3.1.31 \(\int \frac {1}{c^2+x^2} \, dx\) [31]

Optimal. Leaf size=10 \[ \frac {\tan ^{-1}\left (\frac {x}{c}\right )}{c} \]

[Out]

arctan(x/c)/c

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {209} \begin {gather*} \frac {\tan ^{-1}\left (\frac {x}{c}\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c^2 + x^2)^(-1),x]

[Out]

ArcTan[x/c]/c

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

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {x}{c}\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c^2 + x^2)^(-1),x]

[Out]

ArcTan[x/c]/c

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Mathics [C] Result contains complex when optimal does not.
time = 1.97, size = 20, normalized size = 2.00 \begin {gather*} \frac {\frac {I}{2} \left (\text {Log}\left [I c+x\right ]-\text {Log}\left [-I c+x\right ]\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(c^2+x^2),x]')

[Out]

I / 2 (Log[I c + x] - Log[-I c + x]) / c

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Maple [A]
time = 0.03, size = 11, normalized size = 1.10

method	result	size
default	\(\frac {\arctan \left (\frac {x}{c}\right )}{c}\)	\(11\)
risch	\(\frac {\arctan \left (\frac {x}{c}\right )}{c}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x/c)/c

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Maxima [A]
time = 0.35, size = 10, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (\frac {x}{c}\right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x/c)/c

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Fricas [A]
time = 0.32, size = 10, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (\frac {x}{c}\right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x/c)/c

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Sympy [C] Result contains complex when optimal does not.
time = 0.06, size = 20, normalized size = 2.00 \begin {gather*} \frac {- \frac {i \log {\left (- i c + x \right )}}{2} + \frac {i \log {\left (i c + x \right )}}{2}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-I*log(-I*c + x)/2 + I*log(I*c + x)/2)/c

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Giac [A]
time = 0.00, size = 11, normalized size = 1.10 \begin {gather*} \frac {2 \arctan \left (\frac {x}{c}\right )}{2 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c^2+x^2),x)

[Out]

arctan(x/c)/c

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Mupad [B]
time = 0.04, size = 10, normalized size = 1.00 \begin {gather*} \frac {\mathrm {atan}\left (\frac {x}{c}\right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c^2 + x^2),x)

[Out]

atan(x/c)/c

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