Integrand size = 12, antiderivative size = 176 \[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=-\frac {x^2 \sqrt {1-(a+b x)^2}}{2 b \arcsin (a+b x)^2}+\frac {a^2 (a+b x)}{2 b^3 \arcsin (a+b x)}-\frac {2 a (a+b x)^2}{b^3 \arcsin (a+b x)}+\frac {9 a+b x}{8 b^3 \arcsin (a+b x)}-\frac {\left (1+4 a^2\right ) \operatorname {CosIntegral}(\arcsin (a+b x))}{8 b^3}+\frac {9 \operatorname {CosIntegral}(3 \arcsin (a+b x))}{8 b^3}-\frac {3 \sin (3 \arcsin (a+b x))}{8 b^3 \arcsin (a+b x)}+\frac {2 a \text {Si}(2 \arcsin (a+b x))}{b^3} \]
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Time = 0.36 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.49, number of steps used = 24, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4889, 4829, 4717, 4807, 4719, 3383, 4729, 4731, 4491, 12, 3380, 4737} \[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=-\frac {a^2 \operatorname {CosIntegral}(\arcsin (a+b x))}{2 b^3}+\frac {a^2 (a+b x)}{2 b^3 \arcsin (a+b x)}-\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \arcsin (a+b x)^2}-\frac {\operatorname {CosIntegral}(\arcsin (a+b x))}{8 b^3}+\frac {9 \operatorname {CosIntegral}(3 \arcsin (a+b x))}{8 b^3}+\frac {2 a \text {Si}(2 \arcsin (a+b x))}{b^3}+\frac {3 (a+b x)^3}{2 b^3 \arcsin (a+b x)}-\frac {2 a (a+b x)^2}{b^3 \arcsin (a+b x)}-\frac {\sqrt {1-(a+b x)^2} (a+b x)^2}{2 b^3 \arcsin (a+b x)^2}-\frac {a+b x}{b^3 \arcsin (a+b x)}+\frac {a \sqrt {1-(a+b x)^2} (a+b x)}{b^3 \arcsin (a+b x)^2}+\frac {a}{b^3 \arcsin (a+b x)} \]
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Rule 12
Rule 3380
Rule 3383
Rule 4491
Rule 4717
Rule 4719
Rule 4729
Rule 4731
Rule 4737
Rule 4807
Rule 4829
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\arcsin (x)^3} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^2}{b^2 \arcsin (x)^3}-\frac {2 a x}{b^2 \arcsin (x)^3}+\frac {x^2}{b^2 \arcsin (x)^3}\right ) \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {x^2}{\arcsin (x)^3} \, dx,x,a+b x\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int \frac {x}{\arcsin (x)^3} \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \frac {1}{\arcsin (x)^3} \, dx,x,a+b x\right )}{b^3} \\ & = -\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \arcsin (a+b x)^2}+\frac {a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \arcsin (a+b x)^2}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{2 b^3 \arcsin (a+b x)^2}+\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \arcsin (x)^2} \, dx,x,a+b x\right )}{b^3}-\frac {3 \text {Subst}\left (\int \frac {x^3}{\sqrt {1-x^2} \arcsin (x)^2} \, dx,x,a+b x\right )}{2 b^3}-\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \arcsin (x)^2} \, dx,x,a+b x\right )}{b^3}+\frac {(2 a) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} \arcsin (x)^2} \, dx,x,a+b x\right )}{b^3}-\frac {a^2 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \arcsin (x)^2} \, dx,x,a+b x\right )}{2 b^3} \\ & = -\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \arcsin (a+b x)^2}+\frac {a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \arcsin (a+b x)^2}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{2 b^3 \arcsin (a+b x)^2}+\frac {a}{b^3 \arcsin (a+b x)}-\frac {a+b x}{b^3 \arcsin (a+b x)}+\frac {a^2 (a+b x)}{2 b^3 \arcsin (a+b x)}-\frac {2 a (a+b x)^2}{b^3 \arcsin (a+b x)}+\frac {3 (a+b x)^3}{2 b^3 \arcsin (a+b x)}+\frac {\text {Subst}\left (\int \frac {1}{\arcsin (x)} \, dx,x,a+b x\right )}{b^3}-\frac {9 \text {Subst}\left (\int \frac {x^2}{\arcsin (x)} \, dx,x,a+b x\right )}{2 b^3}+\frac {(4 a) \text {Subst}\left (\int \frac {x}{\arcsin (x)} \, dx,x,a+b x\right )}{b^3}-\frac {a^2 \text {Subst}\left (\int \frac {1}{\arcsin (x)} \, dx,x,a+b x\right )}{2 b^3} \\ & = -\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \arcsin (a+b x)^2}+\frac {a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \arcsin (a+b x)^2}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{2 b^3 \arcsin (a+b x)^2}+\frac {a}{b^3 \arcsin (a+b x)}-\frac {a+b x}{b^3 \arcsin (a+b x)}+\frac {a^2 (a+b x)}{2 b^3 \arcsin (a+b x)}-\frac {2 a (a+b x)^2}{b^3 \arcsin (a+b x)}+\frac {3 (a+b x)^3}{2 b^3 \arcsin (a+b x)}+\frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arcsin (a+b x)\right )}{b^3}-\frac {9 \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{x} \, dx,x,\arcsin (a+b x)\right )}{2 b^3}+\frac {(4 a) \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\arcsin (a+b x)\right )}{b^3}-\frac {a^2 \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arcsin (a+b x)\right )}{2 b^3} \\ & = -\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \arcsin (a+b x)^2}+\frac {a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \arcsin (a+b x)^2}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{2 b^3 \arcsin (a+b x)^2}+\frac {a}{b^3 \arcsin (a+b x)}-\frac {a+b x}{b^3 \arcsin (a+b x)}+\frac {a^2 (a+b x)}{2 b^3 \arcsin (a+b x)}-\frac {2 a (a+b x)^2}{b^3 \arcsin (a+b x)}+\frac {3 (a+b x)^3}{2 b^3 \arcsin (a+b x)}+\frac {\operatorname {CosIntegral}(\arcsin (a+b x))}{b^3}-\frac {a^2 \operatorname {CosIntegral}(\arcsin (a+b x))}{2 b^3}-\frac {9 \text {Subst}\left (\int \left (\frac {\cos (x)}{4 x}-\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\arcsin (a+b x)\right )}{2 b^3}+\frac {(4 a) \text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\arcsin (a+b x)\right )}{b^3} \\ & = -\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \arcsin (a+b x)^2}+\frac {a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \arcsin (a+b x)^2}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{2 b^3 \arcsin (a+b x)^2}+\frac {a}{b^3 \arcsin (a+b x)}-\frac {a+b x}{b^3 \arcsin (a+b x)}+\frac {a^2 (a+b x)}{2 b^3 \arcsin (a+b x)}-\frac {2 a (a+b x)^2}{b^3 \arcsin (a+b x)}+\frac {3 (a+b x)^3}{2 b^3 \arcsin (a+b x)}+\frac {\operatorname {CosIntegral}(\arcsin (a+b x))}{b^3}-\frac {a^2 \operatorname {CosIntegral}(\arcsin (a+b x))}{2 b^3}-\frac {9 \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arcsin (a+b x)\right )}{8 b^3}+\frac {9 \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arcsin (a+b x)\right )}{8 b^3}+\frac {(2 a) \text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\arcsin (a+b x)\right )}{b^3} \\ & = -\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \arcsin (a+b x)^2}+\frac {a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \arcsin (a+b x)^2}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{2 b^3 \arcsin (a+b x)^2}+\frac {a}{b^3 \arcsin (a+b x)}-\frac {a+b x}{b^3 \arcsin (a+b x)}+\frac {a^2 (a+b x)}{2 b^3 \arcsin (a+b x)}-\frac {2 a (a+b x)^2}{b^3 \arcsin (a+b x)}+\frac {3 (a+b x)^3}{2 b^3 \arcsin (a+b x)}-\frac {\operatorname {CosIntegral}(\arcsin (a+b x))}{8 b^3}-\frac {a^2 \operatorname {CosIntegral}(\arcsin (a+b x))}{2 b^3}+\frac {9 \operatorname {CosIntegral}(3 \arcsin (a+b x))}{8 b^3}+\frac {2 a \text {Si}(2 \arcsin (a+b x))}{b^3} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.65 \[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=\frac {\frac {4 b x \left (-b x \sqrt {1-a^2-2 a b x-b^2 x^2}+\left (-2+2 a^2+5 a b x+3 b^2 x^2\right ) \arcsin (a+b x)\right )}{\arcsin (a+b x)^2}-\left (1+4 a^2\right ) \operatorname {CosIntegral}(\arcsin (a+b x))+9 \operatorname {CosIntegral}(3 \arcsin (a+b x))+16 a \text {Si}(2 \arcsin (a+b x))}{8 b^3} \]
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Time = 0.70 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {1-\left (b x +a \right )^{2}}}{8 \arcsin \left (b x +a \right )^{2}}+\frac {b x +a}{8 \arcsin \left (b x +a \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (b x +a \right )\right )}{8}+\frac {\cos \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )^{2}}-\frac {3 \sin \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )}+\frac {9 \,\operatorname {Ci}\left (3 \arcsin \left (b x +a \right )\right )}{8}+\frac {a \left (4 \,\operatorname {Si}\left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}+2 \cos \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )+\sin \left (2 \arcsin \left (b x +a \right )\right )\right )}{2 \arcsin \left (b x +a \right )^{2}}-\frac {a^{2} \left (\operatorname {Ci}\left (\arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}-\arcsin \left (b x +a \right ) \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \arcsin \left (b x +a \right )^{2}}}{b^{3}}\) | \(215\) |
default | \(\frac {-\frac {\sqrt {1-\left (b x +a \right )^{2}}}{8 \arcsin \left (b x +a \right )^{2}}+\frac {b x +a}{8 \arcsin \left (b x +a \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (b x +a \right )\right )}{8}+\frac {\cos \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )^{2}}-\frac {3 \sin \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )}+\frac {9 \,\operatorname {Ci}\left (3 \arcsin \left (b x +a \right )\right )}{8}+\frac {a \left (4 \,\operatorname {Si}\left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}+2 \cos \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )+\sin \left (2 \arcsin \left (b x +a \right )\right )\right )}{2 \arcsin \left (b x +a \right )^{2}}-\frac {a^{2} \left (\operatorname {Ci}\left (\arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}-\arcsin \left (b x +a \right ) \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \arcsin \left (b x +a \right )^{2}}}{b^{3}}\) | \(215\) |
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\[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=\int { \frac {x^{2}}{\arcsin \left (b x + a\right )^{3}} \,d x } \]
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\[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=\int \frac {x^{2}}{\operatorname {asin}^{3}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=\int { \frac {x^{2}}{\arcsin \left (b x + a\right )^{3}} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.55 \[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=-\frac {a^{2} \operatorname {Ci}\left (\arcsin \left (b x + a\right )\right )}{2 \, b^{3}} + \frac {{\left (b x + a\right )} a^{2}}{2 \, b^{3} \arcsin \left (b x + a\right )} + \frac {2 \, a \operatorname {Si}\left (2 \, \arcsin \left (b x + a\right )\right )}{b^{3}} + \frac {3 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )}}{2 \, b^{3} \arcsin \left (b x + a\right )} - \frac {2 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} a}{b^{3} \arcsin \left (b x + a\right )} + \frac {9 \, \operatorname {Ci}\left (3 \, \arcsin \left (b x + a\right )\right )}{8 \, b^{3}} - \frac {\operatorname {Ci}\left (\arcsin \left (b x + a\right )\right )}{8 \, b^{3}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a}{b^{3} \arcsin \left (b x + a\right )^{2}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2}}{2 \, b^{3} \arcsin \left (b x + a\right )^{2}} + \frac {b x + a}{2 \, b^{3} \arcsin \left (b x + a\right )} - \frac {a}{b^{3} \arcsin \left (b x + a\right )} + \frac {{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{2 \, b^{3} \arcsin \left (b x + a\right )^{2}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{2 \, b^{3} \arcsin \left (b x + a\right )^{2}} \]
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Timed out. \[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=\int \frac {x^2}{{\mathrm {asin}\left (a+b\,x\right )}^3} \,d x \]
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