\(\int \frac {x \arctan (a x)^n}{c+a^2 c x^2} \, dx\) [1111]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x \arctan (a x)^n}{c+a^2 c x^2} \, dx=\int \frac {x \arctan (a x)^n}{c+a^2 c x^2} \, dx \]

[In]

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

[Out]

(x*ArcTan[a*x]^(1 + n))/(a*c*(1 + n)) - Defer[Int][ArcTan[a*x]^(1 + n), x]/(a*c*(1 + n))

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

Mathematica [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x \arctan (a x)^n}{c+a^2 c x^2} \, dx=\int \frac {x \arctan (a x)^n}{c+a^2 c x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {x \arctan \left (a x \right )^{n}}{a^{2} c \,x^{2}+c}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x \arctan (a x)^n}{c+a^2 c x^2} \, dx=\int { \frac {x \arctan \left (a x\right )^{n}}{a^{2} c x^{2} + c} \,d x } \]

[In]

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

[Out]

integral(x*arctan(a*x)^n/(a^2*c*x^2 + c), x)

Sympy [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {x \arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {\int \frac {x \operatorname {atan}^{n}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x*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 [N/A]

Not integrable

Time = 54.74 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \frac {x \arctan (a x)^n}{c+a^2 c x^2} \, dx=\int { \frac {x \arctan \left (a x\right )^{n}}{a^{2} c x^{2} + c} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x \arctan (a x)^n}{c+a^2 c x^2} \, dx=\int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^n}{c\,a^2\,x^2+c} \,d x \]

[In]

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

[Out]

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