\(\int \frac {1}{x^2 \text {arcsinh}(a x)^3} \, dx\) [66]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^3} \, dx=\text {Int}\left (\frac {1}{x^2 \text {arcsinh}(a x)^3},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 10, 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 = {} \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^3} \, dx=\int \frac {1}{x^2 \text {arcsinh}(a x)^3} \, dx \]

[In]

Int[1/(x^2*ArcSinh[a*x]^3),x]

[Out]

Defer[Int][1/(x^2*ArcSinh[a*x]^3), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 \text {arcsinh}(a x)^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.65 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^3} \, dx=\int \frac {1}{x^2 \text {arcsinh}(a x)^3} \, dx \]

[In]

Integrate[1/(x^2*ArcSinh[a*x]^3),x]

[Out]

Integrate[1/(x^2*ArcSinh[a*x]^3), x]

Maple [N/A] (verified)

Not integrable

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int \frac {1}{x^{2} \operatorname {arcsinh}\left (a x \right )^{3}}d x\]

[In]

int(1/x^2/arcsinh(a*x)^3,x)

[Out]

int(1/x^2/arcsinh(a*x)^3,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^3} \, dx=\int { \frac {1}{x^{2} \operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]

[In]

integrate(1/x^2/arcsinh(a*x)^3,x, algorithm="fricas")

[Out]

integral(1/(x^2*arcsinh(a*x)^3), x)

Sympy [N/A]

Not integrable

Time = 0.58 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^3} \, dx=\int \frac {1}{x^{2} \operatorname {asinh}^{3}{\left (a x \right )}}\, dx \]

[In]

integrate(1/x**2/asinh(a*x)**3,x)

[Out]

Integral(1/(x**2*asinh(a*x)**3), x)

Maxima [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 822, normalized size of antiderivative = 82.20 \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^3} \, dx=\int { \frac {1}{x^{2} \operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]

[In]

integrate(1/x^2/arcsinh(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a^8*x^8 + 3*a^6*x^6 + 3*a^4*x^4 + a^2*x^2 + (a^5*x^5 + a^3*x^3)*(a^2*x^2 + 1)^(3/2) + (3*a^6*x^6 + 5*a^4
*x^4 + 2*a^2*x^2)*(a^2*x^2 + 1) - (a^8*x^8 + 3*a^6*x^6 + 3*a^4*x^4 + a^2*x^2 + (a^5*x^5 + 4*a^3*x^3 + 3*a*x)*(
a^2*x^2 + 1)^(3/2) + (3*a^6*x^6 + 11*a^4*x^4 + 10*a^2*x^2 + 2)*(a^2*x^2 + 1) + (3*a^7*x^7 + 10*a^5*x^5 + 10*a^
3*x^3 + 3*a*x)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1)) + (3*a^7*x^7 + 7*a^5*x^5 + 5*a^3*x^3 + a*x)*sqr
t(a^2*x^2 + 1))/((a^8*x^9 + 3*a^6*x^7 + (a^2*x^2 + 1)^(3/2)*a^5*x^6 + 3*a^4*x^5 + a^2*x^3 + 3*(a^6*x^7 + a^4*x
^5)*(a^2*x^2 + 1) + 3*(a^7*x^8 + 2*a^5*x^6 + a^3*x^4)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))^2) + int
egrate(1/2*(a^10*x^10 + 4*a^8*x^8 + 6*a^6*x^6 + 4*a^4*x^4 + a^2*x^2 + (a^6*x^6 + 12*a^4*x^4 + 15*a^2*x^2)*(a^2
*x^2 + 1)^2 + (4*a^7*x^7 + 40*a^5*x^5 + 57*a^3*x^3 + 18*a*x)*(a^2*x^2 + 1)^(3/2) + 3*(2*a^8*x^8 + 16*a^6*x^6 +
25*a^4*x^4 + 13*a^2*x^2 + 2)*(a^2*x^2 + 1) + (4*a^9*x^9 + 24*a^7*x^7 + 39*a^5*x^5 + 25*a^3*x^3 + 6*a*x)*sqrt(
a^2*x^2 + 1))/((a^10*x^12 + 4*a^8*x^10 + (a^2*x^2 + 1)^2*a^6*x^8 + 6*a^6*x^8 + 4*a^4*x^6 + a^2*x^4 + 4*(a^7*x^
9 + a^5*x^7)*(a^2*x^2 + 1)^(3/2) + 6*(a^8*x^10 + 2*a^6*x^8 + a^4*x^6)*(a^2*x^2 + 1) + 4*(a^9*x^11 + 3*a^7*x^9
+ 3*a^5*x^7 + a^3*x^5)*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 = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^3} \, dx=\int { \frac {1}{x^{2} \operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]

[In]

integrate(1/x^2/arcsinh(a*x)^3,x, algorithm="giac")

[Out]

integrate(1/(x^2*arcsinh(a*x)^3), x)

Mupad [N/A]

Not integrable

Time = 2.51 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^3} \, dx=\int \frac {1}{x^2\,{\mathrm {asinh}\left (a\,x\right )}^3} \,d x \]

[In]

int(1/(x^2*asinh(a*x)^3),x)

[Out]

int(1/(x^2*asinh(a*x)^3), x)