Integrand size = 10, antiderivative size = 41 \[ \int \frac {x^4}{\arcsin (a x)} \, dx=\frac {\operatorname {CosIntegral}(\arcsin (a x))}{8 a^5}-\frac {3 \operatorname {CosIntegral}(3 \arcsin (a x))}{16 a^5}+\frac {\operatorname {CosIntegral}(5 \arcsin (a x))}{16 a^5} \]
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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 = {4731, 4491, 3383} \[ \int \frac {x^4}{\arcsin (a x)} \, dx=\frac {\operatorname {CosIntegral}(\arcsin (a x))}{8 a^5}-\frac {3 \operatorname {CosIntegral}(3 \arcsin (a x))}{16 a^5}+\frac {\operatorname {CosIntegral}(5 \arcsin (a x))}{16 a^5} \]
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Rule 3383
Rule 4491
Rule 4731
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cos (x) \sin ^4(x)}{x} \, dx,x,\arcsin (a x)\right )}{a^5} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\cos (x)}{8 x}-\frac {3 \cos (3 x)}{16 x}+\frac {\cos (5 x)}{16 x}\right ) \, dx,x,\arcsin (a x)\right )}{a^5} \\ & = \frac {\text {Subst}\left (\int \frac {\cos (5 x)}{x} \, dx,x,\arcsin (a x)\right )}{16 a^5}+\frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arcsin (a x)\right )}{8 a^5}-\frac {3 \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arcsin (a x)\right )}{16 a^5} \\ & = \frac {\operatorname {CosIntegral}(\arcsin (a x))}{8 a^5}-\frac {3 \operatorname {CosIntegral}(3 \arcsin (a x))}{16 a^5}+\frac {\operatorname {CosIntegral}(5 \arcsin (a x))}{16 a^5} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76 \[ \int \frac {x^4}{\arcsin (a x)} \, dx=\frac {2 \operatorname {CosIntegral}(\arcsin (a x))-3 \operatorname {CosIntegral}(3 \arcsin (a x))+\operatorname {CosIntegral}(5 \arcsin (a x))}{16 a^5} \]
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Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{8}-\frac {3 \,\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )}{16}+\frac {\operatorname {Ci}\left (5 \arcsin \left (a x \right )\right )}{16}}{a^{5}}\) | \(31\) |
default | \(\frac {\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{8}-\frac {3 \,\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )}{16}+\frac {\operatorname {Ci}\left (5 \arcsin \left (a x \right )\right )}{16}}{a^{5}}\) | \(31\) |
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\[ \int \frac {x^4}{\arcsin (a x)} \, dx=\int { \frac {x^{4}}{\arcsin \left (a x\right )} \,d x } \]
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\[ \int \frac {x^4}{\arcsin (a x)} \, dx=\int \frac {x^{4}}{\operatorname {asin}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^4}{\arcsin (a x)} \, dx=\int { \frac {x^{4}}{\arcsin \left (a x\right )} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \frac {x^4}{\arcsin (a x)} \, dx=\frac {\operatorname {Ci}\left (5 \, \arcsin \left (a x\right )\right )}{16 \, a^{5}} - \frac {3 \, \operatorname {Ci}\left (3 \, \arcsin \left (a x\right )\right )}{16 \, a^{5}} + \frac {\operatorname {Ci}\left (\arcsin \left (a x\right )\right )}{8 \, a^{5}} \]
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Timed out. \[ \int \frac {x^4}{\arcsin (a x)} \, dx=\int \frac {x^4}{\mathrm {asin}\left (a\,x\right )} \,d x \]
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