Optimal. Leaf size=47 \[ \frac {\text {Ci}\left (2 \sin ^{-1}(a+b x)\right )}{2 b}+\frac {\text {Ci}\left (4 \sin ^{-1}(a+b x)\right )}{8 b}+\frac {3 \log \left (\sin ^{-1}(a+b x)\right )}{8 b} \]
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Rubi [A] time = 0.14, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4807, 4661, 3312, 3302} \[ \frac {\text {CosIntegral}\left (2 \sin ^{-1}(a+b x)\right )}{2 b}+\frac {\text {CosIntegral}\left (4 \sin ^{-1}(a+b x)\right )}{8 b}+\frac {3 \log \left (\sin ^{-1}(a+b x)\right )}{8 b} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 3312
Rule 4661
Rule 4807
Rubi steps
\begin {align*} \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\sin ^{-1}(a+b x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2}}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cos ^4(x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cos (2 x)}{2 x}+\frac {\cos (4 x)}{8 x}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac {3 \log \left (\sin ^{-1}(a+b x)\right )}{8 b}+\frac {\operatorname {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b}+\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b}\\ &=\frac {\text {Ci}\left (2 \sin ^{-1}(a+b x)\right )}{2 b}+\frac {\text {Ci}\left (4 \sin ^{-1}(a+b x)\right )}{8 b}+\frac {3 \log \left (\sin ^{-1}(a+b x)\right )}{8 b}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 37, normalized size = 0.79 \[ \frac {4 \text {Ci}\left (2 \sin ^{-1}(a+b x)\right )+\text {Ci}\left (4 \sin ^{-1}(a+b x)\right )+3 \log \left (\sin ^{-1}(a+b x)\right )}{8 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{\arcsin \left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 41, normalized size = 0.87 \[ \frac {\operatorname {Ci}\left (4 \, \arcsin \left (b x + a\right )\right )}{8 \, b} + \frac {\operatorname {Ci}\left (2 \, \arcsin \left (b x + a\right )\right )}{2 \, b} + \frac {3 \, \log \left (\arcsin \left (b x + a\right )\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 42, normalized size = 0.89 \[ \frac {\Ci \left (2 \arcsin \left (b x +a \right )\right )}{2 b}+\frac {\Ci \left (4 \arcsin \left (b x +a \right )\right )}{8 b}+\frac {3 \ln \left (\arcsin \left (b x +a \right )\right )}{8 b} \]
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 \[ \int \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{\arcsin \left (b x + a\right )}\,{d x} \]
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 \[ \int \frac {{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2}}{\mathrm {asin}\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}{\operatorname {asin}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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