3.10.47 \(\int \frac {\sqrt {c+a^2 c x^2}}{x \sqrt {\text {ArcTan}(a x)}} \, dx\) [947]

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {\sqrt {c+a^2 c x^2}}{x \sqrt {\text {ArcTan}(a x)}},x\right ) \]

[Out]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.37, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c+a^2 c x^2}}{x \sqrt {\text {ArcTan}(a x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.88, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} c \,x^{2}+c}}{x \sqrt {\arctan \left (a x \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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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 \sqrt {\operatorname {atan}{\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)**(1/2),x)

[Out]

Integral(sqrt(c*(a**2*x**2 + 1))/(x*sqrt(atan(a*x))), 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)^(1/2),x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {c\,a^2\,x^2+c}}{x\,\sqrt {\mathrm {atan}\left (a\,x\right )}} \,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)^(1/2)),x)

[Out]

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

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