Optimal. Leaf size=15 \[ \text {Int}\left (\frac {1}{x \sinh ^{-1}(a+b x)^2},x\right ) \]
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Rubi [A] time = 0.04, 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 = {} \[ \int \frac {1}{x \sinh ^{-1}(a+b x)^2} \, dx \]
Verification is Not applicable to the result.
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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*}
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Mathematica [A] time = 2.14, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sinh ^{-1}(a+b x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x \operatorname {arsinh}\left (b x + a\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \operatorname {arsinh}\left (b x + a\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \arcsinh \left (b x +a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b + b\right )} x + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} + a}{{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + {\left (a^{2} b + b\right )} x + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b^{2} x^{2} + a b x\right )}\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} - \int \frac {a b^{4} x^{4} + 4 \, a^{2} b^{3} x^{3} + a^{5} + 2 \, a^{3} + 2 \, {\left (3 \, a^{3} b^{2} + a b^{2}\right )} x^{2} + {\left (a b^{2} x^{2} + a^{3} + 2 \, {\left (a^{2} b + b\right )} x + a\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} + 4 \, {\left (a^{4} b + a^{2} b\right )} x + {\left (2 \, a b^{3} x^{3} + 2 \, a^{4} + 2 \, {\left (3 \, a^{2} b^{2} + b^{2}\right )} x^{2} + 3 \, a^{2} + {\left (6 \, a^{3} b + 5 \, a b\right )} x + 1\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + a}{{\left (b^{5} x^{6} + 4 \, a b^{4} x^{5} + 2 \, {\left (3 \, a^{2} b^{3} + b^{3}\right )} x^{4} + 4 \, {\left (a^{3} b^{2} + a b^{2}\right )} x^{3} + {\left (a^{4} b + 2 \, a^{2} b + b\right )} x^{2} + {\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} + 2 \, {\left (b^{4} x^{5} + 3 \, a b^{3} x^{4} + {\left (3 \, a^{2} b^{2} + b^{2}\right )} x^{3} + {\left (a^{3} b + a b\right )} x^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{x\,{\mathrm {asinh}\left (a+b\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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