\(\int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{(d-c^2 d x^2)^{3/2}} \, dx\) [237]

Optimal result

Integrand size = 31, antiderivative size = 31 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Int}\left (\frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 31, 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 = {} \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 4.46 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 3.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.26 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 26.56 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.38 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (f\,x\right )}^m}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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