Optimal. Leaf size=11 \[ \frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5863, 5657, 3301} \[ \frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5657
Rule 5863
Rubi steps
\begin {align*} \int \frac {1}{\sinh ^{-1}(a+b x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ \frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\operatorname {arsinh}\left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {arsinh}\left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 12, normalized size = 1.09 \[ \frac {\Chi \left (\arcsinh \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {arsinh}\left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.09 \[ \int \frac {1}{\mathrm {asinh}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {asinh}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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