\(\int (f x)^m (a+b \text {arccosh}(c x))^n \, dx\) [454]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int (f x)^m (a+b \text {arccosh}(c x))^n \, dx=\text {Int}\left ((f x)^m (a+b \text {arccosh}(c x))^n,x\right ) \]

[Out]

Unintegrable((f*x)^m*(a+b*arccosh(c*x))^n,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int (f x)^m (a+b \text {arccosh}(c x))^n \, dx=\int (f x)^m (a+b \text {arccosh}(c x))^n \, dx \]

[In]

Int[(f*x)^m*(a + b*ArcCosh[c*x])^n,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (f x)^m (a+b \text {arccosh}(c x))^n \, dx=\int (f x)^m (a+b \text {arccosh}(c x))^n \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

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

[In]

int((f*x)^m*(a+b*arccosh(c*x))^n,x)

[Out]

int((f*x)^m*(a+b*arccosh(c*x))^n,x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (f x)^m (a+b \text {arccosh}(c x))^n \, dx=\int { \left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

integral((f*x)^m*(b*arccosh(c*x) + a)^n, x)

Sympy [N/A]

Not integrable

Time = 31.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int (f x)^m (a+b \text {arccosh}(c x))^n \, dx=\int \left (f x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}\, dx \]

[In]

integrate((f*x)**m*(a+b*acosh(c*x))**n,x)

[Out]

Integral((f*x)**m*(a + b*acosh(c*x))**n, x)

Maxima [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (f x)^m (a+b \text {arccosh}(c x))^n \, dx=\int { \left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

integrate((f*x)^m*(b*arccosh(c*x) + a)^n, x)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 2.94 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (f x)^m (a+b \text {arccosh}(c x))^n \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (f\,x\right )}^m \,d x \]

[In]

int((a + b*acosh(c*x))^n*(f*x)^m,x)

[Out]

int((a + b*acosh(c*x))^n*(f*x)^m, x)