Optimal. Leaf size=11 \[ \frac {\log \left (\sinh ^{-1}(a+b x)\right )}{b} \]
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Rubi [A]
time = 0.05, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {5860, 5782}
\begin {gather*} \frac {\log \left (\sinh ^{-1}(a+b x)\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 5782
Rule 5860
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {\log \left (\sinh ^{-1}(a+b x)\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 11, normalized size = 1.00 \begin {gather*} \frac {\log \left (\sinh ^{-1}(a+b x)\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.03, size = 12, normalized size = 1.09
method | result | size |
default | \(\frac {\ln \left (\arcsinh \left (b x +a \right )\right )}{b}\) | \(12\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs.
\(2 (11) = 22\).
time = 0.35, size = 30, normalized size = 2.73 \begin {gather*} \frac {\log \left (\log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 22 vs.
\(2 (8) = 16\).
time = 0.65, size = 22, normalized size = 2.00 \begin {gather*} \begin {cases} \frac {\log {\left (\operatorname {asinh}{\left (a + b x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {a^{2} + 1} \operatorname {asinh}{\left (a \right )}} & \text {otherwise} \end {cases} \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 [B]
time = 0.22, size = 11, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\mathrm {asinh}\left (a+b\,x\right )\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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