\(\int \frac {(f x)^m}{(1-c^2 x^2)^{3/2} (a+b \text {arccosh}(c x))^2} \, dx\) [366]

Optimal result

Integrand size = 30, antiderivative size = 30 \[ \int \frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\text {Int}\left (\frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2},x\right ) \]

[Out]

Unintegrable((f*x)^m/(-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 = 30, 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 = {} \[ \int \frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {(f x)^m}{\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 = 1.67 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.63 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.60 \[ \int \frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\left (f x\right )^{m}}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 596, normalized size of antiderivative = 19.87 \[ \int \frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\left (f x\right )^{m}}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\left (f x\right )^{m}}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {{\left (f\,x\right )}^m}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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