Integrand size = 10, antiderivative size = 41 \[ \int \frac {x^4}{\arccos (a x)} \, dx=-\frac {\text {Si}(\arccos (a x))}{8 a^5}-\frac {3 \text {Si}(3 \arccos (a x))}{16 a^5}-\frac {\text {Si}(5 \arccos (a x))}{16 a^5} \]
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Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4732, 4491, 3380} \[ \int \frac {x^4}{\arccos (a x)} \, dx=-\frac {\text {Si}(\arccos (a x))}{8 a^5}-\frac {3 \text {Si}(3 \arccos (a x))}{16 a^5}-\frac {\text {Si}(5 \arccos (a x))}{16 a^5} \]
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Rule 3380
Rule 4491
Rule 4732
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cos ^4(x) \sin (x)}{x} \, dx,x,\arccos (a x)\right )}{a^5} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {\sin (x)}{8 x}+\frac {3 \sin (3 x)}{16 x}+\frac {\sin (5 x)}{16 x}\right ) \, dx,x,\arccos (a x)\right )}{a^5} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\arccos (a x)\right )}{16 a^5}-\frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arccos (a x)\right )}{8 a^5}-\frac {3 \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arccos (a x)\right )}{16 a^5} \\ & = -\frac {\text {Si}(\arccos (a x))}{8 a^5}-\frac {3 \text {Si}(3 \arccos (a x))}{16 a^5}-\frac {\text {Si}(5 \arccos (a x))}{16 a^5} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76 \[ \int \frac {x^4}{\arccos (a x)} \, dx=-\frac {2 \text {Si}(\arccos (a x))+3 \text {Si}(3 \arccos (a x))+\text {Si}(5 \arccos (a x))}{16 a^5} \]
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Time = 0.66 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {-\frac {3 \,\operatorname {Si}\left (3 \arccos \left (a x \right )\right )}{16}-\frac {\operatorname {Si}\left (5 \arccos \left (a x \right )\right )}{16}-\frac {\operatorname {Si}\left (\arccos \left (a x \right )\right )}{8}}{a^{5}}\) | \(31\) |
default | \(\frac {-\frac {3 \,\operatorname {Si}\left (3 \arccos \left (a x \right )\right )}{16}-\frac {\operatorname {Si}\left (5 \arccos \left (a x \right )\right )}{16}-\frac {\operatorname {Si}\left (\arccos \left (a x \right )\right )}{8}}{a^{5}}\) | \(31\) |
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\[ \int \frac {x^4}{\arccos (a x)} \, dx=\int { \frac {x^{4}}{\arccos \left (a x\right )} \,d x } \]
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\[ \int \frac {x^4}{\arccos (a x)} \, dx=\int \frac {x^{4}}{\operatorname {acos}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^4}{\arccos (a x)} \, dx=\int { \frac {x^{4}}{\arccos \left (a x\right )} \,d x } \]
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \frac {x^4}{\arccos (a x)} \, dx=-\frac {\operatorname {Si}\left (5 \, \arccos \left (a x\right )\right )}{16 \, a^{5}} - \frac {3 \, \operatorname {Si}\left (3 \, \arccos \left (a x\right )\right )}{16 \, a^{5}} - \frac {\operatorname {Si}\left (\arccos \left (a x\right )\right )}{8 \, a^{5}} \]
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Timed out. \[ \int \frac {x^4}{\arccos (a x)} \, dx=\int \frac {x^4}{\mathrm {acos}\left (a\,x\right )} \,d x \]
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