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

Optimal. Leaf size=45 \[ \text {Int}\left (\frac {1}{x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )},x\right )+\frac {c \log \left (a+b \sinh ^{-1}(c x)\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.30, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 1.15, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

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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fricas [A]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} x^{2} + 1}}{b x^{2} \operatorname {arsinh}\left (c x\right ) + a x^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}\,{d x} \]

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)

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maple [A]  time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c^{2} x^{2}+1}}{x^{2} \left (a +b \arcsinh \left (c x \right )\right )}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {c^2\,x^2+1}}{x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c^{2} x^{2} + 1}}{x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}\, dx \]

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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