\(\int \sqrt {d+e x^2} (a+b \text {arccosh}(c x))^2 \, dx\) [530]

Optimal result

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

[Out]

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

[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 \sqrt {d+e x^2} (a+b \text {arccosh}(c x))^2 \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 15.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \sqrt {d+e x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int \sqrt {d+e x^2} (a+b \text {arccosh}(c x))^2 \, dx \]

[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 = 0.67 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \sqrt {d+e x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.64 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \sqrt {d+e x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2} \sqrt {d + e x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((a+b*arccosh(c*x))^2*(e*x^2+d)^(1/2),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(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \sqrt {d+e x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \sqrt {d+e x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {e\,x^2+d} \,d x \]

[In]

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

[Out]

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