\(\int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{(d-c^2 d x^2)^2} \, dx\) [456]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^2} \, dx=\text {Int}\left (\frac {(f x)^m (a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 29, 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 = {} \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.66 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {\left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {\left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 12.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {\left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.42 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (f\,x\right )}^m}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]

[In]

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

[Out]

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