\(\int \frac {\sqrt {d+e x^2}}{(a+b \text {arccosh}(c x))^2} \, dx\) [548]

Optimal result

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

[Out]

Unintegrable((e*x^2+d)^(1/2)/(a+b*arccosh(c*x))^2,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 = {} \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {d+e x^2}}{(a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 10.79 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {d+e x^2}}{(a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.87 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {d + e x^{2}}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.81 (sec) , antiderivative size = 596, normalized size of antiderivative = 27.09 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-(c^3*x^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x)*sqrt(e*x^2 + d)/(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt
(c*x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x - b^2*c)*log(c*x + sqrt(c*x
+ 1)*sqrt(c*x - 1))) + integrate((2*c^5*e*x^6 + (c^5*d - 4*c^3*e)*x^4 + (2*c^3*e*x^4 + c^3*d*x^2 + c*d)*(c*x
+ 1)*(c*x - 1) - 2*(c^3*d - c*e)*x^2 + (4*c^4*e*x^5 + 2*(c^4*d - 2*c^2*e)*x^3 - (c^2*d - e)*x)*sqrt(c*x + 1)*s
qrt(c*x - 1) + c*d)*sqrt(e*x^2 + d)/(a*b*c^5*e*x^6 + (c^5*d - 2*c^3*e)*a*b*x^4 - (2*c^3*d - c*e)*a*b*x^2 + a*b
*c*d + (a*b*c^3*e*x^4 + a*b*c^3*d*x^2)*(c*x + 1)*(c*x - 1) + 2*(a*b*c^4*e*x^5 - a*b*c^2*d*x + (c^4*d - c^2*e)*
a*b*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*e*x^6 + (c^5*d - 2*c^3*e)*b^2*x^4 - (2*c^3*d - c*e)*b^2*x^2 +
b^2*c*d + (b^2*c^3*e*x^4 + b^2*c^3*d*x^2)*(c*x + 1)*(c*x - 1) + 2*(b^2*c^4*e*x^5 - b^2*c^2*d*x + (c^4*d - c^2*
e)*b^2*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {e\,x^2+d}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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