\(\int \frac {(f x)^m (1-c^2 x^2)^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx\) [363]

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 = 1.61 (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 = 2.60 (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 = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {(f x)^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} \left (f x\right )^{m}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(-(c^2*x^2 - 1)*sqrt(-c^2*x^2 + 1)*(f*x)^m/(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2), 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 = 1.95 (sec) , antiderivative size = 577, normalized size of antiderivative = 19.23 \[ \int \frac {(f x)^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} \left (f x\right )^{m}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

((c^4*f^m*x^4 - 2*c^2*f^m*x^2 + f^m)*(c*x + 1)*sqrt(c*x - 1)*x^m + (c^5*f^m*x^5 - 2*c^3*f^m*x^3 + c*f^m*x)*sqr
t(c*x + 1)*x^m)*sqrt(-c*x + 1)/(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + s
qrt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x - b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) - integrate(((c^5*f^m*(m
+ 4)*x^5 - c^3*f^m*(2*m + 3)*x^3 + c*f^m*(m - 1)*x)*(c*x + 1)^(3/2)*(c*x - 1)*x^m + (2*c^6*f^m*(m + 4)*x^6 -
c^4*f^m*(5*m + 12)*x^4 + 4*c^2*f^m*(m + 1)*x^2 - f^m*m)*(c*x + 1)*sqrt(c*x - 1)*x^m + (c^7*f^m*(m + 4)*x^7 - 3
*c^5*f^m*(m + 3)*x^5 + 3*c^3*f^m*(m + 2)*x^3 - c*f^m*(m + 1)*x)*sqrt(c*x + 1)*x^m)*sqrt(-c*x + 1)/(a*b*c^5*x^5
+ (c*x + 1)*(c*x - 1)*a*b*c^3*x^3 - 2*a*b*c^3*x^3 + a*b*c*x + 2*(a*b*c^4*x^4 - a*b*c^2*x^2)*sqrt(c*x + 1)*sqr
t(c*x - 1) + (b^2*c^5*x^5 + (c*x + 1)*(c*x - 1)*b^2*c^3*x^3 - 2*b^2*c^3*x^3 + b^2*c*x + 2*(b^2*c^4*x^4 - b^2*c
^2*x^2)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 3.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {{\left (f\,x\right )}^m\,{\left (1-c^2\,x^2\right )}^{3/2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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