Integrand size = 28, antiderivative size = 282 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}+\frac {\operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{16 b^2 c^3}-\frac {\operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {4 a}{b}\right )}{8 b^2 c^3}-\frac {3 \operatorname {CosIntegral}\left (\frac {6 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {6 a}{b}\right )}{16 b^2 c^3}-\frac {\operatorname {CosIntegral}\left (\frac {8 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {8 a}{b}\right )}{16 b^2 c^3}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 b^2 c^3}+\frac {3 \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {8 a}{b}\right ) \text {Si}\left (\frac {8 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3} \]
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Time = 0.54 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4799, 4809, 4491, 3384, 3380, 3383} \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\frac {\sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3}-\frac {\sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 b^2 c^3}-\frac {3 \sin \left (\frac {6 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {6 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3}-\frac {\sin \left (\frac {8 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {8 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 b^2 c^3}+\frac {3 \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {8 a}{b}\right ) \text {Si}\left (\frac {8 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3}-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 4799
Rule 4809
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}+\frac {2 \int \frac {x \left (1-c^2 x^2\right )^2}{a+b \arcsin (c x)} \, dx}{b c}-\frac {(8 c) \int \frac {x^3 \left (1-c^2 x^2\right )^2}{a+b \arcsin (c x)} \, dx}{b} \\ & = -\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}-\frac {2 \text {Subst}\left (\int \frac {\cos ^5\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 {8 \text {Subst}\left (\int \frac {\cos ^5\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 )^3}{b c (a+b \arcsin (c x))}-\frac {2 \text {Subst}\left (\int \left (\frac {\sin \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{32 x}+\frac {\sin \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 x}+\frac {5 \sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{32 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b^2 c^3}+\frac {8 \text {Subst}\left (\int \left (-\frac {\sin \left (\frac {8 a}{b}-\frac {8 x}{b}\right )}{128 x}-\frac {\sin \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{64 x}+\frac {\sin \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{64 x}+\frac {3 \sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{64 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b^2 c^3} \\ & = -\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}-\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {8 a}{b}-\frac {8 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{16 b^2 c^3}-\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{16 b^2 c^3}-\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{8 b^2 c^3}+\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 )}{8 b^2 c^3}-\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 )}{4 b^2 c^3}-\frac {5 \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{16 b^2 c^3}+\frac {3 \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{8 b^2 c^3} \\ & = -\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}+\frac {\left (5 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{16 b^2 c^3}-\frac {\left (3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{8 b^2 c^3}-\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 )}{8 b^2 c^3}+\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 )}{4 b^2 c^3}+\frac {\cos \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{8 b^2 c^3}+\frac {\cos \left (\frac {8 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {8 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{16 b^2 c^3}-\frac {\left (5 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{16 b^2 c^3}+\frac {\left (3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{8 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 )}{8 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 )}{4 b^2 c^3}-\frac {\sin \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{16 b^2 c^3}-\frac {\sin \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{8 b^2 c^3}-\frac {\sin \left (\frac {8 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {8 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{16 b^2 c^3} \\ & = -\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}+\frac {\operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{16 b^2 c^3}-\frac {\operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {4 a}{b}\right )}{8 b^2 c^3}-\frac {3 \operatorname {CosIntegral}\left (\frac {6 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {6 a}{b}\right )}{16 b^2 c^3}-\frac {\operatorname {CosIntegral}\left (\frac {8 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {8 a}{b}\right )}{16 b^2 c^3}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 b^2 c^3}+\frac {3 \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {8 a}{b}\right ) \text {Si}\left (\frac {8 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3} \\ \end{align*}
Time = 0.95 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.47 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\frac {-16 b c^2 x^2+48 b c^4 x^4-48 b c^6 x^6+16 b c^8 x^8+(a+b \arcsin (c x)) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {2 a}{b}\right )-2 (a+b \arcsin (c x)) \operatorname {CosIntegral}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {4 a}{b}\right )-3 a \operatorname {CosIntegral}\left (6 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {6 a}{b}\right )-3 b \arcsin (c x) \operatorname {CosIntegral}\left (6 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {6 a}{b}\right )-a \operatorname {CosIntegral}\left (8 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {8 a}{b}\right )-b \arcsin (c x) \operatorname {CosIntegral}\left (8 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {8 a}{b}\right )-a \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-b \arcsin (c x) \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+2 a \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+2 b \arcsin (c x) \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+3 a \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (6 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+3 b \arcsin (c x) \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (6 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+a \cos \left (\frac {8 a}{b}\right ) \text {Si}\left (8 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+b \arcsin (c x) \cos \left (\frac {8 a}{b}\right ) \text {Si}\left (8 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{16 b^2 c^3 (a+b \arcsin (c x))} \]
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Time = 0.13 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.70
method | result | size |
default | \(\frac {16 \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 -16 \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 +8 \arcsin \left (c x \right ) \operatorname {Si}\left (8 \arcsin \left (c x \right )+\frac {8 a}{b}\right ) \cos \left (\frac {8 a}{b}\right ) b -8 \arcsin \left (c x \right ) \operatorname {Ci}\left (8 \arcsin \left (c x \right )+\frac {8 a}{b}\right ) \sin \left (\frac {8 a}{b}\right ) b -8 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b +8 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b +24 \arcsin \left (c x \right ) \operatorname {Si}\left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \cos \left (\frac {6 a}{b}\right ) b -24 \arcsin \left (c x \right ) \operatorname {Ci}\left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \sin \left (\frac {6 a}{b}\right ) b +16 \,\operatorname {Si}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a -16 \,\operatorname {Ci}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \sin \left (\frac {4 a}{b}\right ) a +8 \,\operatorname {Si}\left (8 \arcsin \left (c x \right )+\frac {8 a}{b}\right ) \cos \left (\frac {8 a}{b}\right ) a -8 \,\operatorname {Ci}\left (8 \arcsin \left (c x \right )+\frac {8 a}{b}\right ) \sin \left (\frac {8 a}{b}\right ) a -8 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a +8 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a +24 \,\operatorname {Si}\left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \cos \left (\frac {6 a}{b}\right ) a -24 \,\operatorname {Ci}\left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \sin \left (\frac {6 a}{b}\right ) a +4 \cos \left (4 \arcsin \left (c x \right )\right ) b +\cos \left (8 \arcsin \left (c x \right )\right ) b -4 \cos \left (2 \arcsin \left (c x \right )\right ) b +4 \cos \left (6 \arcsin \left (c x \right )\right ) b -5 b}{128 c^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\) | \(478\) |
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\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x^{2} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 2461 vs. \(2 (264) = 528\).
Time = 0.44 (sec) , antiderivative size = 2461, normalized size of antiderivative = 8.73 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x^2\,{\left (1-c^2\,x^2\right )}^{5/2}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]
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