Integrand size = 10, antiderivative size = 158 \[ \int \frac {x^4}{\arccos (a x)^4} \, dx=\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {2 x^3}{3 a^2 \arccos (a x)^2}+\frac {5 x^5}{6 \arccos (a x)^2}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \arccos (a x)}-\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \arccos (a x)}+\frac {\operatorname {CosIntegral}(\arccos (a x))}{48 a^5}+\frac {27 \operatorname {CosIntegral}(3 \arccos (a x))}{32 a^5}+\frac {125 \operatorname {CosIntegral}(5 \arccos (a x))}{96 a^5} \]
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Time = 0.22 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4730, 4808, 4728, 3383} \[ \int \frac {x^4}{\arccos (a x)^4} \, dx=\frac {\operatorname {CosIntegral}(\arccos (a x))}{48 a^5}+\frac {27 \operatorname {CosIntegral}(3 \arccos (a x))}{32 a^5}+\frac {125 \operatorname {CosIntegral}(5 \arccos (a x))}{96 a^5}-\frac {2 x^3}{3 a^2 \arccos (a x)^2}-\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \arccos (a x)}+\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \arccos (a x)}+\frac {5 x^5}{6 \arccos (a x)^2} \]
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Rule 3383
Rule 4728
Rule 4730
Rule 4808
Rubi steps \begin{align*} \text {integral}& = \frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {4 \int \frac {x^3}{\sqrt {1-a^2 x^2} \arccos (a x)^3} \, dx}{3 a}+\frac {1}{3} (5 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \arccos (a x)^3} \, dx \\ & = \frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {2 x^3}{3 a^2 \arccos (a x)^2}+\frac {5 x^5}{6 \arccos (a x)^2}-\frac {25}{6} \int \frac {x^4}{\arccos (a x)^2} \, dx+\frac {2 \int \frac {x^2}{\arccos (a x)^2} \, dx}{a^2} \\ & = \frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {2 x^3}{3 a^2 \arccos (a x)^2}+\frac {5 x^5}{6 \arccos (a x)^2}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \arccos (a x)}-\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \arccos (a x)}+\frac {2 \text {Subst}\left (\int \left (-\frac {\cos (x)}{4 x}-\frac {3 \cos (3 x)}{4 x}\right ) \, dx,x,\arccos (a x)\right )}{a^5}-\frac {25 \text {Subst}\left (\int \left (-\frac {\cos (x)}{8 x}-\frac {9 \cos (3 x)}{16 x}-\frac {5 \cos (5 x)}{16 x}\right ) \, dx,x,\arccos (a x)\right )}{6 a^5} \\ & = \frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {2 x^3}{3 a^2 \arccos (a x)^2}+\frac {5 x^5}{6 \arccos (a x)^2}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \arccos (a x)}-\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \arccos (a x)}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arccos (a x)\right )}{2 a^5}+\frac {25 \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arccos (a x)\right )}{48 a^5}+\frac {125 \text {Subst}\left (\int \frac {\cos (5 x)}{x} \, dx,x,\arccos (a x)\right )}{96 a^5}-\frac {3 \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arccos (a x)\right )}{2 a^5}+\frac {75 \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arccos (a x)\right )}{32 a^5} \\ & = \frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {2 x^3}{3 a^2 \arccos (a x)^2}+\frac {5 x^5}{6 \arccos (a x)^2}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \arccos (a x)}-\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \arccos (a x)}+\frac {\operatorname {CosIntegral}(\arccos (a x))}{48 a^5}+\frac {27 \operatorname {CosIntegral}(3 \arccos (a x))}{32 a^5}+\frac {125 \operatorname {CosIntegral}(5 \arccos (a x))}{96 a^5} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.01 \[ \int \frac {x^4}{\arccos (a x)^4} \, dx=\frac {32 a^4 x^4 \sqrt {1-a^2 x^2}-64 a^3 x^3 \arccos (a x)+80 a^5 x^5 \arccos (a x)+192 a^2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2-400 a^4 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^2+2 \arccos (a x)^3 \operatorname {CosIntegral}(\arccos (a x))+81 \arccos (a x)^3 \operatorname {CosIntegral}(3 \arccos (a x))+125 \arccos (a x)^3 \operatorname {CosIntegral}(5 \arccos (a x))}{96 a^5 \arccos (a x)^3} \]
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Time = 0.69 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arccos \left (a x \right )^{3}}+\frac {a x}{48 \arccos \left (a x \right )^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{48 \arccos \left (a x \right )}+\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{48}+\frac {\sin \left (3 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )^{3}}+\frac {3 \cos \left (3 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )^{2}}-\frac {9 \sin \left (3 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )}+\frac {27 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{32}+\frac {\sin \left (5 \arccos \left (a x \right )\right )}{48 \arccos \left (a x \right )^{3}}+\frac {5 \cos \left (5 \arccos \left (a x \right )\right )}{96 \arccos \left (a x \right )^{2}}-\frac {25 \sin \left (5 \arccos \left (a x \right )\right )}{96 \arccos \left (a x \right )}+\frac {125 \,\operatorname {Ci}\left (5 \arccos \left (a x \right )\right )}{96}}{a^{5}}\) | \(171\) |
default | \(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arccos \left (a x \right )^{3}}+\frac {a x}{48 \arccos \left (a x \right )^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{48 \arccos \left (a x \right )}+\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{48}+\frac {\sin \left (3 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )^{3}}+\frac {3 \cos \left (3 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )^{2}}-\frac {9 \sin \left (3 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )}+\frac {27 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{32}+\frac {\sin \left (5 \arccos \left (a x \right )\right )}{48 \arccos \left (a x \right )^{3}}+\frac {5 \cos \left (5 \arccos \left (a x \right )\right )}{96 \arccos \left (a x \right )^{2}}-\frac {25 \sin \left (5 \arccos \left (a x \right )\right )}{96 \arccos \left (a x \right )}+\frac {125 \,\operatorname {Ci}\left (5 \arccos \left (a x \right )\right )}{96}}{a^{5}}\) | \(171\) |
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\[ \int \frac {x^4}{\arccos (a x)^4} \, dx=\int { \frac {x^{4}}{\arccos \left (a x\right )^{4}} \,d x } \]
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\[ \int \frac {x^4}{\arccos (a x)^4} \, dx=\int \frac {x^{4}}{\operatorname {acos}^{4}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^4}{\arccos (a x)^4} \, dx=\int { \frac {x^{4}}{\arccos \left (a x\right )^{4}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.87 \[ \int \frac {x^4}{\arccos (a x)^4} \, dx=\frac {5 \, x^{5}}{6 \, \arccos \left (a x\right )^{2}} - \frac {25 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{6 \, a \arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{4}}{3 \, a \arccos \left (a x\right )^{3}} - \frac {2 \, x^{3}}{3 \, a^{2} \arccos \left (a x\right )^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{3} \arccos \left (a x\right )} + \frac {125 \, \operatorname {Ci}\left (5 \, \arccos \left (a x\right )\right )}{96 \, a^{5}} + \frac {27 \, \operatorname {Ci}\left (3 \, \arccos \left (a x\right )\right )}{32 \, a^{5}} + \frac {\operatorname {Ci}\left (\arccos \left (a x\right )\right )}{48 \, a^{5}} \]
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Timed out. \[ \int \frac {x^4}{\arccos (a x)^4} \, dx=\int \frac {x^4}{{\mathrm {acos}\left (a\,x\right )}^4} \,d x \]
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