3.705 \(\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^3 (c+a^2 c x^2)} \, dx\)

Optimal. Leaf size=72 \[ -\frac{i a^2 \text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{x (a x+i)},x\right )}{c}+\frac{\text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{x^3},x\right )}{c}+\frac{2 i a^2 \tan ^{-1}(a x)^{3/2}}{3 c} \]

[Out]

(((2*I)/3)*a^2*ArcTan[a*x]^(3/2))/c + Unintegrable[Sqrt[ArcTan[a*x]]/x^3, x]/c - (I*a^2*Unintegrable[Sqrt[ArcT
an[a*x]]/(x*(I + a*x)), x])/c

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Rubi [A]  time = 0.193768, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx \]

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*}

Mathematica [A]  time = 2.15093, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx \]

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 = 0.605, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ({a}^{2}c{x}^{2}+c \right ) }\sqrt{\arctan \left ( ax \right ) }}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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: UnboundLocalError

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

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{3}}\,{d x} \end{align*}

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]

integrate(sqrt(arctan(a*x))/((a^2*c*x^2 + c)*x^3), x)