\(\int \frac {1}{(1-c^2 x^2)^{5/2} (a+b \text {arccosh}(c x))^2} \, dx\) [360]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x}}{b c \left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))}-\frac {4 c \sqrt {-1+c x} \text {Int}\left (\frac {x}{\left (-1+c^2 x^2\right )^3 (a+b \text {arccosh}(c x))},x\right )}{b \sqrt {1-c x}} \]

[Out]

-(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))-4*c*(c*x-1)^(1/2)*Unintegrable(x/(c^2*x
^2-1)^3/(a+b*arccosh(c*x)),x)/b/(-c*x+1)^(1/2)

Rubi [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

-((Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x]))) - (4*c*Sqrt[-1 + c*x]*Defer[I
nt][x/((-1 + c^2*x^2)^3*(a + b*ArcCosh[c*x])), x])/(b*Sqrt[1 - c*x])

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {-1+c x} \sqrt {1+c x}}{b c \left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))}-\frac {\left (4 c \sqrt {-1+c x}\right ) \int \frac {x}{(-1+c x)^3 (1+c x)^3 (a+b \text {arccosh}(c x))} \, dx}{b \sqrt {1-c x}} \\ & = -\frac {\sqrt {-1+c x} \sqrt {1+c x}}{b c \left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))}-\frac {\left (4 c \sqrt {-1+c x}\right ) \int \frac {x}{\left (-1+c^2 x^2\right )^3 (a+b \text {arccosh}(c x))} \, dx}{b \sqrt {1-c x}} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.65 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

int(1/(-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 = 141, normalized size of antiderivative = 5.64 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 1.33 (sec) , antiderivative size = 609, normalized size of antiderivative = 24.36 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(((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)) - integr
ate((4*c^4*x^4 - 3*c^2*x^2 + (4*c^2*x^2 - 1)*(c*x + 1)*(c*x - 1) + 4*(2*c^3*x^3 - c*x)*sqrt(c*x + 1)*sqrt(c*x
- 1) - 1)/(((b^2*c^6*x^6 - 2*b^2*c^4*x^4 + b^2*c^2*x^2)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(b^2*c^7*x^7 - 3*b^2*c^5
*x^5 + 3*b^2*c^3*x^3 - b^2*c*x)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^8*x^8 - 4*b^2*c^6*x^6 + 6*b^2*c^4*x^4 - 4*b^2
*c^2*x^2 + b^2)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((a*b*c^6*x^6 - 2*a*b*c
^4*x^4 + a*b*c^2*x^2)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(a*b*c^7*x^7 - 3*a*b*c^5*x^5 + 3*a*b*c^3*x^3 - a*b*c*x)*(c
*x + 1)*sqrt(c*x - 1) + (a*b*c^8*x^8 - 4*a*b*c^6*x^6 + 6*a*b*c^4*x^4 - 4*a*b*c^2*x^2 + a*b)*sqrt(c*x + 1))*sqr
t(-c*x + 1)), x)

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.51 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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