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

Optimal. Leaf size=46 \[ \frac {x \tan ^{-1}(a x)^{n+1}}{a c (n+1)}-\frac {\text {Int}\left (\tan ^{-1}(a x)^{n+1},x\right )}{a c (n+1)} \]

[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 = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.95, size = 0, normalized size = 0.00 \[ \int \frac {x \tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx \]

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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fricas [A]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \arctan \left (a x\right )^{n}}{a^{2} c x^{2} + c}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [A]  time = 0.80, size = 0, normalized size = 0.00 \[ \int \frac {x \arctan \left (a x \right )^{n}}{a^{2} c \,x^{2}+c}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^n}{c\,a^2\,x^2+c} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x \operatorname {atan}^{n}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]

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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