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

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

[Out]

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

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Rubi [A]  time = 0.101692, 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 = {} \[ \int \frac{1}{\left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.4745, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.08, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({a}^{2}c{x}^{2}+c \right ) \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)/arcsinh(a*x)^2,x)

[Out]

int(1/(a^2*c*x^2+c)/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^{3} c x^{2} + \sqrt{a^{2} x^{2} + 1} a^{2} c x + a c\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )} - \int \frac{a^{4} x^{4} +{\left (a^{2} x^{2} + 1\right )}^{2} +{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt{a^{2} x^{2} + 1} - 1}{{\left (a^{6} c x^{6} + 3 \, a^{4} c x^{4} + 3 \, a^{2} c x^{2} +{\left (a^{4} c x^{4} + a^{2} c x^{2}\right )}{\left (a^{2} x^{2} + 1\right )} + 2 \,{\left (a^{5} c x^{5} + 2 \, a^{3} c x^{3} + a c x\right )} \sqrt{a^{2} x^{2} + 1} + c\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)/arcsinh(a*x)^2,x, algorithm="maxima")

[Out]

-(a*x + sqrt(a^2*x^2 + 1))/((a^3*c*x^2 + sqrt(a^2*x^2 + 1)*a^2*c*x + a*c)*log(a*x + sqrt(a^2*x^2 + 1))) - inte
grate((a^4*x^4 + (a^2*x^2 + 1)^2 + (2*a^3*x^3 + a*x)*sqrt(a^2*x^2 + 1) - 1)/((a^6*c*x^6 + 3*a^4*c*x^4 + 3*a^2*
c*x^2 + (a^4*c*x^4 + a^2*c*x^2)*(a^2*x^2 + 1) + 2*(a^5*c*x^5 + 2*a^3*c*x^3 + a*c*x)*sqrt(a^2*x^2 + 1) + c)*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^{2} c x^{2} + c\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)/arcsinh(a*x)^2,x, algorithm="fricas")

[Out]

integral(1/((a^2*c*x^2 + c)*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^{2} x^{2} \operatorname{asinh}^{2}{\left (a x \right )} + \operatorname{asinh}^{2}{\left (a x \right )}}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} \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)/arcsinh(a*x)^2,x, algorithm="giac")

[Out]

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