c -
a*d), Int[(e*x)^(m - n)/(a + b*x^n), x], x] + Dist[(c*e^n)/(b*c - a*d), Int[(e*x)^(m - n)/(c + d*x^n), x], x]
/; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1]

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 ) \left (b^2+x^2\right )} \, dx &=\frac{a^2 \int \frac{1}{a^2+x^2} \, dx}{a^2-b^2}-\frac{b^2 \int \frac{1}{b^2+x^2} \, dx}{a^2-b^2}\\ &=\frac{a \tan ^{-1}\left (\frac{x}{a}\right )}{a^2-b^2}-\frac{b \tan ^{-1}\left (\frac{x}{b}\right )}{a^2-b^2}\\ \end{align*}

Mathematica [A]  time = 0.0140833, size = 30, normalized size = 0.75 \[ \frac{a \tan ^{-1}\left (\frac{x}{a}\right )-b \tan ^{-1}\left (\frac{x}{b}\right )}{a^2-b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*ArcTan[x/a] - b*ArcTan[x/b])/(a^2 - b^2)

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Maple [A]  time = 0.007, size = 41, normalized size = 1. \begin{align*}{\frac{a}{{a}^{2}-{b}^{2}}\arctan \left ({\frac{x}{a}} \right ) }-{\frac{b}{{a}^{2}-{b}^{2}}\arctan \left ({\frac{x}{b}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a^2+x^2)/(b^2+x^2),x)

[Out]

a*arctan(x/a)/(a^2-b^2)-b*arctan(x/b)/(a^2-b^2)

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Maxima [A]  time = 1.41942, size = 54, normalized size = 1.35 \begin{align*} \frac{a \arctan \left (\frac{x}{a}\right )}{a^{2} - b^{2}} - \frac{b \arctan \left (\frac{x}{b}\right )}{a^{2} - b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2+x^2)/(b^2+x^2),x, algorithm="maxima")

[Out]

a*arctan(x/a)/(a^2 - b^2) - b*arctan(x/b)/(a^2 - b^2)

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Fricas [A]  time = 2.07345, size = 61, normalized size = 1.52 \begin{align*} \frac{a \arctan \left (\frac{x}{a}\right ) - b \arctan \left (\frac{x}{b}\right )}{a^{2} - b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2+x^2)/(b^2+x^2),x, algorithm="fricas")

[Out]

(a*arctan(x/a) - b*arctan(x/b))/(a^2 - b^2)

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Sympy [C]  time = 0.943608, size = 393, normalized size = 9.82 \begin{align*} - \frac{i a \log{\left (- \frac{2 i a^{7}}{\left (a - b\right )^{3} \left (a + b\right )^{3}} + \frac{4 i a^{5} b^{2}}{\left (a - b\right )^{3} \left (a + b\right )^{3}} - \frac{2 i a^{3} b^{4}}{\left (a - b\right )^{3} \left (a + b\right )^{3}} + \frac{i a^{3}}{\left (a - b\right ) \left (a + b\right )} + \frac{i a b^{2}}{\left (a - b\right ) \left (a + b\right )} + x \right )}}{2 \left (a - b\right ) \left (a + b\right )} + \frac{i a \log{\left (\frac{2 i a^{7}}{\left (a - b\right )^{3} \left (a + b\right )^{3}} - \frac{4 i a^{5} b^{2}}{\left (a - b\right )^{3} \left (a + b\right )^{3}} + \frac{2 i a^{3} b^{4}}{\left (a - b\right )^{3} \left (a + b\right )^{3}} - \frac{i a^{3}}{\left (a - b\right ) \left (a + b\right )} - \frac{i a b^{2}}{\left (a - b\right ) \left (a + b\right )} + x \right )}}{2 \left (a - b\right ) \left (a + b\right )} - \frac{i b \log{\left (- \frac{2 i a^{4} b^{3}}{\left (a - b\right )^{3} \left (a + b\right )^{3}} + \frac{4 i a^{2} b^{5}}{\left (a - b\right )^{3} \left (a + b\right )^{3}} + \frac{i a^{2} b}{\left (a - b\right ) \left (a + b\right )} - \frac{2 i b^{7}}{\left (a - b\right )^{3} \left (a + b\right )^{3}} + \frac{i b^{3}}{\left (a - b\right ) \left (a + b\right )} + x \right )}}{2 \left (a - b\right ) \left (a + b\right )} + \frac{i b \log{\left (\frac{2 i a^{4} b^{3}}{\left (a - b\right )^{3} \left (a + b\right )^{3}} - \frac{4 i a^{2} b^{5}}{\left (a - b\right )^{3} \left (a + b\right )^{3}} - \frac{i a^{2} b}{\left (a - b\right ) \left (a + b\right )} + \frac{2 i b^{7}}{\left (a - b\right )^{3} \left (a + b\right )^{3}} - \frac{i b^{3}}{\left (a - b\right ) \left (a + b\right )} + x \right )}}{2 \left (a - b\right ) \left (a + b\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a**2+x**2)/(b**2+x**2),x)

[Out]

-I*a*log(-2*I*a**7/((a - b)**3*(a + b)**3) + 4*I*a**5*b**2/((a - b)**3*(a + b)**3) - 2*I*a**3*b**4/((a - b)**3
*(a + b)**3) + I*a**3/((a - b)*(a + b)) + I*a*b**2/((a - b)*(a + b)) + x)/(2*(a - b)*(a + b)) + I*a*log(2*I*a*
*7/((a - b)**3*(a + b)**3) - 4*I*a**5*b**2/((a - b)**3*(a + b)**3) + 2*I*a**3*b**4/((a - b)**3*(a + b)**3) - I
*a**3/((a - b)*(a + b)) - I*a*b**2/((a - b)*(a + b)) + x)/(2*(a - b)*(a + b)) - I*b*log(-2*I*a**4*b**3/((a - b
)**3*(a + b)**3) + 4*I*a**2*b**5/((a - b)**3*(a + b)**3) + I*a**2*b/((a - b)*(a + b)) - 2*I*b**7/((a - b)**3*(
a + b)**3) + I*b**3/((a - b)*(a + b)) + x)/(2*(a - b)*(a + b)) + I*b*log(2*I*a**4*b**3/((a - b)**3*(a + b)**3)
- 4*I*a**2*b**5/((a - b)**3*(a + b)**3) - I*a**2*b/((a - b)*(a + b)) + 2*I*b**7/((a - b)**3*(a + b)**3) - I*b
**3/((a - b)*(a + b)) + x)/(2*(a - b)*(a + b))

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Giac [A]  time = 1.07482, size = 54, normalized size = 1.35 \begin{align*} \frac{a \arctan \left (\frac{x}{a}\right )}{a^{2} - b^{2}} - \frac{b \arctan \left (\frac{x}{b}\right )}{a^{2} - b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2+x^2)/(b^2+x^2),x, algorithm="giac")

[Out]

a*arctan(x/a)/(a^2 - b^2) - b*arctan(x/b)/(a^2 - b^2)