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

Optimal. Leaf size=14 \[ \text{Unintegrable}\left (\frac{1}{x \sinh ^{-1}(a+b x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0386165, 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 = {} \[ \int \frac{1}{x \sinh ^{-1}(a+b x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*ArcSinh[a + b*x]^2),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.04346, size = 0, normalized size = 0. \[ \int \frac{1}{x \sinh ^{-1}(a+b x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*ArcSinh[a + b*x]^2),x]

[Out]

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

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Maple [A]  time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(x*arcsinh(b*x + a)^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*asinh(a + b*x)**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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