Integrand size = 33, antiderivative size = 128 \[ \int \frac {\arcsin (a+b x)^3}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=-\frac {i \arcsin (a+b x)^3}{b}+\frac {(a+b x) \arcsin (a+b x)^3}{b \sqrt {1-(a+b x)^2}}+\frac {3 \arcsin (a+b x)^2 \log \left (1+e^{2 i \arcsin (a+b x)}\right )}{b}-\frac {3 i \arcsin (a+b x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a+b x)}\right )}{b}+\frac {3 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (a+b x)}\right )}{2 b} \]
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Time = 0.15 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4891, 4745, 4765, 3800, 2221, 2611, 2320, 6724} \[ \int \frac {\arcsin (a+b x)^3}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=-\frac {3 i \arcsin (a+b x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a+b x)}\right )}{b}+\frac {3 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (a+b x)}\right )}{2 b}+\frac {(a+b x) \arcsin (a+b x)^3}{b \sqrt {1-(a+b x)^2}}-\frac {i \arcsin (a+b x)^3}{b}+\frac {3 \arcsin (a+b x)^2 \log \left (1+e^{2 i \arcsin (a+b x)}\right )}{b} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 4745
Rule 4765
Rule 4891
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\arcsin (x)^3}{\left (1-x^2\right )^{3/2}} \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \arcsin (a+b x)^3}{b \sqrt {1-(a+b x)^2}}-\frac {3 \text {Subst}\left (\int \frac {x \arcsin (x)^2}{1-x^2} \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \arcsin (a+b x)^3}{b \sqrt {1-(a+b x)^2}}-\frac {3 \text {Subst}\left (\int x^2 \tan (x) \, dx,x,\arcsin (a+b x)\right )}{b} \\ & = -\frac {i \arcsin (a+b x)^3}{b}+\frac {(a+b x) \arcsin (a+b x)^3}{b \sqrt {1-(a+b x)^2}}+\frac {(6 i) \text {Subst}\left (\int \frac {e^{2 i x} x^2}{1+e^{2 i x}} \, dx,x,\arcsin (a+b x)\right )}{b} \\ & = -\frac {i \arcsin (a+b x)^3}{b}+\frac {(a+b x) \arcsin (a+b x)^3}{b \sqrt {1-(a+b x)^2}}+\frac {3 \arcsin (a+b x)^2 \log \left (1+e^{2 i \arcsin (a+b x)}\right )}{b}-\frac {6 \text {Subst}\left (\int x \log \left (1+e^{2 i x}\right ) \, dx,x,\arcsin (a+b x)\right )}{b} \\ & = -\frac {i \arcsin (a+b x)^3}{b}+\frac {(a+b x) \arcsin (a+b x)^3}{b \sqrt {1-(a+b x)^2}}+\frac {3 \arcsin (a+b x)^2 \log \left (1+e^{2 i \arcsin (a+b x)}\right )}{b}-\frac {3 i \arcsin (a+b x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a+b x)}\right )}{b}+\frac {(3 i) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \, dx,x,\arcsin (a+b x)\right )}{b} \\ & = -\frac {i \arcsin (a+b x)^3}{b}+\frac {(a+b x) \arcsin (a+b x)^3}{b \sqrt {1-(a+b x)^2}}+\frac {3 \arcsin (a+b x)^2 \log \left (1+e^{2 i \arcsin (a+b x)}\right )}{b}-\frac {3 i \arcsin (a+b x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a+b x)}\right )}{b}+\frac {3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 i \arcsin (a+b x)}\right )}{2 b} \\ & = -\frac {i \arcsin (a+b x)^3}{b}+\frac {(a+b x) \arcsin (a+b x)^3}{b \sqrt {1-(a+b x)^2}}+\frac {3 \arcsin (a+b x)^2 \log \left (1+e^{2 i \arcsin (a+b x)}\right )}{b}-\frac {3 i \arcsin (a+b x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a+b x)}\right )}{b}+\frac {3 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (a+b x)}\right )}{2 b} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.12 \[ \int \frac {\arcsin (a+b x)^3}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=\frac {2 \arcsin (a+b x)^2 \left (\frac {\left (a+b x-i \sqrt {1-a^2-2 a b x-b^2 x^2}\right ) \arcsin (a+b x)}{\sqrt {1-a^2-2 a b x-b^2 x^2}}+3 \log \left (1+e^{2 i \arcsin (a+b x)}\right )\right )-6 i \arcsin (a+b x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a+b x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (a+b x)}\right )}{2 b} \]
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Time = 3.27 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.79
method | result | size |
default | \(\frac {\left (-x b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+i b^{2} x^{2}-a \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+2 i a b x +i a^{2}-i\right ) \arcsin \left (b x +a \right )^{3}}{b \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}+\frac {-4 i \arcsin \left (b x +a \right )^{3}+6 \arcsin \left (b x +a \right )^{2} \ln \left (1+\left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}\right )-6 i \arcsin \left (b x +a \right ) \operatorname {polylog}\left (2, -\left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}\right )+3 \operatorname {polylog}\left (3, -\left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}\right )}{2 b}\) | \(229\) |
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\[ \int \frac {\arcsin (a+b x)^3}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=\int { \frac {\arcsin \left (b x + a\right )^{3}}{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\arcsin (a+b x)^3}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {asin}^{3}{\left (a + b x \right )}}{\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.48 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.07 \[ \int \frac {\arcsin (a+b x)^3}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=\frac {3}{2} \, b {\left (\frac {\log \left (b x + a + 1\right )}{b^{2}} + \frac {\log \left (b x + a - 1\right )}{b^{2}}\right )} \arcsin \left (b x + a\right )^{2} + {\left (\frac {b^{2} x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} + \frac {a b}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}\right )} \arcsin \left (b x + a\right )^{3} \]
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\[ \int \frac {\arcsin (a+b x)^3}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=\int { \frac {\arcsin \left (b x + a\right )^{3}}{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\arcsin (a+b x)^3}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {asin}\left (a+b\,x\right )}^3}{{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2}} \,d x \]
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