Integrand size = 10, antiderivative size = 141 \[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {x}{3 a^2 \arccos (a x)^2}+\frac {x^3}{2 \arccos (a x)^2}+\frac {\sqrt {1-a^2 x^2}}{3 a^3 \arccos (a x)}-\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)}+\frac {\operatorname {CosIntegral}(\arccos (a x))}{24 a^3}+\frac {9 \operatorname {CosIntegral}(3 \arccos (a x))}{8 a^3} \]
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Time = 0.21 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4730, 4808, 4728, 3383, 4718, 4810} \[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\frac {\operatorname {CosIntegral}(\arccos (a x))}{24 a^3}+\frac {9 \operatorname {CosIntegral}(3 \arccos (a x))}{8 a^3}-\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)}+\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {x}{3 a^2 \arccos (a x)^2}+\frac {\sqrt {1-a^2 x^2}}{3 a^3 \arccos (a x)}+\frac {x^3}{2 \arccos (a x)^2} \]
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Rule 3383
Rule 4718
Rule 4728
Rule 4730
Rule 4808
Rule 4810
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {2 \int \frac {x}{\sqrt {1-a^2 x^2} \arccos (a x)^3} \, dx}{3 a}+a \int \frac {x^3}{\sqrt {1-a^2 x^2} \arccos (a x)^3} \, dx \\ & = \frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {x}{3 a^2 \arccos (a x)^2}+\frac {x^3}{2 \arccos (a x)^2}-\frac {3}{2} \int \frac {x^2}{\arccos (a x)^2} \, dx+\frac {\int \frac {1}{\arccos (a x)^2} \, dx}{3 a^2} \\ & = \frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {x}{3 a^2 \arccos (a x)^2}+\frac {x^3}{2 \arccos (a x)^2}+\frac {\sqrt {1-a^2 x^2}}{3 a^3 \arccos (a x)}-\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)}-\frac {3 \text {Subst}\left (\int \left (-\frac {\cos (x)}{4 x}-\frac {3 \cos (3 x)}{4 x}\right ) \, dx,x,\arccos (a x)\right )}{2 a^3}+\frac {\int \frac {x}{\sqrt {1-a^2 x^2} \arccos (a x)} \, dx}{3 a} \\ & = \frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {x}{3 a^2 \arccos (a x)^2}+\frac {x^3}{2 \arccos (a x)^2}+\frac {\sqrt {1-a^2 x^2}}{3 a^3 \arccos (a x)}-\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arccos (a x)\right )}{3 a^3}+\frac {3 \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arccos (a x)\right )}{8 a^3}+\frac {9 \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arccos (a x)\right )}{8 a^3} \\ & = \frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {x}{3 a^2 \arccos (a x)^2}+\frac {x^3}{2 \arccos (a x)^2}+\frac {\sqrt {1-a^2 x^2}}{3 a^3 \arccos (a x)}-\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)}+\frac {\operatorname {CosIntegral}(\arccos (a x))}{24 a^3}+\frac {9 \operatorname {CosIntegral}(3 \arccos (a x))}{8 a^3} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\frac {\frac {8 a^2 x^2 \sqrt {1-a^2 x^2}}{\arccos (a x)^3}+\frac {4 a x \left (-2+3 a^2 x^2\right )}{\arccos (a x)^2}-\frac {4 \sqrt {1-a^2 x^2} \left (-2+9 a^2 x^2\right )}{\arccos (a x)}-80 \operatorname {CosIntegral}(\arccos (a x))+27 (3 \operatorname {CosIntegral}(\arccos (a x))+\operatorname {CosIntegral}(3 \arccos (a x)))}{24 a^3} \]
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Time = 0.64 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{12 \arccos \left (a x \right )^{3}}+\frac {a x}{24 \arccos \left (a x \right )^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arccos \left (a x \right )}+\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{24}+\frac {\sin \left (3 \arccos \left (a x \right )\right )}{12 \arccos \left (a x \right )^{3}}+\frac {\cos \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}-\frac {3 \sin \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )}+\frac {9 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{8}}{a^{3}}\) | \(117\) |
default | \(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{12 \arccos \left (a x \right )^{3}}+\frac {a x}{24 \arccos \left (a x \right )^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arccos \left (a x \right )}+\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{24}+\frac {\sin \left (3 \arccos \left (a x \right )\right )}{12 \arccos \left (a x \right )^{3}}+\frac {\cos \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}-\frac {3 \sin \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )}+\frac {9 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{8}}{a^{3}}\) | \(117\) |
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\[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\int { \frac {x^{2}}{\arccos \left (a x\right )^{4}} \,d x } \]
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\[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\int \frac {x^{2}}{\operatorname {acos}^{4}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\int { \frac {x^{2}}{\arccos \left (a x\right )^{4}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86 \[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\frac {x^{3}}{2 \, \arccos \left (a x\right )^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{2 \, a \arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{3 \, a \arccos \left (a x\right )^{3}} + \frac {9 \, \operatorname {Ci}\left (3 \, \arccos \left (a x\right )\right )}{8 \, a^{3}} + \frac {\operatorname {Ci}\left (\arccos \left (a x\right )\right )}{24 \, a^{3}} - \frac {x}{3 \, a^{2} \arccos \left (a x\right )^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, a^{3} \arccos \left (a x\right )} \]
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Timed out. \[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\int \frac {x^2}{{\mathrm {acos}\left (a\,x\right )}^4} \,d x \]
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