3.8.4 \(\int \frac {\sqrt {\text {ArcTan}(a x)}}{x^2 (c+a^2 c x^2)} \, dx\) [704]

Optimal. Leaf size=36 \[ -\frac {2 a \text {ArcTan}(a x)^{3/2}}{3 c}+\frac {\text {Int}\left (\frac {\sqrt {\text {ArcTan}(a x)}}{x^2},x\right )}{c} \]

[Out]

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

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Rubi [A]
time = 0.07, 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^2 \left (c+a^2 c x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\text {ArcTan}(a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.38, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\arctan \left (a x \right )}}{x^{2} \left (a^{2} c \,x^{2}+c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(arctan(a*x)^(1/2)/x^2/(a^2*c*x^2+c),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^2/(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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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^2/(a^2*c*x^2+c),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^{2} x^{4} + x^{2}}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^2/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x^2\,\left (c\,a^2\,x^2+c\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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