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

Optimal. Leaf size=44 \[ \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )},x\right )+\frac{c \log \left (a+b \sinh ^{-1}(c x)\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.299652, 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 = {} \[ \int \frac{\sqrt{1+c^2 x^2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.10183, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1+c^2 x^2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.162, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) }\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)),x)

[Out]

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

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Maxima [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 )} 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)),x, algorithm="maxima")

[Out]

integrate(sqrt(c^2*x^2 + 1)/((b*arcsinh(c*x) + a)*x^2), 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 x^{2} \operatorname{arsinh}\left (c x\right ) + a 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)),x, algorithm="fricas")

[Out]

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

[Out]

Integral(sqrt(c**2*x**2 + 1)/(x**2*(a + b*asinh(c*x))), 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 )} 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)),x, algorithm="giac")

[Out]

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