\(\int \frac {1}{(1-c^2 x^2)^{3/2} (a+b \text {arccosh}(c x))} \, dx\) [307]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))} \, dx=\text {Int}\left (\frac {1}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {1}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {1}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {1}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.92 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integral(sqrt(-c^2*x^2 + 1)/(a*c^4*x^4 - 2*a*c^2*x^2 + (b*c^4*x^4 - 2*b*c^2*x^2 + b)*arccosh(c*x) + a), x)

Sympy [N/A]

Not integrable

Time = 6.76 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {1}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.72 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {1}{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (1-c^2\,x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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