\(\int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx\) [59]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx=\text {Int}\left (\sqrt {c+d x^2} \cot ^{-1}(a x),x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx=\int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 4.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx=\int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.76 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

\[\int \sqrt {d \,x^{2}+c}\, \operatorname {arccot}\left (a x \right )d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx=\int { \sqrt {d x^{2} + c} \operatorname {arccot}\left (a x\right ) \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 4.39 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx=\int \sqrt {c + d x^{2}} \operatorname {acot}{\left (a x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx=\text {Exception raised: ValueError} \]

[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 detail

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx=\int { \sqrt {d x^{2} + c} \operatorname {arccot}\left (a x\right ) \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {c+d x^2} \cot ^{-1}(a x) \, dx=\int \mathrm {acot}\left (a\,x\right )\,\sqrt {d\,x^2+c} \,d x \]

[In]

int(acot(a*x)*(c + d*x^2)^(1/2),x)

[Out]

int(acot(a*x)*(c + d*x^2)^(1/2), x)