3.6.69 \(\int \frac {\sqrt {c+a^2 c x^2}}{x \text {ArcTan}(a x)^2} \, dx\) [569]

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

[Out]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c+a^2 c x^2}}{x \text {ArcTan}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 1.81, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} c \,x^{2}+c}}{x \arctan \left (a x \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c*(a**2*x**2 + 1))/(x*atan(a*x)**2), 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((a^2*c*x^2+c)^(1/2)/x/arctan(a*x)^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 {\sqrt {c\,a^2\,x^2+c}}{x\,{\mathrm {atan}\left (a\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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