\(\int \frac {1}{x^2 (1-c^2 x^2)^{3/2} (a+b \text {arccosh}(c x))^2} \, dx\) [355]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\text {Int}\left (\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2},x\right ) \]

[Out]

Unintegrable(1/x^2/(-c^2*x^2+1)^(3/2)/(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 = {} \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 17.59 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.07 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 1.29 (sec) , antiderivative size = 538, normalized size of antiderivative = 19.21 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.57 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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