3.418 \(\int \frac{\sqrt{1+c^2 x^2}}{x^4 (a+b \sinh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=29 \[ \text{Unintegrable}\left (\frac{\sqrt{c^2 x^2+1}}{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable[Sqrt[1 + c^2*x^2]/(x^4*(a + b*ArcSinh[c*x])^2), x]

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Rubi [A]  time = 0.130909, 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 = {} \[ \int \frac{\sqrt{1+c^2 x^2}}{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[1 + c^2*x^2]/(x^4*(a + b*ArcSinh[c*x])^2),x]

[Out]

Defer[Int][Sqrt[1 + c^2*x^2]/(x^4*(a + b*ArcSinh[c*x])^2), x]

Rubi steps

\begin{align*} \int \frac{\sqrt{1+c^2 x^2}}{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=\int \frac{\sqrt{1+c^2 x^2}}{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 3.51667, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1+c^2 x^2}}{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[1 + c^2*x^2]/(x^4*(a + b*ArcSinh[c*x])^2),x]

[Out]

Integrate[Sqrt[1 + c^2*x^2]/(x^4*(a + b*ArcSinh[c*x])^2), x]

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Maple [A]  time = 0.472, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}\sqrt{{c}^{2}{x}^{2}+1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(1/2)/x^4/(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

-((c^2*x^2 + 1)^2 + (c^3*x^3 + c*x)*sqrt(c^2*x^2 + 1))/(a*b*c^3*x^6 + sqrt(c^2*x^2 + 1)*a*b*c^2*x^5 + a*b*c*x^
4 + (b^2*c^3*x^6 + sqrt(c^2*x^2 + 1)*b^2*c^2*x^5 + b^2*c*x^4)*log(c*x + sqrt(c^2*x^2 + 1))) - integrate(((2*c^
3*x^3 + 5*c*x)*(c^2*x^2 + 1)^(3/2) + 2*(2*c^4*x^4 + 5*c^2*x^2 + 2)*(c^2*x^2 + 1) + (2*c^5*x^5 + 5*c^3*x^3 + 3*
c*x)*sqrt(c^2*x^2 + 1))/(a*b*c^5*x^9 + (c^2*x^2 + 1)*a*b*c^3*x^7 + 2*a*b*c^3*x^7 + a*b*c*x^5 + (b^2*c^5*x^9 +
(c^2*x^2 + 1)*b^2*c^3*x^7 + 2*b^2*c^3*x^7 + b^2*c*x^5 + 2*(b^2*c^4*x^8 + b^2*c^2*x^6)*sqrt(c^2*x^2 + 1))*log(c
*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^8 + a*b*c^2*x^6)*sqrt(c^2*x^2 + 1)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c^{2} x^{2} + 1}}{b^{2} x^{4} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname{arsinh}\left (c x\right ) + a^{2} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(1/2)/x^4/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(1/2)/x^4/(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

integrate(sqrt(c^2*x^2 + 1)/((b*arcsinh(c*x) + a)^2*x^4), x)