Optimal. Leaf size=13 \[ -\frac {1}{b \text {ArcSin}(a+b x)} \]
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Rubi [A]
time = 0.05, antiderivative size = 13, 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, 4737}
\begin {gather*} -\frac {1}{b \text {ArcSin}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 4737
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)^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {1}{b \sin ^{-1}(a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{b \text {ArcSin}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 14, normalized size = 1.08
method | result | size |
default | \(-\frac {1}{b \arcsin \left (b x +a \right )}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs.
\(2 (13) = 26\).
time = 0.73, size = 33, normalized size = 2.54 \begin {gather*} -\frac {1}{b \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.61, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{b \arcsin \left (b x + a\right )} \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. 26 vs.
\(2 (10) = 20\).
time = 0.65, size = 26, normalized size = 2.00 \begin {gather*} \begin {cases} - \frac {1}{b \operatorname {asin}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {1 - a^{2}} \operatorname {asin}^{2}{\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 = 0.48, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{b \arcsin \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{b\,\mathrm {asin}\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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