\(\int \frac {1}{(c-a^2 c x^2)^{5/2} \sqrt {\text {arccosh}(a x)}} \, dx\) [407]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\text {arccosh}(a x)}} \, dx=\text {Int}\left (\frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\text {arccosh}(a x)}},x\right ) \]

[Out]

Unintegrable(1/(-a^2*c*x^2+c)^(5/2)/arccosh(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\text {arccosh}(a x)}} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\text {arccosh}(a x)}} \, dx \]

[In]

Int[1/((c - a^2*c*x^2)^(5/2)*Sqrt[ArcCosh[a*x]]),x]

[Out]

Defer[Int][1/((c - a^2*c*x^2)^(5/2)*Sqrt[ArcCosh[a*x]]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\text {arccosh}(a x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 5.84 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\text {arccosh}(a x)}} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\text {arccosh}(a x)}} \, dx \]

[In]

Integrate[1/((c - a^2*c*x^2)^(5/2)*Sqrt[ArcCosh[a*x]]),x]

[Out]

Integrate[1/((c - a^2*c*x^2)^(5/2)*Sqrt[ArcCosh[a*x]]), x]

Maple [N/A] (verified)

Not integrable

Time = 1.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

\[\int \frac {1}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {\operatorname {arccosh}\left (a x \right )}}d x\]

[In]

int(1/(-a^2*c*x^2+c)^(5/2)/arccosh(a*x)^(1/2),x)

[Out]

int(1/(-a^2*c*x^2+c)^(5/2)/arccosh(a*x)^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\text {arccosh}(a x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(-a^2*c*x^2+c)^(5/2)/arccosh(a*x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\text {arccosh}(a x)}} \, dx=\text {Timed out} \]

[In]

integrate(1/(-a**2*c*x**2+c)**(5/2)/acosh(a*x)**(1/2),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\text {arccosh}(a x)}} \, dx=\int { \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\operatorname {arcosh}\left (a x\right )}} \,d x } \]

[In]

integrate(1/(-a^2*c*x^2+c)^(5/2)/arccosh(a*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((-a^2*c*x^2 + c)^(5/2)*sqrt(arccosh(a*x))), x)

Giac [N/A]

Not integrable

Time = 2.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\text {arccosh}(a x)}} \, dx=\int { \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\operatorname {arcosh}\left (a x\right )}} \,d x } \]

[In]

integrate(1/(-a^2*c*x^2+c)^(5/2)/arccosh(a*x)^(1/2),x, algorithm="giac")

[Out]

integrate(1/((-a^2*c*x^2 + c)^(5/2)*sqrt(arccosh(a*x))), x)

Mupad [N/A]

Not integrable

Time = 2.88 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\text {arccosh}(a x)}} \, dx=\int \frac {1}{\sqrt {\mathrm {acosh}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]

[In]

int(1/(acosh(a*x)^(1/2)*(c - a^2*c*x^2)^(5/2)),x)

[Out]

int(1/(acosh(a*x)^(1/2)*(c - a^2*c*x^2)^(5/2)), x)