3.59 \(\int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx\)

Optimal. Leaf size=19 \[ \text {Int}\left (\cot ^{-1}(a x) \sqrt {c+d x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, 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 = {} \[ \int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[c + d*x^2]*ArcCot[a*x],x]

[Out]

Defer[Int][Sqrt[c + d*x^2]*ArcCot[a*x], x]

Rubi steps

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

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Mathematica [A]  time = 6.84, size = 0, normalized size = 0.00 \[ \int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[c + d*x^2]*ArcCot[a*x],x]

[Out]

Integrate[Sqrt[c + d*x^2]*ArcCot[a*x], x]

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fricas [A]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {d x^{2} + c} \operatorname {arccot}\left (a x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*x^2 + c)*arccot(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*x^2 + c)*arccot(a*x), x)

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maple [A]  time = 1.40, size = 0, normalized size = 0.00 \[ \int \sqrt {d \,x^{2}+c}\, \mathrm {arccot}\left (a x \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(d-a^2*c>0)', see `assume?` for
more details)Is d-a^2*c positive or negative?

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \mathrm {acot}\left (a\,x\right )\,\sqrt {d\,x^2+c} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(a*x)*(c + d*x^2)^(1/2),x)

[Out]

int(acot(a*x)*(c + d*x^2)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c + d*x**2)*acot(a*x), x)

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