Optimal. Leaf size=50 \[ \frac {\log \left (1-(a+b x)^2\right )}{2 b}+\frac {(a+b x) \sin ^{-1}(a+b x)}{b \sqrt {1-(a+b x)^2}} \]
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Rubi [A] time = 0.06, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {4807, 4651, 260} \[ \frac {\log \left (1-(a+b x)^2\right )}{2 b}+\frac {(a+b x) \sin ^{-1}(a+b x)}{b \sqrt {1-(a+b x)^2}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 4651
Rule 4807
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a+b x)}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)}{\left (1-x^2\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sin ^{-1}(a+b x)}{b \sqrt {1-(a+b x)^2}}-\frac {\operatorname {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sin ^{-1}(a+b x)}{b \sqrt {1-(a+b x)^2}}+\frac {\log \left (1-(a+b x)^2\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 66, normalized size = 1.32 \[ \frac {\log \left (-a^2-2 a b x-b^2 x^2+1\right )+\frac {2 (a+b x) \sin ^{-1}(a+b x)}{\sqrt {-a^2-2 a b x-b^2 x^2+1}}}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 99, normalized size = 1.98 \[ -\frac {2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right ) - {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + {\left (a^{2} - 1\right )} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.17, size = 83, normalized size = 1.66 \[ -\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (x + \frac {a}{b}\right )} \arcsin \left (b x + a\right )}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1} + \frac {\log \left ({\left | b x + a + 1 \right |}\right )}{2 \, b} + \frac {\log \left ({\left | b x + a - 1 \right |}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 155, normalized size = 3.10 \[ -\frac {-\ln \left (1-\left (b x +a \right )^{2}\right ) x^{2} b^{2}+2 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, x b -2 \ln \left (1-\left (b x +a \right )^{2}\right ) x a b +2 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -\ln \left (1-\left (b x +a \right )^{2}\right ) a^{2}+\ln \left (1-\left (b x +a \right )^{2}\right )}{2 b \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 160, normalized size = 3.20 \[ -\frac {1}{2} \, {\left (a {\left (\frac {\log \left (b x + a + 1\right )}{b^{2}} - \frac {\log \left (b x + a - 1\right )}{b^{2}}\right )} - \frac {{\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{2}} + \frac {{\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{2}}\right )} b + {\left (\frac {b^{2} x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} + \frac {a b}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}\right )} \arcsin \left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {asin}\left (a+b\,x\right )}{{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {asin}{\left (a + b x \right )}}{\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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