\(\int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx\) [1112]

Optimal result

Integrand size = 19, antiderivative size = 20 \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {\arctan (a x)^{1+n}}{a c (1+n)} \]

[Out]

arctan(a*x)^(1+n)/a/c/(1+n)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, 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 = {5004} \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {\arctan (a x)^{n+1}}{a c (n+1)} \]

[In]

Int[ArcTan[a*x]^n/(c + a^2*c*x^2),x]

[Out]

ArcTan[a*x]^(1 + n)/(a*c*(1 + n))

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\arctan (a x)^{1+n}}{a c (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {\arctan (a x)^{1+n}}{a c (1+n)} \]

[In]

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

[Out]

ArcTan[a*x]^(1 + n)/(a*c*(1 + n))

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05

method	result	size
default	\(\frac {\arctan \left (a x \right )^{1+n}}{a c \left (1+n \right )}\)	\(21\)
parallelrisch	\(\frac {\arctan \left (a x \right )^{n} \arctan \left (a x \right )}{c a \left (1+n \right )}\)	\(23\)
risch	\(\frac {i \left (\ln \left (-i a x +1\right )-\ln \left (i a x +1\right )\right ) \left (\frac {i \left (\ln \left (-i a x +1\right )-\ln \left (i a x +1\right )\right )}{2}\right )^{n}}{2 c a \left (1+n \right )}\)	\(58\)

[In]

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

[Out]

arctan(a*x)^(1+n)/a/c/(1+n)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {\arctan \left (a x\right )^{n} \arctan \left (a x\right )}{a c n + a c} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {\int \frac {\operatorname {atan}^{n}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]

[In]

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

[Out]

Integral(atan(a*x)**n/(a**2*x**2 + 1), x)/c

Maxima [F(-2)]

Exception generated. \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {\arctan \left (a x\right )^{n + 1}}{a c {\left (n + 1\right )}} \]

[In]

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

[Out]

arctan(a*x)^(n + 1)/(a*c*(n + 1))

Mupad [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {{\mathrm {atan}\left (a\,x\right )}^{n+1}}{a\,c\,\left (n+1\right )} \]

[In]

int(atan(a*x)^n/(c + a^2*c*x^2),x)

[Out]

atan(a*x)^(n + 1)/(a*c*(n + 1))