Integrand size = 10, antiderivative size = 43 \[ \int \frac {x^5}{\arccos (a x)} \, dx=-\frac {5 \text {Si}(2 \arccos (a x))}{32 a^6}-\frac {\text {Si}(4 \arccos (a x))}{8 a^6}-\frac {\text {Si}(6 \arccos (a x))}{32 a^6} \]
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Time = 0.05 (sec) , antiderivative size = 43, 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} \[ \int \frac {x^5}{\arccos (a x)} \, dx=-\frac {5 \text {Si}(2 \arccos (a x))}{32 a^6}-\frac {\text {Si}(4 \arccos (a x))}{8 a^6}-\frac {\text {Si}(6 \arccos (a x))}{32 a^6} \]
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Rule 3380
Rule 4491
Rule 4732
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cos ^5(x) \sin (x)}{x} \, dx,x,\arccos (a x)\right )}{a^6} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {5 \sin (2 x)}{32 x}+\frac {\sin (4 x)}{8 x}+\frac {\sin (6 x)}{32 x}\right ) \, dx,x,\arccos (a x)\right )}{a^6} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (6 x)}{x} \, dx,x,\arccos (a x)\right )}{32 a^6}-\frac {\text {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\arccos (a x)\right )}{8 a^6}-\frac {5 \text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\arccos (a x)\right )}{32 a^6} \\ & = -\frac {5 \text {Si}(2 \arccos (a x))}{32 a^6}-\frac {\text {Si}(4 \arccos (a x))}{8 a^6}-\frac {\text {Si}(6 \arccos (a x))}{32 a^6} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.77 \[ \int \frac {x^5}{\arccos (a x)} \, dx=-\frac {5 \text {Si}(2 \arccos (a x))+4 \text {Si}(4 \arccos (a x))+\text {Si}(6 \arccos (a x))}{32 a^6} \]
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Time = 0.66 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {-\frac {5 \,\operatorname {Si}\left (2 \arccos \left (a x \right )\right )}{32}-\frac {\operatorname {Si}\left (4 \arccos \left (a x \right )\right )}{8}-\frac {\operatorname {Si}\left (6 \arccos \left (a x \right )\right )}{32}}{a^{6}}\) | \(33\) |
default | \(\frac {-\frac {5 \,\operatorname {Si}\left (2 \arccos \left (a x \right )\right )}{32}-\frac {\operatorname {Si}\left (4 \arccos \left (a x \right )\right )}{8}-\frac {\operatorname {Si}\left (6 \arccos \left (a x \right )\right )}{32}}{a^{6}}\) | \(33\) |
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\[ \int \frac {x^5}{\arccos (a x)} \, dx=\int { \frac {x^{5}}{\arccos \left (a x\right )} \,d x } \]
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\[ \int \frac {x^5}{\arccos (a x)} \, dx=\int \frac {x^{5}}{\operatorname {acos}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^5}{\arccos (a x)} \, dx=\int { \frac {x^{5}}{\arccos \left (a x\right )} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {x^5}{\arccos (a x)} \, dx=-\frac {\operatorname {Si}\left (6 \, \arccos \left (a x\right )\right )}{32 \, a^{6}} - \frac {\operatorname {Si}\left (4 \, \arccos \left (a x\right )\right )}{8 \, a^{6}} - \frac {5 \, \operatorname {Si}\left (2 \, \arccos \left (a x\right )\right )}{32 \, a^{6}} \]
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Timed out. \[ \int \frac {x^5}{\arccos (a x)} \, dx=\int \frac {x^5}{\mathrm {acos}\left (a\,x\right )} \,d x \]
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