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

Optimal. Leaf size=57 \[ -\frac{3 a \text{Unintegrable}\left (\frac{x}{\left (a^2 x^2+1\right )^{5/2} \sinh ^{-1}(a x)},x\right )}{c^2}-\frac{1}{a c^2 \left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)} \]

[Out]

-(1/(a*c^2*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x])) - (3*a*Unintegrable[x/((1 + a^2*x^2)^(5/2)*ArcSinh[a*x]), x])/c^
2

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Rubi [A]  time = 0.101213, 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 = {} \[ \int \frac{1}{\left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(1/(a*c^2*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x])) - (3*a*Defer[Int][x/((1 + a^2*x^2)^(5/2)*ArcSinh[a*x]), x])/c^2

Rubi steps

\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)^2} \, dx &=-\frac{1}{a c^2 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)}-\frac{(3 a) \int \frac{x}{\left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)} \, dx}{c^2}\\ \end{align*}

Mathematica [A]  time = 3.63577, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.132, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({a}^{2}c{x}^{2}+c \right ) ^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a x + \sqrt{a^{2} x^{2} + 1}}{{\left (a^{5} c^{2} x^{4} + 2 \, a^{3} c^{2} x^{2} + a c^{2} +{\left (a^{4} c^{2} x^{3} + a^{2} c^{2} x\right )} \sqrt{a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )} - \int \frac{3 \, a^{4} x^{4} + 2 \, a^{2} x^{2} +{\left (3 \, a^{2} x^{2} + 1\right )}{\left (a^{2} x^{2} + 1\right )} + 3 \,{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt{a^{2} x^{2} + 1} - 1}{{\left (a^{8} c^{2} x^{8} + 4 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} + 4 \, a^{2} c^{2} x^{2} +{\left (a^{6} c^{2} x^{6} + 2 \, a^{4} c^{2} x^{4} + a^{2} c^{2} x^{2}\right )}{\left (a^{2} x^{2} + 1\right )} + c^{2} + 2 \,{\left (a^{7} c^{2} x^{7} + 3 \, a^{5} c^{2} x^{5} + 3 \, a^{3} c^{2} x^{3} + a c^{2} x\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/(a^2*c*x^2+c)^2/arcsinh(a*x)^2,x, algorithm="maxima")

[Out]

-(a*x + sqrt(a^2*x^2 + 1))/((a^5*c^2*x^4 + 2*a^3*c^2*x^2 + a*c^2 + (a^4*c^2*x^3 + a^2*c^2*x)*sqrt(a^2*x^2 + 1)
)*log(a*x + sqrt(a^2*x^2 + 1))) - integrate((3*a^4*x^4 + 2*a^2*x^2 + (3*a^2*x^2 + 1)*(a^2*x^2 + 1) + 3*(2*a^3*
x^3 + a*x)*sqrt(a^2*x^2 + 1) - 1)/((a^8*c^2*x^8 + 4*a^6*c^2*x^6 + 6*a^4*c^2*x^4 + 4*a^2*c^2*x^2 + (a^6*c^2*x^6
+ 2*a^4*c^2*x^4 + a^2*c^2*x^2)*(a^2*x^2 + 1) + c^2 + 2*(a^7*c^2*x^7 + 3*a^5*c^2*x^5 + 3*a^3*c^2*x^3 + a*c^2*x
)*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}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \operatorname{arsinh}\left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{4} x^{4} \operatorname{asinh}^{2}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname{asinh}^{2}{\left (a x \right )} + \operatorname{asinh}^{2}{\left (a x \right )}}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{2} \operatorname{arsinh}\left (a x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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