\(\int \frac {1}{(d+e x^2) (a+b \text {arcsinh}(c x))} \, dx\) [622]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))} \, dx=\text {Int}\left (\frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.48 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integral(1/(a*e*x^2 + a*d + (b*e*x^2 + b*d)*arcsinh(c*x)), x)

Sympy [N/A]

Not integrable

Time = 3.58 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.62 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )} \,d x \]

[In]

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

[Out]

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