3.65 \(\int \frac{1}{x \sinh ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=12 \[ \text{Unintegrable}\left (\frac{1}{x \sinh ^{-1}(a x)^3},x\right ) \]

[Out]

Unintegrable[1/(x*ArcSinh[a*x]^3), x]

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Rubi [A]  time = 0.0131602, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x \sinh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][1/(x*ArcSinh[a*x]^3), x]

Rubi steps

\begin{align*} \int \frac{1}{x \sinh ^{-1}(a x)^3} \, dx &=\int \frac{1}{x \sinh ^{-1}(a x)^3} \, dx\\ \end{align*}

Mathematica [A]  time = 0.540267, size = 0, normalized size = 0. \[ \int \frac{1}{x \sinh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[1/(x*ArcSinh[a*x]^3), x]

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Maple [A]  time = 0.06, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{8} x^{8} + 3 \, a^{6} x^{6} + 3 \, a^{4} x^{4} + a^{2} x^{2} +{\left (a^{5} x^{5} + a^{3} x^{3}\right )}{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} +{\left (3 \, a^{6} x^{6} + 5 \, a^{4} x^{4} + 2 \, a^{2} x^{2}\right )}{\left (a^{2} x^{2} + 1\right )} -{\left (2 \,{\left (a^{3} x^{3} + a x\right )}{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} +{\left (4 \, a^{4} x^{4} + 5 \, a^{2} x^{2} + 1\right )}{\left (a^{2} x^{2} + 1\right )} +{\left (2 \, a^{5} x^{5} + 3 \, a^{3} x^{3} + a x\right )} \sqrt{a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) +{\left (3 \, a^{7} x^{7} + 7 \, a^{5} x^{5} + 5 \, a^{3} x^{3} + a x\right )} \sqrt{a^{2} x^{2} + 1}}{2 \,{\left (a^{8} x^{8} + 3 \, a^{6} x^{6} +{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{5} x^{5} + 3 \, a^{4} x^{4} + a^{2} x^{2} + 3 \,{\left (a^{6} x^{6} + a^{4} x^{4}\right )}{\left (a^{2} x^{2} + 1\right )} + 3 \,{\left (a^{7} x^{7} + 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} \sqrt{a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}} + \int \frac{4 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2}\right )}{\left (a^{2} x^{2} + 1\right )}^{2} +{\left (12 \, a^{5} x^{5} + 22 \, a^{3} x^{3} + 7 \, a x\right )}{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 2 \,{\left (6 \, a^{6} x^{6} + 10 \, a^{4} x^{4} + 5 \, a^{2} x^{2} + 1\right )}{\left (a^{2} x^{2} + 1\right )} +{\left (4 \, a^{7} x^{7} + 6 \, a^{5} x^{5} + 3 \, a^{3} x^{3} + a x\right )} \sqrt{a^{2} x^{2} + 1}}{2 \,{\left (a^{10} x^{11} + 4 \, a^{8} x^{9} +{\left (a^{2} x^{2} + 1\right )}^{2} a^{6} x^{7} + 6 \, a^{6} x^{7} + 4 \, a^{4} x^{5} + a^{2} x^{3} + 4 \,{\left (a^{7} x^{8} + a^{5} x^{6}\right )}{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 6 \,{\left (a^{8} x^{9} + 2 \, a^{6} x^{7} + a^{4} x^{5}\right )}{\left (a^{2} x^{2} + 1\right )} + 4 \,{\left (a^{9} x^{10} + 3 \, a^{7} x^{8} + 3 \, a^{5} x^{6} + a^{3} x^{4}\right )} \sqrt{a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/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) - (2*(a^3*x^3 + a*x)*(a^2*x^2 + 1)^(3/2) + (4*a^4*x^4 + 5*a^2*x^2 + 1)*(a^2*x^
2 + 1) + (2*a^5*x^5 + 3*a^3*x^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)*sqrt(a^2*x^2 + 1))/((a^8*x^8 + 3*a^6*x^6 + (a^2*x^2 + 1)^(3/2)*a^5*x^5 + 3*a^4*x^4 + a^2*
x^2 + 3*(a^6*x^6 + a^4*x^4)*(a^2*x^2 + 1) + 3*(a^7*x^7 + 2*a^5*x^5 + a^3*x^3)*sqrt(a^2*x^2 + 1))*log(a*x + sqr
t(a^2*x^2 + 1))^2) + integrate(1/2*(4*(a^4*x^4 + 2*a^2*x^2)*(a^2*x^2 + 1)^2 + (12*a^5*x^5 + 22*a^3*x^3 + 7*a*x
)*(a^2*x^2 + 1)^(3/2) + 2*(6*a^6*x^6 + 10*a^4*x^4 + 5*a^2*x^2 + 1)*(a^2*x^2 + 1) + (4*a^7*x^7 + 6*a^5*x^5 + 3*
a^3*x^3 + a*x)*sqrt(a^2*x^2 + 1))/((a^10*x^11 + 4*a^8*x^9 + (a^2*x^2 + 1)^2*a^6*x^7 + 6*a^6*x^7 + 4*a^4*x^5 +
a^2*x^3 + 4*(a^7*x^8 + a^5*x^6)*(a^2*x^2 + 1)^(3/2) + 6*(a^8*x^9 + 2*a^6*x^7 + a^4*x^5)*(a^2*x^2 + 1) + 4*(a^9
*x^10 + 3*a^7*x^8 + 3*a^5*x^6 + a^3*x^4)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{x \operatorname{arsinh}\left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(x*arcsinh(a*x)^3), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \operatorname{asinh}^{3}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*asinh(a*x)**3), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \operatorname{arsinh}\left (a x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x*arcsinh(a*x)^3), x)