\(\int \sqrt {c+d x^2} \coth ^{-1}(a x) \, dx\) [42]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int \sqrt {c+d x^2} \coth ^{-1}(a x) \, dx=\int \sqrt {c+d x^2} \coth ^{-1}(a x) \, dx \]

[In]

Int[Sqrt[c + d*x^2]*ArcCoth[a*x],x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 4.33 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \sqrt {c+d x^2} \coth ^{-1}(a x) \, dx=\int \sqrt {c+d x^2} \coth ^{-1}(a x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.39 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {c+d x^2} \coth ^{-1}(a x) \, dx=\int { \sqrt {d x^{2} + c} \operatorname {arcoth}\left (a x\right ) \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \sqrt {c+d x^2} \coth ^{-1}(a x) \, dx=\int \sqrt {c + d x^{2}} \operatorname {acoth}{\left (a x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {c+d x^2} \coth ^{-1}(a x) \, dx=\int { \sqrt {d x^{2} + c} \operatorname {arcoth}\left (a x\right ) \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {c+d x^2} \coth ^{-1}(a x) \, dx=\int { \sqrt {d x^{2} + c} \operatorname {arcoth}\left (a x\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {c+d x^2} \coth ^{-1}(a x) \, dx=\int \mathrm {acoth}\left (a\,x\right )\,\sqrt {d\,x^2+c} \,d x \]

[In]

int(acoth(a*x)*(c + d*x^2)^(1/2),x)

[Out]

int(acoth(a*x)*(c + d*x^2)^(1/2), x)