\(\int \frac {(f x)^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx\) [365]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.13 (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 = {} \[ \int \frac {(f x)^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {(f x)^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {(f x)^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.36 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.73 \[ \int \frac {(f x)^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\left (f x\right )^{m}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 24.71 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {(f x)^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (f x\right )^{m}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 544, normalized size of antiderivative = 18.13 \[ \int \frac {(f x)^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\left (f x\right )^{m}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-((c^2*f^m*x^2 - f^m)*sqrt(c*x + 1)*sqrt(c*x - 1)*x^m + (c^3*f^m*x^3 - c*f^m*x)*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*
x^3 - c*f^m*(m - 1)*x)*(c*x + 1)*(c*x - 1)*x^m + (2*c^4*f^m*m*x^4 - 3*c^2*f^m*m*x^2 + f^m*m)*sqrt(c*x + 1)*sqr
t(c*x - 1)*x^m + (c^5*f^m*m*x^5 - c^3*f^m*(2*m + 1)*x^3 + c*f^m*(m + 1)*x)*x^m)/(((c*x + 1)^(3/2)*(c*x - 1)*b^
2*c^3*x^3 + 2*(b^2*c^4*x^4 - b^2*c^2*x^2)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^5 - 2*b^2*c^3*x^3 + b^2*c*x)*sq
rt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)^(3/2)*(c*x - 1)*a*b*c^3*x^3 +
2*(a*b*c^4*x^4 - a*b*c^2*x^2)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c^5*x^5 - 2*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 = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\left (f x\right )^{m}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.71 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {{\left (f\,x\right )}^m}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \]

[In]

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

[Out]

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