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

Optimal. Leaf size=67 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right )}{2 a}-\frac{\sqrt{a^2 c x^2+c}}{2 a c x \tan ^{-1}(a x)^2} \]

[Out]

-Sqrt[c + a^2*c*x^2]/(2*a*c*x*ArcTan[a*x]^2) - Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(2*a
)

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Rubi [A]  time = 0.216458, 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 = {} \[ \int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3),x]

[Out]

-Sqrt[c + a^2*c*x^2]/(2*a*c*x*ArcTan[a*x]^2) - Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(2*a)

Rubi steps

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

Mathematica [A]  time = 3.57479, size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3),x]

[Out]

Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]

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Maple [A]  time = 0.838, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( \arctan \left ( ax \right ) \right ) ^{3}}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a^{2} c x^{2} + c} x \arctan \left (a x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c}}{{\left (a^{2} c x^{3} + c x\right )} \arctan \left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{c \left (a^{2} x^{2} + 1\right )} \operatorname{atan}^{3}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a^{2} c x^{2} + c} x \arctan \left (a x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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