\(\int \frac {e^{n \arctan (a x)}}{c+a^2 c x^2} \, dx\) [343]

Optimal result

Integrand size = 21, antiderivative size = 18 \[ \int \frac {e^{n \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{n \arctan (a x)}}{a c n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5179} \[ \int \frac {e^{n \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{n \arctan (a x)}}{a c n} \]

[In]

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

[Out]

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

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^{n \arctan (a x)}}{a c n} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.33 \[ \int \frac {e^{n \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a c n} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

method	result	size
gosper	\(\frac {{\mathrm e}^{n \arctan \left (a x \right )}}{a c n}\)	\(18\)
parallelrisch	\(\frac {{\mathrm e}^{n \arctan \left (a x \right )}}{a c n}\)	\(18\)
risch	\(\frac {\left (-i a x +1\right )^{\frac {i n}{2}} \left (i a x +1\right )^{-\frac {i n}{2}}}{c a n}\)	\(35\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {e^{n \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{\left (n \arctan \left (a x\right )\right )}}{a c n} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).

Time = 0.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {e^{n \arctan (a x)}}{c+a^2 c x^2} \, dx=\begin {cases} \frac {x}{c} & \text {for}\: a = 0 \wedge \left (a = 0 \vee n = 0\right ) \\\frac {\operatorname {atan}{\left (a x \right )}}{a c} & \text {for}\: n = 0 \\\frac {e^{n \operatorname {atan}{\left (a x \right )}}}{a c n} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x/c, Eq(a, 0) & (Eq(a, 0) | Eq(n, 0))), (atan(a*x)/(a*c), Eq(n, 0)), (exp(n*atan(a*x))/(a*c*n), Tru
e))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {e^{n \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{\left (n \arctan \left (a x\right )\right )}}{a c n} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {e^{n \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{\left (n \arctan \left (a x\right )\right )}}{a c n} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {e^{n \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{a\,c\,n} \]

[In]

int(exp(n*atan(a*x))/(c + a^2*c*x^2),x)

[Out]

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