3.8.5 \(\int \frac {\sqrt {\text {ArcTan}(a x)}}{x^3 (c+a^2 c x^2)} \, dx\) [705]

Optimal. Leaf size=74 \[ \frac {2 i a^2 \text {ArcTan}(a x)^{3/2}}{3 c}+\frac {\text {Int}\left (\frac {\sqrt {\text {ArcTan}(a x)}}{x^3},x\right )}{c}-\frac {i a^2 \text {Int}\left (\frac {\sqrt {\text {ArcTan}(a x)}}{x (i+a x)},x\right )}{c} \]

[Out]

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

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Rubi [A]
time = 0.13, 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^3 \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^3*(c + a^2*c*x^2)),x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.59, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\text {ArcTan}(a x)}}{x^3 \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^3*(c + a^2*c*x^2)),x]

[Out]

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

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Maple [A]
time = 2.48, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\arctan \left (a x \right )}}{x^{3} \left (a^{2} c \,x^{2}+c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(arctan(a*x)^(1/2)/x^3/(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^3/(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^3/(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^{5} + x^{3}}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(atan(a*x))/(a**2*x**5 + x**3), 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^3/(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 = -0.01 \begin {gather*} \int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x^3\,\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^3*(c + a^2*c*x^2)),x)

[Out]

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

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