\(\int \frac {1}{(1-c^2 x^2) (a+b \text {arcsinh}(\frac {\sqrt {1-c x}}{\sqrt {1+c x}}))} \, dx\) [346]

Optimal result

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

[Out]

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

[In]

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

[Out]

Defer[Int][1/((1 - c^2*x^2)*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), 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 )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.34 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=\int { -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 163.88 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.52 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=- \int \frac {1}{a c^{2} x^{2} - a + b c^{2} x^{2} \operatorname {asinh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )} - b \operatorname {asinh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=\int { -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 5.65 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=\int { -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.86 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=-\int \frac {1}{\left (a+b\,\mathrm {asinh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )\,\left (c^2\,x^2-1\right )} \,d x \]

[In]

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

[Out]

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