3.116 \(\int \frac{1}{(1+x^2) (2+x^2) (3+x^2) (4+x^2)} \, dx\)

Optimal. Leaf size=51 \[ -\frac{1}{12} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{6} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

-ArcTan[x/2]/12 + ArcTan[x]/6 - ArcTan[x/Sqrt[2]]/(2*Sqrt[2]) + ArcTan[x/Sqrt[3]]/(2*Sqrt[3])

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Rubi [A]  time = 0.284273, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {6725, 203} \[ -\frac{1}{12} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{6} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + x^2)*(2 + x^2)*(3 + x^2)*(4 + x^2)),x]

[Out]

-ArcTan[x/2]/12 + ArcTan[x]/6 - ArcTan[x/Sqrt[2]]/(2*Sqrt[2]) + ArcTan[x/Sqrt[3]]/(2*Sqrt[3])

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 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{1}{\left (1+x^2\right ) \left (2+x^2\right ) \left (3+x^2\right ) \left (4+x^2\right )} \, dx &=\int \left (\frac{1}{6 \left (1+x^2\right )}-\frac{1}{2 \left (2+x^2\right )}+\frac{1}{2 \left (3+x^2\right )}-\frac{1}{6 \left (4+x^2\right )}\right ) \, dx\\ &=\frac{1}{6} \int \frac{1}{1+x^2} \, dx-\frac{1}{6} \int \frac{1}{4+x^2} \, dx-\frac{1}{2} \int \frac{1}{2+x^2} \, dx+\frac{1}{2} \int \frac{1}{3+x^2} \, dx\\ &=-\frac{1}{12} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{6} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{2 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0235758, size = 47, normalized size = 0.92 \[ \frac{1}{12} \left (-\tan ^{-1}\left (\frac{x}{2}\right )+2 \tan ^{-1}(x)-3 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + x^2)*(2 + x^2)*(3 + x^2)*(4 + x^2)),x]

[Out]

(-ArcTan[x/2] + 2*ArcTan[x] - 3*Sqrt[2]*ArcTan[x/Sqrt[2]] + 2*Sqrt[3]*ArcTan[x/Sqrt[3]])/12

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Maple [A]  time = 0.009, size = 36, normalized size = 0.7 \begin{align*} -{\frac{1}{12}\arctan \left ({\frac{x}{2}} \right ) }+{\frac{\arctan \left ( x \right ) }{6}}-{\frac{\sqrt{2}}{4}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{x\sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+1)/(x^2+2)/(x^2+3)/(x^2+4),x)

[Out]

-1/12*arctan(1/2*x)+1/6*arctan(x)-1/4*arctan(1/2*x*2^(1/2))*2^(1/2)+1/6*arctan(1/3*x*3^(1/2))*3^(1/2)

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Maxima [A]  time = 1.41324, size = 47, normalized size = 0.92 \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1}{12} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{6} \, \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)/(x^2+2)/(x^2+3)/(x^2+4),x, algorithm="maxima")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*x) - 1/4*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/12*arctan(1/2*x) + 1/6*arctan(x)

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Fricas [A]  time = 2.23834, size = 146, normalized size = 2.86 \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1}{12} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{6} \, \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)/(x^2+2)/(x^2+3)/(x^2+4),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*x) - 1/4*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/12*arctan(1/2*x) + 1/6*arctan(x)

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Sympy [A]  time = 0.356608, size = 44, normalized size = 0.86 \begin{align*} - \frac{\operatorname{atan}{\left (\frac{x}{2} \right )}}{12} + \frac{\operatorname{atan}{\left (x \right )}}{6} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{4} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+1)/(x**2+2)/(x**2+3)/(x**2+4),x)

[Out]

-atan(x/2)/12 + atan(x)/6 - sqrt(2)*atan(sqrt(2)*x/2)/4 + sqrt(3)*atan(sqrt(3)*x/3)/6

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Giac [A]  time = 1.06052, size = 47, normalized size = 0.92 \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1}{12} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{6} \, \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)/(x^2+2)/(x^2+3)/(x^2+4),x, algorithm="giac")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*x) - 1/4*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/12*arctan(1/2*x) + 1/6*arctan(x)