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

Optimal. Leaf size=55 \[ -\frac{2 \text{Unintegrable}\left (\frac{1}{x^3 \left (a+b \sinh ^{-1}(c x)\right )},x\right )}{b c}-\frac{c^2 x^2+1}{b c x^2 \left (a+b \sinh ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.154345, 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 = {} \[ \int \frac{\sqrt{1+c^2 x^2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.67017, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1+c^2 x^2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.177, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \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^2/(a+b*arcsinh(c*x))^2,x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c^2*x^2 + 1)/(b^2*x^2*arcsinh(c*x)^2 + 2*a*b*x^2*arcsinh(c*x) + a^2*x^2), 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^{2} \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**2/(a+b*asinh(c*x))**2,x)

[Out]

Integral(sqrt(c**2*x**2 + 1)/(x**2*(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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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