Integrand size = 10, antiderivative size = 83 \[ \int \frac {x^3}{\arccos (a x)^3} \, dx=\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {3 x^2}{2 a^2 \arccos (a x)}+\frac {2 x^4}{\arccos (a x)}+\frac {\text {Si}(2 \arccos (a x))}{2 a^4}+\frac {\text {Si}(4 \arccos (a x))}{a^4} \]
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Time = 0.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4730, 4808, 4732, 4491, 3380, 12} \[ \int \frac {x^3}{\arccos (a x)^3} \, dx=\frac {\text {Si}(2 \arccos (a x))}{2 a^4}+\frac {\text {Si}(4 \arccos (a x))}{a^4}-\frac {3 x^2}{2 a^2 \arccos (a x)}+\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}+\frac {2 x^4}{\arccos (a x)} \]
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Rule 12
Rule 3380
Rule 4491
Rule 4730
Rule 4732
Rule 4808
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2} \arccos (a x)^2} \, dx}{2 a}+(2 a) \int \frac {x^4}{\sqrt {1-a^2 x^2} \arccos (a x)^2} \, dx \\ & = \frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {3 x^2}{2 a^2 \arccos (a x)}+\frac {2 x^4}{\arccos (a x)}-8 \int \frac {x^3}{\arccos (a x)} \, dx+\frac {3 \int \frac {x}{\arccos (a x)} \, dx}{a^2} \\ & = \frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {3 x^2}{2 a^2 \arccos (a x)}+\frac {2 x^4}{\arccos (a x)}-\frac {3 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\arccos (a x)\right )}{a^4}+\frac {8 \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\arccos (a x)\right )}{a^4} \\ & = \frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {3 x^2}{2 a^2 \arccos (a x)}+\frac {2 x^4}{\arccos (a x)}-\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\arccos (a x)\right )}{a^4}+\frac {8 \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 x}+\frac {\sin (4 x)}{8 x}\right ) \, dx,x,\arccos (a x)\right )}{a^4} \\ & = \frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {3 x^2}{2 a^2 \arccos (a x)}+\frac {2 x^4}{\arccos (a x)}+\frac {\text {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\arccos (a x)\right )}{a^4}-\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\arccos (a x)\right )}{2 a^4}+\frac {2 \text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\arccos (a x)\right )}{a^4} \\ & = \frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {3 x^2}{2 a^2 \arccos (a x)}+\frac {2 x^4}{\arccos (a x)}+\frac {\text {Si}(2 \arccos (a x))}{2 a^4}+\frac {\text {Si}(4 \arccos (a x))}{a^4} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.84 \[ \int \frac {x^3}{\arccos (a x)^3} \, dx=\frac {\frac {a^2 x^2 \left (a x \sqrt {1-a^2 x^2}+\left (-3+4 a^2 x^2\right ) \arccos (a x)\right )}{\arccos (a x)^2}+\text {Si}(2 \arccos (a x))+2 \text {Si}(4 \arccos (a x))}{2 a^4} \]
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Time = 0.60 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.99
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\frac {\operatorname {Si}\left (2 \arccos \left (a x \right )\right )}{2}+\frac {\sin \left (4 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )^{2}}+\frac {\cos \left (4 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\operatorname {Si}\left (4 \arccos \left (a x \right )\right )}{a^{4}}\) | \(82\) |
default | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\frac {\operatorname {Si}\left (2 \arccos \left (a x \right )\right )}{2}+\frac {\sin \left (4 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )^{2}}+\frac {\cos \left (4 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\operatorname {Si}\left (4 \arccos \left (a x \right )\right )}{a^{4}}\) | \(82\) |
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\[ \int \frac {x^3}{\arccos (a x)^3} \, dx=\int { \frac {x^{3}}{\arccos \left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {x^3}{\arccos (a x)^3} \, dx=\int \frac {x^{3}}{\operatorname {acos}^{3}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^3}{\arccos (a x)^3} \, dx=\int { \frac {x^{3}}{\arccos \left (a x\right )^{3}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{\arccos (a x)^3} \, dx=\frac {2 \, x^{4}}{\arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{2 \, a \arccos \left (a x\right )^{2}} - \frac {3 \, x^{2}}{2 \, a^{2} \arccos \left (a x\right )} + \frac {\operatorname {Si}\left (4 \, \arccos \left (a x\right )\right )}{a^{4}} + \frac {\operatorname {Si}\left (2 \, \arccos \left (a x\right )\right )}{2 \, a^{4}} \]
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Timed out. \[ \int \frac {x^3}{\arccos (a x)^3} \, dx=\int \frac {x^3}{{\mathrm {acos}\left (a\,x\right )}^3} \,d x \]
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