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

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

[Out]

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

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Rubi [A]  time = 0.10, 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 = {} \[ \int \frac {\sqrt {c+a^2 c x^2}}{x \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.36, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c+a^2 c x^2}}{x \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c}}{x \arctan \left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 1.66, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a^{2} c \,x^{2}+c}}{x \arctan \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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