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

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {\sqrt {a^2 c x^2+c}}{\tan ^{-1}(a x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {\sqrt {c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c}}{\arctan \left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

sage0*x

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maple [A]  time = 1.25, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a^{2} c \,x^{2}+c}}{\arctan \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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