\(\int \frac {(a+b \text {arccosh}(c x))^n}{x (d-c^2 d x^2)^{3/2}} \, dx\) [449]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Int}\left (\frac {(a+b \text {arccosh}(c x))^n}{x \left (d-c^2 d x^2\right )^{3/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.11 (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 = {} \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {(a+b \text {arccosh}(c x))^n}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.67 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {(a+b \text {arccosh}(c x))^n}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.52 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

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

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)^n/(c^4*d^2*x^5 - 2*c^2*d^2*x^3 + d^2*x), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 12.58 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.70 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{x\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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