3.399 \(\int \frac{1}{x (1+c^2 x^2)^{3/2} (a+b \sinh ^{-1}(c x))} \, dx\)

Optimal. Leaf size=29 \[ \text{Unintegrable}\left (\frac{1}{x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.137728, 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 = {} \[ \int \frac{1}{x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.45323, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.249, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) } \left ({c}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x)

[Out]

int(1/x/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

integrate(1/((c^2*x^2 + 1)^(3/2)*(b*arcsinh(c*x) + a)*x), 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}}{a c^{4} x^{5} + 2 \, a c^{2} x^{3} + a x +{\left (b c^{4} x^{5} + 2 \, b c^{2} x^{3} + b x\right )} \operatorname{arsinh}\left (c x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*x^2 + 1)/(a*c^4*x^5 + 2*a*c^2*x^3 + a*x + (b*c^4*x^5 + 2*b*c^2*x^3 + b*x)*arcsinh(c*x)), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (a + b \operatorname{asinh}{\left (c x \right )}\right ) \left (c^{2} x^{2} + 1\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c**2*x**2+1)**(3/2)/(a+b*asinh(c*x)),x)

[Out]

Integral(1/(x*(a + b*asinh(c*x))*(c**2*x**2 + 1)**(3/2)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

integrate(1/((c^2*x^2 + 1)^(3/2)*(b*arcsinh(c*x) + a)*x), x)