\(\int \frac {(a+b \text {arccosh}(c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx\) [445]

Optimal result

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

[Out]

Unintegrable((a+b*arccosh(c*x))^n/x^2/(-c^2*d*x^2+d)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.10 (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 = {} \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {(a+b \text {arccosh}(c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {(a+b \text {arccosh}(c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.47 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

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

[In]

int((a+b*arccosh(c*x))^n/x^2/(-c^2*d*x^2+d)^(1/2),x)

[Out]

int((a+b*arccosh(c*x))^n/x^2/(-c^2*d*x^2+d)^(1/2),x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt {-c^{2} d x^{2} + d} x^{2}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))^n/x^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 31.69 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}}{x^{2} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt {-c^{2} d x^{2} + d} x^{2}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))^n/x^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 12.59 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt {-c^{2} d x^{2} + d} x^{2}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))^n/x^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{x^2\,\sqrt {d-c^2\,d\,x^2}} \,d x \]

[In]

int((a + b*acosh(c*x))^n/(x^2*(d - c^2*d*x^2)^(1/2)),x)

[Out]

int((a + b*acosh(c*x))^n/(x^2*(d - c^2*d*x^2)^(1/2)), x)