3.1111 \(\int \frac{x \tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx\)

Optimal. Leaf size=45 \[ \frac{x \tan ^{-1}(a x)^{n+1}}{a c (n+1)}-\frac{\text{Unintegrable}\left (\tan ^{-1}(a x)^{n+1},x\right )}{a c (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.084999, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x \tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx \]

Verification is Not applicable to the result.

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

Mathematica [A]  time = 0.89217, size = 0, normalized size = 0. \[ \int \frac{x \tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 0.602, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ( \arctan \left ( ax \right ) \right ) ^{n}}{{a}^{2}c{x}^{2}+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x \arctan \left (a x\right )^{n}}{a^{2} c x^{2} + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x \operatorname{atan}^{n}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \arctan \left (a x\right )^{n}}{a^{2} c x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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