\(\int \frac {1}{(c+a^2 c x^2)^2 \text {arcsinh}(a x)^2} \, dx\) [410]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^2} \, dx=-\frac {1}{a c^2 \left (1+a^2 x^2\right )^{3/2} \text {arcsinh}(a x)}-\frac {3 a \text {Int}\left (\frac {x}{\left (1+a^2 x^2\right )^{5/2} \text {arcsinh}(a x)},x\right )}{c^2} \]

[Out]

-1/a/c^2/(a^2*x^2+1)^(3/2)/arcsinh(a*x)-3*a*Unintegrable(x/(a^2*x^2+1)^(5/2)/arcsinh(a*x),x)/c^2

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^2} \, dx=\int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^2} \, dx \]

[In]

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

[Out]

-(1/(a*c^2*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x])) - (3*a*Defer[Int][x/((1 + a^2*x^2)^(5/2)*ArcSinh[a*x]), x])/c^2

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{a c^2 \left (1+a^2 x^2\right )^{3/2} \text {arcsinh}(a x)}-\frac {(3 a) \int \frac {x}{\left (1+a^2 x^2\right )^{5/2} \text {arcsinh}(a x)} \, dx}{c^2} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^2} \, dx=\int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.46 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.16 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^2} \, dx=\frac {\int \frac {1}{a^{4} x^{4} \operatorname {asinh}^{2}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {asinh}^{2}{\left (a x \right )} + \operatorname {asinh}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 325, normalized size of antiderivative = 17.11 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} \operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

-(a*x + sqrt(a^2*x^2 + 1))/((a^5*c^2*x^4 + 2*a^3*c^2*x^2 + a*c^2 + (a^4*c^2*x^3 + a^2*c^2*x)*sqrt(a^2*x^2 + 1)
)*log(a*x + sqrt(a^2*x^2 + 1))) - integrate((3*a^4*x^4 + 2*a^2*x^2 + (3*a^2*x^2 + 1)*(a^2*x^2 + 1) + 3*(2*a^3*
x^3 + a*x)*sqrt(a^2*x^2 + 1) - 1)/((a^8*c^2*x^8 + 4*a^6*c^2*x^6 + 6*a^4*c^2*x^4 + 4*a^2*c^2*x^2 + (a^6*c^2*x^6
+ 2*a^4*c^2*x^4 + a^2*c^2*x^2)*(a^2*x^2 + 1) + c^2 + 2*(a^7*c^2*x^7 + 3*a^5*c^2*x^5 + 3*a^3*c^2*x^3 + a*c^2*x
)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))), x)

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} \operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.91 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^2} \, dx=\int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

[In]

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

[Out]

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