\(\int \frac {1}{1+x^2} \, dx\) [88]

Optimal result

Integrand size = 7, antiderivative size = 2 \[ \int \frac {1}{1+x^2} \, dx=\arctan (x) \]

[Out]

arctan(x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {209} \[ \int \frac {1}{1+x^2} \, dx=\arctan (x) \]

[In]

Int[(1 + x^2)^(-1),x]

[Out]

ArcTan[x]

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])

Rubi steps \begin{align*} \text {integral}& = \arctan (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+x^2} \, dx=\arctan (x) \]

[In]

Integrate[(1 + x^2)^(-1),x]

[Out]

ArcTan[x]

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.50

method	result	size
default	\(\arctan \left (x \right )\)	\(3\)
meijerg	\(\arctan \left (x \right )\)	\(3\)
risch	\(\arctan \left (x \right )\)	\(3\)
parallelrisch	\(\frac {i \ln \left (i+x \right )}{2}-\frac {i \ln \left (x -i\right )}{2}\)	\(18\)

[In]

int(1/(x^2+1),x,method=_RETURNVERBOSE)

[Out]

arctan(x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+x^2} \, dx=\arctan \left (x\right ) \]

[In]

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

[Out]

arctan(x)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+x^2} \, dx=\operatorname {atan}{\left (x \right )} \]

[In]

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

[Out]

atan(x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+x^2} \, dx=\arctan \left (x\right ) \]

[In]

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

[Out]

arctan(x)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+x^2} \, dx=\arctan \left (x\right ) \]

[In]

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

[Out]

arctan(x)

Mupad [B] (verification not implemented)

Time = 0.00 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+x^2} \, dx=\mathrm {atan}\left (x\right ) \]

[In]

int(1/(x^2 + 1),x)

[Out]

atan(x)