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 = {5858, 5774,
3382} \begin {gather*} \frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 5774
Rule 5858
Rubi steps
\begin {align*} \int \frac {1}{\sinh ^{-1}(a+b x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {\text {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 \begin {gather*} \frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.60, size = 12, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {\hyperbolicCosineIntegral \left (\arcsinh \left (b x +a \right )\right )}{b}\) | \(12\) |
default | \(\frac {\hyperbolicCosineIntegral \left (\arcsinh \left (b x +a \right )\right )}{b}\) | \(12\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\operatorname {asinh}{\left (a + b x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\mathrm {asinh}\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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