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

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

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Rubi [A]  time = 0.28, antiderivative size = 51, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.02, 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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IntegrateAlgebraic [A]  time = 0.03, size = 51, normalized size = 1.00 \[ -\frac {1}{12} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {1}{6} \tan ^{-1}(x)-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*ArcTan[x/2] + ArcTan[x]/6 - ArcTan[x/Sqrt[2]]/(2*Sqrt[2]) + ArcTan[x/Sqrt[3]]/(2*Sqrt[3])

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fricas [A]  time = 0.91, size = 35, normalized size = 0.69 \[ \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{12} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{6} \, \arctan \relax (x) \]

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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giac [A]  time = 0.86, size = 35, normalized size = 0.69 \[ \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{12} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{6} \, \arctan \relax (x) \]

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)

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maple [A]  time = 0.32, size = 36, normalized size = 0.71

method	result	size
default	\(-\frac {\arctan \left (\frac {x}{2}\right )}{12}+\frac {\arctan \relax (x )}{6}-\frac {\arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{4}+\frac {\arctan \left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{6}\)	\(36\)
risch	\(-\frac {\arctan \left (\frac {x}{2}\right )}{12}+\frac {\arctan \relax (x )}{6}-\frac {\arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{4}+\frac {\arctan \left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{6}\)	\(36\)

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,method=_RETURNVERBOSE)

[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 = 0.99, size = 35, normalized size = 0.69 \[ \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{12} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{6} \, \arctan \relax (x) \]

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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mupad [B]  time = 0.24, size = 35, normalized size = 0.69 \[ \frac {\mathrm {atan}\relax (x)}{6}-\frac {\mathrm {atan}\left (\frac {x}{2}\right )}{12}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{4}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,x}{3}\right )}{6} \]

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]

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

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sympy [A]  time = 0.44, size = 44, normalized size = 0.86 \[ - \frac {\operatorname {atan}{\left (\frac {x}{2} \right )}}{12} + \frac {\operatorname {atan}{\relax (x )}}{6} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{4} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} x}{3} \right )}}{6} \]

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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