\(\int \frac {1}{(c-a^2 c x^2)^{3/2} \text {arccosh}(a x)^{3/2}} \, dx\) [412]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \left (c-a^2 c x^2\right )^{3/2} \sqrt {\text {arccosh}(a x)}}+\frac {4 a \sqrt {-1+a x} \sqrt {1+a x} \text {Int}\left (\frac {x}{\left (-1+a^2 x^2\right )^2 \sqrt {\text {arccosh}(a x)}},x\right )}{c \sqrt {c-a^2 c x^2}} \]

[Out]

-2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(1/2)+4*a*(a*x-1)^(1/2)*(a*x+1)^(1/2)*Unint
egrable(x/(a^2*x^2-1)^2/arccosh(a*x)^(1/2),x)/c/(-a^2*c*x^2+c)^(1/2)

Rubi [N/A]

Not integrable

Time = 0.14 (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 = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{3/2}} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{3/2}} \, dx \]

[In]

Int[1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2)),x]

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*(c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]]) + (4*a*Sqrt[-1 + a*x]*Sqrt[1 +
a*x]*Defer[Int][x/((-1 + a^2*x^2)^2*Sqrt[ArcCosh[a*x]]), x])/(c*Sqrt[c - a^2*c*x^2])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \left (c-a^2 c x^2\right )^{3/2} \sqrt {\text {arccosh}(a x)}}+\frac {\left (4 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{(-1+a x)^2 (1+a x)^2 \sqrt {\text {arccosh}(a x)}} \, dx}{c \sqrt {c-a^2 c x^2}} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \left (c-a^2 c x^2\right )^{3/2} \sqrt {\text {arccosh}(a x)}}+\frac {\left (4 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\left (-1+a^2 x^2\right )^2 \sqrt {\text {arccosh}(a x)}} \, dx}{c \sqrt {c-a^2 c x^2}} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.76 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{3/2}} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{3/2}} \, dx \]

[In]

Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2)),x]

[Out]

Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2)), x]

Maple [N/A] (verified)

Not integrable

Time = 1.40 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

\[\int \frac {1}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}}}d x\]

[In]

int(1/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(3/2),x)

[Out]

int(1/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(3/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 170.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{3/2}} \, dx=\int \frac {1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(1/(-a**2*c*x**2+c)**(3/2)/acosh(a*x)**(3/2),x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((-a^2*c*x^2 + c)^(3/2)*arccosh(a*x)^(3/2)), x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((-a^2*c*x^2 + c)^(3/2)*arccosh(a*x)^(3/2)), x)

Mupad [N/A]

Not integrable

Time = 3.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}\,{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]

[In]

int(1/(acosh(a*x)^(3/2)*(c - a^2*c*x^2)^(3/2)),x)

[Out]

int(1/(acosh(a*x)^(3/2)*(c - a^2*c*x^2)^(3/2)), x)