\(\int \frac {e^{\arctan (a x)}}{c+a^2 c x^2} \, dx\) [248]

Optimal result

Integrand size = 19, antiderivative size = 13 \[ \int \frac {e^{\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{\arctan (a x)}}{a c} \]

[Out]

exp(arctan(a*x))/a/c

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {5179} \[ \int \frac {e^{\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{\arctan (a x)}}{a c} \]

[In]

Int[E^ArcTan[a*x]/(c + a^2*c*x^2),x]

[Out]

E^ArcTan[a*x]/(a*c)

Rule 5179

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcTan[a*x])/(a*c*n), x] /; Fre
eQ[{a, c, d, n}, x] && EqQ[d, a^2*c]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{\arctan (a x)}}{a c} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.69 \[ \int \frac {e^{\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {(1-i a x)^{\frac {i}{2}} (1+i a x)^{-\frac {i}{2}}}{a c} \]

[In]

Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2),x]

[Out]

(1 - I*a*x)^(I/2)/(a*c*(1 + I*a*x)^(I/2))

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00

method	result	size
gosper	\(\frac {{\mathrm e}^{\arctan \left (a x \right )}}{a c}\)	\(13\)
parallelrisch	\(\frac {{\mathrm e}^{\arctan \left (a x \right )}}{a c}\)	\(13\)
risch	\(\frac {\left (-i a x +1\right )^{\frac {i}{2}} \left (i a x +1\right )^{-\frac {i}{2}}}{a c}\)	\(28\)

[In]

int(exp(arctan(a*x))/(a^2*c*x^2+c),x,method=_RETURNVERBOSE)

[Out]

exp(arctan(a*x))/a/c

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{\left (\arctan \left (a x\right )\right )}}{a c} \]

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c),x, algorithm="fricas")

[Out]

e^(arctan(a*x))/(a*c)

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\arctan (a x)}}{c+a^2 c x^2} \, dx=\begin {cases} \frac {e^{\operatorname {atan}{\left (a x \right )}}}{a c} & \text {for}\: a \neq 0 \\\frac {x}{c} & \text {otherwise} \end {cases} \]

[In]

integrate(exp(atan(a*x))/(a**2*c*x**2+c),x)

[Out]

Piecewise((exp(atan(a*x))/(a*c), Ne(a, 0)), (x/c, True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{\left (\arctan \left (a x\right )\right )}}{a c} \]

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c),x, algorithm="maxima")

[Out]

e^(arctan(a*x))/(a*c)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{\left (\arctan \left (a x\right )\right )}}{a c} \]

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

e^(arctan(a*x))/(a*c)

Mupad [B] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}}{a\,c} \]

[In]

int(exp(atan(a*x))/(c + a^2*c*x^2),x)

[Out]

exp(atan(a*x))/(a*c)