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

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{x \left (a^2 c x^2+c\right )^2},x\right ) \]

[Out]

Unintegrable(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^2,x)

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {\sqrt {\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt {\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^2} \, dx &=\int \frac {\sqrt {\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 1.84, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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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(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 3.16, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\arctan \left (a x \right )}}{x \left (a^{2} c \,x^{2}+c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^2,x)

[Out]

int(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^2,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(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^2,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 = -0.04 \[ \int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a*x)^(1/2)/(x*(c + a^2*c*x^2)^2),x)

[Out]

int(atan(a*x)^(1/2)/(x*(c + a^2*c*x^2)^2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{4} x^{5} + 2 a^{2} x^{3} + x}\, dx}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(atan(a*x))/(a**4*x**5 + 2*a**2*x**3 + x), x)/c**2

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