\(\int \frac {x^2}{(1-c^2 x^2)^{5/2} (a+b \text {arccosh}(c x))^2} \, dx\) [358]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\text {Int}\left (\frac {x^2}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2},x\right ) \]

[Out]

Unintegrable(x^2/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^2}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

Int[x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]

[Out]

Defer[Int][x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 8.99 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^2}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

Integrate[x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]

[Out]

Integrate[x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 1.43 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int \frac {x^{2}}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}d x\]

[In]

int(x^2/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x)

[Out]

int(x^2/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 144, normalized size of antiderivative = 5.14 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{2}}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*x^2 + 1)*x^2/(a^2*c^6*x^6 - 3*a^2*c^4*x^4 + 3*a^2*c^2*x^2 + (b^2*c^6*x^6 - 3*b^2*c^4*x^4 +
3*b^2*c^2*x^2 - b^2)*arccosh(c*x)^2 - a^2 + 2*(a*b*c^6*x^6 - 3*a*b*c^4*x^4 + 3*a*b*c^2*x^2 - a*b)*arccosh(c*x
)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\text {Timed out} \]

[In]

integrate(x**2/(-c**2*x**2+1)**(5/2)/(a+b*acosh(c*x))**2,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 1.40 (sec) , antiderivative size = 635, normalized size of antiderivative = 22.68 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{2}}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

-(c*x^3 + sqrt(c*x + 1)*sqrt(c*x - 1)*x^2)/(((b^2*c^4*x^3 - b^2*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^4
- 2*b^2*c^3*x^2 + b^2*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((a*b*c^4*x^3
- a*b*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c^5*x^4 - 2*a*b*c^3*x^2 + a*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)) -
integrate((2*c^5*x^6 + c^3*x^4 + (2*c^3*x^4 + c*x^2)*(c*x + 1)*(c*x - 1) - 3*c*x^2 + 2*(2*c^4*x^5 + c^2*x^3 -
x)*sqrt(c*x + 1)*sqrt(c*x - 1))/(((b^2*c^7*x^6 - 2*b^2*c^5*x^4 + b^2*c^3*x^2)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(b
^2*c^8*x^7 - 3*b^2*c^6*x^5 + 3*b^2*c^4*x^3 - b^2*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^9*x^8 - 4*b^2*c^7*x^6
+ 6*b^2*c^5*x^4 - 4*b^2*c^3*x^2 + b^2*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))
+ ((a*b*c^7*x^6 - 2*a*b*c^5*x^4 + a*b*c^3*x^2)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(a*b*c^8*x^7 - 3*a*b*c^6*x^5 + 3
*a*b*c^4*x^3 - a*b*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c^9*x^8 - 4*a*b*c^7*x^6 + 6*a*b*c^5*x^4 - 4*a*b*c^3*x
^2 + a*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)), x)

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{2}}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

integrate(x^2/((-c^2*x^2 + 1)^(5/2)*(b*arccosh(c*x) + a)^2), x)

Mupad [N/A]

Not integrable

Time = 3.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{5/2}} \,d x \]

[In]

int(x^2/((a + b*acosh(c*x))^2*(1 - c^2*x^2)^(5/2)),x)

[Out]

int(x^2/((a + b*acosh(c*x))^2*(1 - c^2*x^2)^(5/2)), x)