Optimal. Leaf size=46 \[ \frac {\sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt {c-c (a+b x)^2}} \]
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Rubi [A] time = 0.16, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {247, 217, 203, 4643, 4641} \[ \frac {\sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt {c-c (a+b x)^2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 247
Rule 4641
Rule 4643
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a+b x)}{\sqrt {c-c (a+b x)^2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {c-c x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\sqrt {1-(a+b x)^2} \operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b \sqrt {c-c (a+b x)^2}}\\ &=\frac {\sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt {c-c (a+b x)^2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 46, normalized size = 1.00 \[ \frac {\sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt {-c \left ((a+b x)^2-1\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b^{2} c x^{2} - 2 \, a b c x - {\left (a^{2} - 1\right )} c} \arcsin \left (b x + a\right )}{b^{2} c x^{2} + 2 \, a b c x + {\left (a^{2} - 1\right )} c}, x\right ) \]
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 \[ \int \frac {\arcsin \left (b x + a\right )}{\sqrt {-{\left (b x + a\right )}^{2} c + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 80, normalized size = 1.74 \[ -\frac {\sqrt {-c \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{2}}{2 b \left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 206, normalized size = 4.48 \[ \frac {\sqrt {c} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )^{2}}{2 \, \sqrt {a^{2} b^{2} c^{2} - {\left (a^{2} c - c\right )} b^{2} c}} - \frac {\arcsin \left (b x + a\right ) \arcsin \left (-\frac {b^{2} c x + a b c}{\sqrt {a^{2} b^{2} c^{2} - {\left (a^{2} c - c\right )} b^{2} c}}\right )}{b \sqrt {c}} - \frac {\arcsin \left (-\frac {b^{2} c x + a b c}{\sqrt {a^{2} b^{2} c^{2} - {\left (a^{2} c - c\right )} b^{2} c}}\right ) \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b \sqrt {c}} \]
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 \[ \int \frac {\mathrm {asin}\left (a+b\,x\right )}{\sqrt {c-c\,{\left (a+b\,x\right )}^2}} \,d x \]
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 \[ \int \frac {\operatorname {asin}{\left (a + b x \right )}}{\sqrt {- c \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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