3.8.20 \(\int \frac {\sqrt {\text {ArcTan}(a x)}}{x (c+a^2 c x^2)^3} \, dx\) [720]

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {\sqrt {\text {ArcTan}(a x)}}{x \left (c+a^2 c x^2\right )^3},x\right ) \]

[Out]

Unintegrable(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^3,x)

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Rubi [A]
time = 0.04, 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 = {} \begin {gather*} \int \frac {\sqrt {\text {ArcTan}(a x)}}{x \left (c+a^2 c x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^3),x]

[Out]

Defer[Int][Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^3), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^3} \, dx &=\int \frac {\sqrt {\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^3} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\text {ArcTan}(a x)}}{x \left (c+a^2 c x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^3),x]

[Out]

Integrate[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^3), x]

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Maple [A]
time = 1.13, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\arctan \left (a x \right )}}{x \left (a^{2} c \,x^{2}+c \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^3,x)

[Out]

int(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^3,x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^3,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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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{6} x^{7} + 3 a^{4} x^{5} + 3 a^{2} x^{3} + x}\, dx}{c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)**(1/2)/x/(a**2*c*x**2+c)**3,x)

[Out]

Integral(sqrt(atan(a*x))/(a**6*x**7 + 3*a**4*x**5 + 3*a**2*x**3 + x), x)/c**3

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(1/2)/x/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a*x)^(1/2)/(x*(c + a^2*c*x^2)^3),x)

[Out]

int(atan(a*x)^(1/2)/(x*(c + a^2*c*x^2)^3), x)

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