3.7.56 \(\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3} \, dx\) [656]

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3},x\right ) \]

[Out]

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

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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 {1}{x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.46, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \arctan \left (a x \right )^{3} \sqrt {a^{2} c \,x^{2}+c}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/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^2*arctan(a*x)^3), x)

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Fricas [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(1/x^2/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^4 + c*x^2)*arctan(a*x)^3), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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