Optimal. Leaf size=39 \[ -\frac {1-(a+b x)^2}{b \text {ArcSin}(a+b x)}-\frac {\text {Si}(2 \text {ArcSin}(a+b x))}{b} \]
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Rubi [A]
time = 0.09, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4891, 4751,
4731, 4491, 12, 3380} \begin {gather*} -\frac {\text {Si}(2 \text {ArcSin}(a+b x))}{b}-\frac {1-(a+b x)^2}{b \text {ArcSin}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3380
Rule 4491
Rule 4731
Rule 4751
Rule 4891
Rubi steps
\begin {align*} \int \frac {\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 {\sqrt {1-x^2}}{\sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {1-(a+b x)^2}{b \sin ^{-1}(a+b x)}-\frac {2 \text {Subst}\left (\int \frac {x}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=-\frac {1-(a+b x)^2}{b \sin ^{-1}(a+b x)}-\frac {2 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac {1-(a+b x)^2}{b \sin ^{-1}(a+b x)}-\frac {2 \text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac {1-(a+b x)^2}{b \sin ^{-1}(a+b x)}-\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac {1-(a+b x)^2}{b \sin ^{-1}(a+b x)}-\frac {\text {Si}\left (2 \sin ^{-1}(a+b x)\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 46, normalized size = 1.18 \begin {gather*} \frac {-1+a^2+2 a b x+b^2 x^2-\text {ArcSin}(a+b x) \text {Si}(2 \text {ArcSin}(a+b x))}{b \text {ArcSin}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 42, normalized size = 1.08
method | result | size |
default | \(-\frac {2 \sinIntegral \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )+\cos \left (2 \arcsin \left (b x +a \right )\right )+1}{2 b \arcsin \left (b x +a \right )}\) | \(42\) |
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 {\sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}{\operatorname {asin}^{2}{\left (a + b x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 44, normalized size = 1.13 \begin {gather*} -\frac {\operatorname {Si}\left (2 \, \arcsin \left (b x + a\right )\right )}{b} + \frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 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 [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}}{{\mathrm {asin}\left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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