Integrand size = 33, antiderivative size = 47 \[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)} \, dx=\frac {\operatorname {CosIntegral}(2 \arcsin (a+b x))}{2 b}+\frac {\operatorname {CosIntegral}(4 \arcsin (a+b x))}{8 b}+\frac {3 \log (\arcsin (a+b x))}{8 b} \]
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Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4891, 4753, 3393, 3383} \[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)} \, dx=\frac {\operatorname {CosIntegral}(2 \arcsin (a+b x))}{2 b}+\frac {\operatorname {CosIntegral}(4 \arcsin (a+b x))}{8 b}+\frac {3 \log (\arcsin (a+b x))}{8 b} \]
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Rule 3383
Rule 3393
Rule 4753
Rule 4891
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2}}{\arcsin (x)} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\cos ^4(x)}{x} \, dx,x,\arcsin (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cos (2 x)}{2 x}+\frac {\cos (4 x)}{8 x}\right ) \, dx,x,\arcsin (a+b x)\right )}{b} \\ & = \frac {3 \log (\arcsin (a+b x))}{8 b}+\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\arcsin (a+b x)\right )}{8 b}+\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\arcsin (a+b x)\right )}{2 b} \\ & = \frac {\operatorname {CosIntegral}(2 \arcsin (a+b x))}{2 b}+\frac {\operatorname {CosIntegral}(4 \arcsin (a+b x))}{8 b}+\frac {3 \log (\arcsin (a+b x))}{8 b} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.79 \[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)} \, dx=\frac {4 \operatorname {CosIntegral}(2 \arcsin (a+b x))+\operatorname {CosIntegral}(4 \arcsin (a+b x))+3 \log (\arcsin (a+b x))}{8 b} \]
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Time = 1.79 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {3 \ln \left (\arcsin \left (b x +a \right )\right )+4 \,\operatorname {Ci}\left (2 \arcsin \left (b x +a \right )\right )+\operatorname {Ci}\left (4 \arcsin \left (b x +a \right )\right )}{8 b}\) | \(36\) |
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\[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)} \, dx=\int { \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{\arcsin \left (b x + a\right )} \,d x } \]
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\[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)} \, dx=\int \frac {\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}{\operatorname {asin}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)} \, dx=\int { \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{\arcsin \left (b x + a\right )} \,d x } \]
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none
Time = 0.34 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)} \, dx=\frac {\operatorname {Ci}\left (4 \, \arcsin \left (b x + a\right )\right )}{8 \, b} + \frac {\operatorname {Ci}\left (2 \, \arcsin \left (b x + a\right )\right )}{2 \, b} + \frac {3 \, \log \left (\arcsin \left (b x + a\right )\right )}{8 \, b} \]
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Timed out. \[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)} \, dx=\int \frac {{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2}}{\mathrm {asin}\left (a+b\,x\right )} \,d x \]
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