\(\int \frac {1}{(1-c^2 x^2) (a+b \text {arcsinh}(\frac {\sqrt {1-c x}}{\sqrt {1+c x}}))^2} \, dx\) [347]

Optimal result

Integrand size = 40, antiderivative size = 40 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 40, 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 = {} \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.28 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int { -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 474, normalized size of antiderivative = 11.85 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int { -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-4*(sqrt(2) + sqrt(-c*x + 1))/(2*sqrt(2)*a*b*c^2*x - 2*sqrt(2)*a*b*c - 4*sqrt(-c*x + 1)*a*b*c - (sqrt(2)*b^2*c
^2*x - sqrt(2)*b^2*c - 2*sqrt(-c*x + 1)*b^2*c)*log(c*x + 1) + 2*(sqrt(2)*b^2*c^2*x - sqrt(2)*b^2*c - 2*sqrt(-c
*x + 1)*b^2*c)*log(sqrt(2) + sqrt(-c*x + 1))) - integrate((4*c*x + (sqrt(2)*c*x - 3*sqrt(2))*sqrt(-c*x + 1) -
4)/(2*a*b*c^3*x^3 - 6*a*b*c^2*x^2 + 6*a*b*c*x - 4*(a*b*c*x - a*b)*(c*x - 1) - 2*a*b - (b^2*c^3*x^3 - 3*b^2*c^2
*x^2 + 3*b^2*c*x - 2*(b^2*c*x - b^2)*(c*x - 1) - b^2 - 2*(sqrt(2)*b^2*c^2*x^2 - 2*sqrt(2)*b^2*c*x + sqrt(2)*b^
2)*sqrt(-c*x + 1))*log(c*x + 1) + 2*(b^2*c^3*x^3 - 3*b^2*c^2*x^2 + 3*b^2*c*x - 2*(b^2*c*x - b^2)*(c*x - 1) - b
^2 - 2*(sqrt(2)*b^2*c^2*x^2 - 2*sqrt(2)*b^2*c*x + sqrt(2)*b^2)*sqrt(-c*x + 1))*log(sqrt(2) + sqrt(-c*x + 1)) -
4*(sqrt(2)*a*b*c^2*x^2 - 2*sqrt(2)*a*b*c*x + sqrt(2)*a*b)*sqrt(-c*x + 1)), x)

Giac [N/A]

Not integrable

Time = 20.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int { -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integrate(-1/((c^2*x^2 - 1)*(b*arcsinh(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a)^2), x)

Mupad [N/A]

Not integrable

Time = 3.63 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=-\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2\,\left (c^2\,x^2-1\right )} \,d x \]

[In]

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

[Out]

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