\(\int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{d-c^2 d x^2} \, dx\) [455]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{d-c^2 d x^2} \, dx=\int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{d-c^2 d x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.55 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{d-c^2 d x^2} \, dx=\int { -\frac {\left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{c^{2} d x^{2} - d} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 79.71 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{d-c^2 d x^2} \, dx=- \frac {\int \frac {\left (f x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}}{c^{2} x^{2} - 1}\, dx}{d} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{d-c^2 d x^2} \, dx=\int { -\frac {\left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{c^{2} d x^{2} - d} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 13.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{d-c^2 d x^2} \, dx=\int { -\frac {\left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{c^{2} d x^{2} - d} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))^n}{d-c^2 d x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (f\,x\right )}^m}{d-c^2\,d\,x^2} \,d x \]

[In]

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

[Out]

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