\(\int \frac {1}{(d+e x^2)^2 (a+b \text {arccosh}(c x))^{3/2}} \, dx\) [569]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^{3/2}} \, dx=\text {Int}\left (\frac {1}{\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^{3/2}},x\right ) \]

[Out]

Unintegrable(1/(e*x^2+d)^2/(a+b*arccosh(c*x))^(3/2),x)

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^{3/2}} \, dx \]

[In]

Int[1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])^(3/2)),x]

[Out]

Defer[Int][1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])^(3/2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^{3/2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^{3/2}} \, dx \]

[In]

Integrate[1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])^(3/2)),x]

[Out]

Integrate[1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])^(3/2)), x]

Maple [N/A] (verified)

Not integrable

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

\[\int \frac {1}{\left (e \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]

[In]

int(1/(e*x^2+d)^2/(a+b*arccosh(c*x))^(3/2),x)

[Out]

int(1/(e*x^2+d)^2/(a+b*arccosh(c*x))^(3/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(e*x^2+d)^2/(a+b*arccosh(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x**2+d)**2/(a+b*acosh(c*x))**(3/2),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(e*x^2+d)^2/(a+b*arccosh(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((e*x^2 + d)^2*(b*arccosh(c*x) + a)^(3/2)), x)

Giac [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(e*x^2+d)^2/(a+b*arccosh(c*x))^(3/2),x, algorithm="giac")

[Out]

integrate(1/((e*x^2 + d)^2*(b*arccosh(c*x) + a)^(3/2)), x)

Mupad [N/A]

Not integrable

Time = 3.78 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}\,{\left (e\,x^2+d\right )}^2} \,d x \]

[In]

int(1/((a + b*acosh(c*x))^(3/2)*(d + e*x^2)^2),x)

[Out]

int(1/((a + b*acosh(c*x))^(3/2)*(d + e*x^2)^2), x)