\(\int \frac {(a+b \text {arccosh}(c x))^n}{(d-c^2 d x^2)^{3/2}} \, dx\) [448]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Int}\left (\frac {(a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^{3/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {(a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {(a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {(a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.38 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

Sympy [N/A]

Not integrable

Time = 49.66 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral((a + b*acosh(c*x))**n/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)

Maxima [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))^n/(-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)

Giac [N/A]

Not integrable

Time = 12.62 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))^n/(-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)

Mupad [N/A]

Not integrable

Time = 3.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c x))^n}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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