3.706 \(\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4 (c+a^2 c x^2)} \, dx\)

Optimal. Leaf size=59 \[ -\frac{a^2 \text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{x^2},x\right )}{c}+\frac{\text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{x^4},x\right )}{c}+\frac{2 a^3 \tan ^{-1}(a x)^{3/2}}{3 c} \]

[Out]

(2*a^3*ArcTan[a*x]^(3/2))/(3*c) + Unintegrable[Sqrt[ArcTan[a*x]]/x^4, x]/c - (a^2*Unintegrable[Sqrt[ArcTan[a*x
]]/x^2, x])/c

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Rubi [A]  time = 0.187223, 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 = {} \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[ArcTan[a*x]]/(x^4*(c + a^2*c*x^2)),x]

[Out]

(2*a^3*ArcTan[a*x]^(3/2))/(3*c) + Defer[Int][Sqrt[ArcTan[a*x]]/x^4, x]/c - (a^2*Defer[Int][Sqrt[ArcTan[a*x]]/x
^2, x])/c

Rubi steps

\begin{align*} \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx\right )+\frac{\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4} \, dx}{c}\\ &=a^4 \int \frac{\sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx+\frac{\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4} \, dx}{c}-\frac{a^2 \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2} \, dx}{c}\\ &=\frac{2 a^3 \tan ^{-1}(a x)^{3/2}}{3 c}+\frac{\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4} \, dx}{c}-\frac{a^2 \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2} \, dx}{c}\\ \end{align*}

Mathematica [A]  time = 6.02801, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[ArcTan[a*x]]/(x^4*(c + a^2*c*x^2)),x]

[Out]

Integrate[Sqrt[ArcTan[a*x]]/(x^4*(c + a^2*c*x^2)), x]

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Maple [A]  time = 0.5, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \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^4/(a^2*c*x^2+c),x)

[Out]

int(arctan(a*x)^(1/2)/x^4/(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^4/(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^4/(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^{6} + x^{4}}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)**(1/2)/x**4/(a**2*c*x**2+c),x)

[Out]

Integral(sqrt(atan(a*x))/(a**2*x**6 + x**4), 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^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(1/2)/x^4/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

integrate(sqrt(arctan(a*x))/((a^2*c*x^2 + c)*x^4), x)