Optimal. Leaf size=11 \[ \frac {\log (\text {ArcSin}(a+b x))}{b} \]
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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 = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {4891, 4735}
\begin {gather*} \frac {\log (\text {ArcSin}(a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 4735
Rule 4891
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-a^2-2 a b x-b^2 x^2} \sin ^{-1}(a+b x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {\log \left (\sin ^{-1}(a+b x)\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 11, normalized size = 1.00 \begin {gather*} \frac {\log (\text {ArcSin}(a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 12, normalized size = 1.09
method | result | size |
default | \(\frac {\ln \left (\arcsin \left (b x +a \right )\right )}{b}\) | \(12\) |
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 [A]
time = 2.27, size = 13, normalized size = 1.18 \begin {gather*} \frac {\log \left (-\arcsin \left (b x + a\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 = 0.51, size = 22, normalized size = 2.00 \begin {gather*} \begin {cases} \frac {\log {\left (\operatorname {asin}{\left (a + b x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {1 - a^{2}} \operatorname {asin}{\left (a \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 12, normalized size = 1.09 \begin {gather*} \frac {\log \left ({\left | \arcsin \left (b x + a\right ) \right |}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 11, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\mathrm {asin}\left (a+b\,x\right )\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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