Integrand size = 28, antiderivative size = 245 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{a+b \arcsin (c x)} \, dx=-\frac {3 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{64 b c^4}-\frac {3 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{64 b c^4}+\frac {\operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{64 b c^4}+\frac {\operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{64 b c^4}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{64 b c^4}+\frac {3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 b c^4}-\frac {\cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 b c^4}-\frac {\cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 b c^4} \]
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Time = 0.30 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {4809, 4491, 3384, 3380, 3383} \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{a+b \arcsin (c x)} \, dx=-\frac {3 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{64 b c^4}-\frac {3 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 b c^4}+\frac {\sin \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 b c^4}+\frac {\sin \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 b c^4}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{64 b c^4}+\frac {3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 b c^4}-\frac {\cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 b c^4}-\frac {\cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 b c^4} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 4809
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cos ^4\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 c^4} \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {\sin \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{64 x}-\frac {\sin \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{64 x}+\frac {3 \sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{64 x}+\frac {3 \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{64 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b c^4} \\ & = \frac {\text {Subst}\left (\int \frac {\sin \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^4}+\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^4}-\frac {3 \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^4}-\frac {3 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^4} \\ & = \frac {\left (3 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^4}+\frac {\left (3 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^4}-\frac {\cos \left (\frac {5 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^4}-\frac {\cos \left (\frac {7 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^4}-\frac {\left (3 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^4}-\frac {\left (3 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^4}+\frac {\sin \left (\frac {5 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^4}+\frac {\sin \left (\frac {7 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^4} \\ & = -\frac {3 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{64 b c^4}-\frac {3 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{64 b c^4}+\frac {\operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{64 b c^4}+\frac {\operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{64 b c^4}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{64 b c^4}+\frac {3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 b c^4}-\frac {\cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 b c^4}-\frac {\cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 b c^4} \\ \end{align*}
Time = 0.76 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{a+b \arcsin (c x)} \, dx=\frac {-3 \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )-3 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+\operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {5 a}{b}\right )+\operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {7 a}{b}\right )+3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-\cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-\cos \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{64 b c^4} \]
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Time = 0.13 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.75
method | result | size |
default | \(-\frac {\operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right )-\operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right )-3 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )+3 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )-3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )+3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )+\operatorname {Si}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right )-\operatorname {Ci}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{64 c^{4} b}\) | \(184\) |
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\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{b \arcsin \left (c x\right ) + a} \,d x } \]
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\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{a+b \arcsin (c x)} \, dx=\int \frac {x^{3} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]
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\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{b \arcsin \left (c x\right ) + a} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (229) = 458\).
Time = 0.34 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.51 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{a+b \arcsin (c x)} \, dx=\frac {\cos \left (\frac {a}{b}\right )^{6} \operatorname {Ci}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b c^{4}} - \frac {\cos \left (\frac {a}{b}\right )^{7} \operatorname {Si}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right )}{b c^{4}} - \frac {5 \, \cos \left (\frac {a}{b}\right )^{4} \operatorname {Ci}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, b c^{4}} + \frac {\cos \left (\frac {a}{b}\right )^{4} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, b c^{4}} + \frac {7 \, \cos \left (\frac {a}{b}\right )^{5} \operatorname {Si}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right )}{4 \, b c^{4}} - \frac {\cos \left (\frac {a}{b}\right )^{5} \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{4 \, b c^{4}} + \frac {3 \, \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{8 \, b c^{4}} - \frac {3 \, \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{16 \, b c^{4}} - \frac {3 \, \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{16 \, b c^{4}} - \frac {7 \, \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right )}{8 \, b c^{4}} + \frac {5 \, \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{16 \, b c^{4}} + \frac {3 \, \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{16 \, b c^{4}} - \frac {\operatorname {Ci}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{64 \, b c^{4}} + \frac {\operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{64 \, b c^{4}} + \frac {3 \, \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{64 \, b c^{4}} - \frac {3 \, \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{64 \, b c^{4}} + \frac {7 \, \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right )}{64 \, b c^{4}} - \frac {5 \, \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{64 \, b c^{4}} - \frac {9 \, \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{64 \, b c^{4}} + \frac {3 \, \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{64 \, b c^{4}} \]
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Timed out. \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{a+b \arcsin (c x)} \, dx=\int \frac {x^3\,{\left (1-c^2\,x^2\right )}^{3/2}}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]
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