Integrand size = 28, antiderivative size = 245 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=-\frac {3 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{128 b c^4}-\frac {\operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{32 b c^4}+\frac {3 \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{256 b c^4}+\frac {\operatorname {CosIntegral}\left (\frac {9 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {9 a}{b}\right )}{256 b c^4}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{128 b c^4}+\frac {\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{32 b c^4}-\frac {3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{256 b c^4}-\frac {\cos \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 (a+b \arcsin (c x))}{b}\right )}{256 b c^4} \]
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Time = 0.28 (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} \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=-\frac {3 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{128 b c^4}-\frac {\sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{32 b c^4}+\frac {3 \sin \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{256 b c^4}+\frac {\sin \left (\frac {9 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {9 (a+b \arcsin (c x))}{b}\right )}{256 b c^4}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{128 b c^4}+\frac {\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{32 b c^4}-\frac {3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{256 b c^4}-\frac {\cos \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 (a+b \arcsin (c x))}{b}\right )}{256 b c^4} \]
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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 ^6\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 {9 a}{b}-\frac {9 x}{b}\right )}{256 x}-\frac {3 \sin \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{256 x}+\frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{32 x}+\frac {3 \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{128 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b c^4} \\ & = \frac {\text {Subst}\left (\int \frac {\sin \left (\frac {9 a}{b}-\frac {9 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{256 b c^4}+\frac {3 \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 )}{256 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 )}{128 b c^4}-\frac {\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 )}{32 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 )}{128 b c^4}+\frac {\cos \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{32 b c^4}-\frac {\left (3 \cos \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{256 b c^4}-\frac {\cos \left (\frac {9 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {9 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{256 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 )}{128 b c^4}-\frac {\sin \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{32 b c^4}+\frac {\left (3 \sin \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{256 b c^4}+\frac {\sin \left (\frac {9 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {9 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{256 b c^4} \\ & = -\frac {3 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{128 b c^4}-\frac {\operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{32 b c^4}+\frac {3 \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{256 b c^4}+\frac {\operatorname {CosIntegral}\left (\frac {9 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {9 a}{b}\right )}{256 b c^4}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{128 b c^4}+\frac {\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{32 b c^4}-\frac {3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{256 b c^4}-\frac {\cos \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 (a+b \arcsin (c x))}{b}\right )}{256 b c^4} \\ \end{align*}
Time = 0.97 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\frac {-6 \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )-8 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+3 \operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {7 a}{b}\right )+\operatorname {CosIntegral}\left (9 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {9 a}{b}\right )+6 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+8 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-\cos \left (\frac {9 a}{b}\right ) \text {Si}\left (9 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{256 b c^4} \]
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Time = 0.13 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {6 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )-6 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+8 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )-8 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )-3 \,\operatorname {Si}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right )-\operatorname {Si}\left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \cos \left (\frac {9 a}{b}\right )+\operatorname {Ci}\left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \sin \left (\frac {9 a}{b}\right )+3 \,\operatorname {Ci}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{256 c^{4} b}\) | \(185\) |
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\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{b \arcsin \left (c x\right ) + a} \,d x } \]
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\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\int \frac {x^{3} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]
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\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{b \arcsin \left (c x\right ) + a} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (229) = 458\).
Time = 0.35 (sec) , antiderivative size = 746, normalized size of antiderivative = 3.04 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\int \frac {x^3\,{\left (1-c^2\,x^2\right )}^{5/2}}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]
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