Integrand size = 28, antiderivative size = 94 \[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=-\frac {x^2 \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}-\frac {\operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {4 a}{b}\right )}{2 b^2 c^3}+\frac {\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{2 b^2 c^3} \]
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Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4799, 4731, 4491, 12, 3384, 3380, 3383} \[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=-\frac {\sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{2 b^2 c^3}+\frac {\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{2 b^2 c^3}-\frac {x^2 \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))} \]
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Rule 12
Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 4731
Rule 4799
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}+\frac {2 \int \frac {x}{a+b \arcsin (c x)} \, dx}{b c}-\frac {(4 c) \int \frac {x^3}{a+b \arcsin (c x)} \, dx}{b} \\ & = -\frac {x^2 \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}-\frac {2 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b^2 c^3}+\frac {4 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b^2 c^3} \\ & = -\frac {x^2 \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}-\frac {2 \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x} \, dx,x,a+b \arcsin (c x)\right )}{b^2 c^3}+\frac {4 \text {Subst}\left (\int \left (-\frac {\sin \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 x}+\frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{4 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b^2 c^3} \\ & = -\frac {x^2 \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}-\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b^2 c^3} \\ & = -\frac {x^2 \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}+\frac {\cos \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b^2 c^3}-\frac {\sin \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b^2 c^3} \\ & = -\frac {x^2 \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}-\frac {\operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {4 a}{b}\right )}{2 b^2 c^3}+\frac {\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{2 b^2 c^3} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.87 \[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=\frac {\frac {2 b c^2 x^2 \left (-1+c^2 x^2\right )}{a+b \arcsin (c x)}-\operatorname {CosIntegral}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {4 a}{b}\right )+\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{2 b^2 c^3} \]
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Time = 0.12 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.45
method | result | size |
default | \(\frac {4 \arcsin \left (c x \right ) \operatorname {Si}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) b -4 \arcsin \left (c x \right ) \operatorname {Ci}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \sin \left (\frac {4 a}{b}\right ) b +4 \,\operatorname {Si}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a -4 \,\operatorname {Ci}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \sin \left (\frac {4 a}{b}\right ) a +\cos \left (4 \arcsin \left (c x \right )\right ) b -b}{8 c^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\) | \(136\) |
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\[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x^{2} \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (88) = 176\).
Time = 0.41 (sec) , antiderivative size = 563, normalized size of antiderivative = 5.99 \[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=-\frac {4 \, b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {4 \, b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right )^{4} \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac {4 \, a \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {4 \, a \cos \left (\frac {a}{b}\right )^{4} \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {2 \, b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac {4 \, b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {2 \, a \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac {4 \, a \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {b \arcsin \left (c x\right ) \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{2 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} + \frac {{\left (c^{2} x^{2} - 1\right )} b}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {a \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{2 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} \]
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Timed out. \[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x^2\,\sqrt {1-c^2\,x^2}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]
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