\(\int \frac {1}{(c+a^2 c x^2) \text {arcsinh}(a x)^2} \, dx\) [409]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^2} \, dx=\int \frac {1}{\left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.66 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^2} \, dx=\int \frac {1}{\left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^2} \, dx=\frac {\int \frac {1}{a^{2} x^{2} \operatorname {asinh}^{2}{\left (a x \right )} + \operatorname {asinh}^{2}{\left (a x \right )}}\, dx}{c} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 226, normalized size of antiderivative = 11.89 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )} \operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

-(a*x + sqrt(a^2*x^2 + 1))/((a^3*c*x^2 + sqrt(a^2*x^2 + 1)*a^2*c*x + a*c)*log(a*x + sqrt(a^2*x^2 + 1))) - inte
grate((a^4*x^4 + (a^2*x^2 + 1)^2 + (2*a^3*x^3 + a*x)*sqrt(a^2*x^2 + 1) - 1)/((a^6*c*x^6 + 3*a^4*c*x^4 + 3*a^2*
c*x^2 + (a^4*c*x^4 + a^2*c*x^2)*(a^2*x^2 + 1) + 2*(a^5*c*x^5 + 2*a^3*c*x^3 + a*c*x)*sqrt(a^2*x^2 + 1) + c)*log
(a*x + sqrt(a^2*x^2 + 1))), x)

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )} \operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.84 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^2} \, dx=\int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[In]

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

[Out]

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