3.2 \(\int \frac {1}{1+x^2} \, dx\)

Optimal. Leaf size=2 \[ \tan ^{-1}(x) \]

[Out]

arctan(x)

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Rubi [A]  time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {203} \[ \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)^(-1),x]

[Out]

ArcTan[x]

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}{1+x^2} \, dx &=\tan ^{-1}(x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 2, normalized size = 1.00 \[ \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)^(-1),x]

[Out]

ArcTan[x]

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fricas [A]  time = 0.40, size = 2, normalized size = 1.00 \[ \arctan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x)

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giac [A]  time = 0.79, size = 2, normalized size = 1.00 \[ \arctan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x)

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maple [A]  time = 0.00, size = 3, normalized size = 1.50 \[ \arctan \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x)

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maxima [A]  time = 0.96, size = 2, normalized size = 1.00 \[ \arctan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x)

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mupad [B]  time = 0.04, size = 2, normalized size = 1.00 \[ \mathrm {atan}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2 + 1),x)

[Out]

atan(x)

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sympy [A]  time = 0.09, size = 2, normalized size = 1.00 \[ \operatorname {atan}{\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(x)

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