\(\int \frac {1}{(c+a^2 c x^2)^2 \text {arcsinh}(a x)} \, dx\) [354]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)} \, dx=\text {Int}\left (\frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 19, 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 = {} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)} \, dx=\int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)} \, dx=\int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} \operatorname {arsinh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.98 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)} \, dx=\frac {\int \frac {1}{a^{4} x^{4} \operatorname {asinh}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {asinh}{\left (a x \right )} + \operatorname {asinh}{\left (a x \right )}}\, dx}{c^{2}} \]

[In]

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

[Out]

Integral(1/(a**4*x**4*asinh(a*x) + 2*a**2*x**2*asinh(a*x) + asinh(a*x)), x)/c**2

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} \operatorname {arsinh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} \operatorname {arsinh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.54 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)} \, dx=\int \frac {1}{\mathrm {asinh}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

[In]

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

[Out]

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