3.354 \(\int \frac {1}{(c+a^2 c x^2)^2 \sinh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=22 \[ \text {Int}\left (\frac {1}{\left (a^2 c x^2+c\right )^2 \sinh ^{-1}(a x)},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)} \, dx &=\int \frac {1}{\left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.61, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \operatorname {arsinh}\left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a^{2} c \,x^{2}+c \right )^{2} \arcsinh \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{\mathrm {asinh}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{4} x^{4} \operatorname {asinh}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {asinh}{\left (a x \right )} + \operatorname {asinh}{\left (a x \right )}}\, dx}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________