3.66 \(\int \frac {1}{x^2 \sinh ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=13 \[ \text {Int}\left (\frac {1}{x^2 \sinh ^{-1}(a x)^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x^2 \sinh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac {1}{x^2 \sinh ^{-1}(a x)^3} \, dx &=\int \frac {1}{x^2 \sinh ^{-1}(a x)^3} \, dx\\ \end {align*}

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Mathematica [A]  time = 5.32, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \sinh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[1/(x^2*ArcSinh[a*x]^3), x]

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x^{2} \operatorname {arsinh}\left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(x^2*arcsinh(a*x)^3), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \operatorname {arsinh}\left (a x\right )^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x^2*arcsinh(a*x)^3), x)

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maple [A]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \arcsinh \left (a x \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\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 (a^{8} x^{8} + 3 \, a^{6} x^{6} + 3 \, a^{4} x^{4} + a^{2} x^{2} + {\left (a^{5} x^{5} + 4 \, a^{3} x^{3} + 3 \, a x\right )} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} + {\left (3 \, a^{6} x^{6} + 11 \, a^{4} x^{4} + 10 \, a^{2} x^{2} + 2\right )} {\left (a^{2} x^{2} + 1\right )} + {\left (3 \, a^{7} x^{7} + 10 \, a^{5} x^{5} + 10 \, a^{3} x^{3} + 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^{9} + 3 \, a^{6} x^{7} + {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{5} x^{6} + 3 \, a^{4} x^{5} + a^{2} x^{3} + 3 \, {\left (a^{6} x^{7} + a^{4} x^{5}\right )} {\left (a^{2} x^{2} + 1\right )} + 3 \, {\left (a^{7} x^{8} + 2 \, 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 )^{2}} + \int \frac {a^{10} x^{10} + 4 \, a^{8} x^{8} + 6 \, a^{6} x^{6} + 4 \, a^{4} x^{4} + a^{2} x^{2} + {\left (a^{6} x^{6} + 12 \, a^{4} x^{4} + 15 \, a^{2} x^{2}\right )} {\left (a^{2} x^{2} + 1\right )}^{2} + {\left (4 \, a^{7} x^{7} + 40 \, a^{5} x^{5} + 57 \, a^{3} x^{3} + 18 \, a x\right )} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 3 \, {\left (2 \, a^{8} x^{8} + 16 \, a^{6} x^{6} + 25 \, a^{4} x^{4} + 13 \, a^{2} x^{2} + 2\right )} {\left (a^{2} x^{2} + 1\right )} + {\left (4 \, a^{9} x^{9} + 24 \, a^{7} x^{7} + 39 \, a^{5} x^{5} + 25 \, a^{3} x^{3} + 6 \, a x\right )} \sqrt {a^{2} x^{2} + 1}}{2 \, {\left (a^{10} x^{12} + 4 \, a^{8} x^{10} + {\left (a^{2} x^{2} + 1\right )}^{2} a^{6} x^{8} + 6 \, a^{6} x^{8} + 4 \, a^{4} x^{6} + a^{2} x^{4} + 4 \, {\left (a^{7} x^{9} + a^{5} x^{7}\right )} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 6 \, {\left (a^{8} x^{10} + 2 \, a^{6} x^{8} + a^{4} x^{6}\right )} {\left (a^{2} x^{2} + 1\right )} + 4 \, {\left (a^{9} x^{11} + 3 \, a^{7} x^{9} + 3 \, a^{5} x^{7} + a^{3} x^{5}\right )} \sqrt {a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {1}{x^2\,{\mathrm {asinh}\left (a\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(x^2*asinh(a*x)^3), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \operatorname {asinh}^{3}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**2*asinh(a*x)**3), x)

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