\(\int \frac {\sqrt {d+e x^2}}{a+b \text {arccosh}(c x)} \, dx\) [539]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arccosh}(c x)} \, dx=\text {Int}\left (\frac {\sqrt {d+e x^2}}{a+b \text {arccosh}(c x)},x\right ) \]

[Out]

Unintegrable((e*x^2+d)^(1/2)/(a+b*arccosh(c*x)),x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {\sqrt {d+e x^2}}{a+b \text {arccosh}(c x)} \, dx \]

[In]

Int[Sqrt[d + e*x^2]/(a + b*ArcCosh[c*x]),x]

[Out]

Defer[Int][Sqrt[d + e*x^2]/(a + b*ArcCosh[c*x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {d+e x^2}}{a+b \text {arccosh}(c x)} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.98 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {\sqrt {d+e x^2}}{a+b \text {arccosh}(c x)} \, dx \]

[In]

Integrate[Sqrt[d + e*x^2]/(a + b*ArcCosh[c*x]),x]

[Out]

Integrate[Sqrt[d + e*x^2]/(a + b*ArcCosh[c*x]), x]

Maple [N/A] (verified)

Not integrable

Time = 1.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

int((e*x^2+d)^(1/2)/(a+b*arccosh(c*x)),x)

[Out]

int((e*x^2+d)^(1/2)/(a+b*arccosh(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

integral(sqrt(e*x^2 + d)/(b*arccosh(c*x) + a), x)

Sympy [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {\sqrt {d + e x^{2}}}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]

[In]

integrate((e*x**2+d)**(1/2)/(a+b*acosh(c*x)),x)

[Out]

Integral(sqrt(d + e*x**2)/(a + b*acosh(c*x)), x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

integrate(sqrt(e*x^2 + d)/(b*arccosh(c*x) + a), x)

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

integrate(sqrt(e*x^2 + d)/(b*arccosh(c*x) + a), x)

Mupad [N/A]

Not integrable

Time = 2.79 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {\sqrt {e\,x^2+d}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]

[In]

int((d + e*x^2)^(1/2)/(a + b*acosh(c*x)),x)

[Out]

int((d + e*x^2)^(1/2)/(a + b*acosh(c*x)), x)