\(\int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx\) [395]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\text {Int}\left (\frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 27, 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 = {} \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {1}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.93 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \sqrt {c^{2} x^{2} + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {1}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {1}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.80 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {c^2\,x^2+1}} \,d x \]

[In]

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

[Out]

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