Integrand size = 24, antiderivative size = 41 \[ \int \frac {x^4}{\sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=-\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{2 a^5}+\frac {\operatorname {CosIntegral}(4 \arcsin (a x))}{8 a^5}+\frac {3 \log (\arcsin (a x))}{8 a^5} \]
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Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4809, 3393, 3383} \[ \int \frac {x^4}{\sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=-\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{2 a^5}+\frac {\operatorname {CosIntegral}(4 \arcsin (a x))}{8 a^5}+\frac {3 \log (\arcsin (a x))}{8 a^5} \]
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Rule 3383
Rule 3393
Rule 4809
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sin ^4(x)}{x} \, dx,x,\arcsin (a x)\right )}{a^5} \\ & = \frac {\text {Subst}\left (\int \left (\frac {3}{8 x}-\frac {\cos (2 x)}{2 x}+\frac {\cos (4 x)}{8 x}\right ) \, dx,x,\arcsin (a x)\right )}{a^5} \\ & = \frac {3 \log (\arcsin (a x))}{8 a^5}+\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\arcsin (a x)\right )}{8 a^5}-\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\arcsin (a x)\right )}{2 a^5} \\ & = -\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{2 a^5}+\frac {\operatorname {CosIntegral}(4 \arcsin (a x))}{8 a^5}+\frac {3 \log (\arcsin (a x))}{8 a^5} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80 \[ \int \frac {x^4}{\sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=-\frac {4 \operatorname {CosIntegral}(2 \arcsin (a x))-\operatorname {CosIntegral}(4 \arcsin (a x))-3 \log (\arcsin (a x))}{8 a^5} \]
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Time = 0.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {3 \ln \left (\arcsin \left (a x \right )\right )-4 \,\operatorname {Ci}\left (2 \arcsin \left (a x \right )\right )+\operatorname {Ci}\left (4 \arcsin \left (a x \right )\right )}{8 a^{5}}\) | \(30\) |
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\[ \int \frac {x^4}{\sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=\int { \frac {x^{4}}{\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )} \,d x } \]
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\[ \int \frac {x^4}{\sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=\int \frac {x^{4}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {asin}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^4}{\sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=\int { \frac {x^{4}}{\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )} \,d x } \]
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none
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \frac {x^4}{\sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=\frac {\operatorname {Ci}\left (4 \, \arcsin \left (a x\right )\right )}{8 \, a^{5}} - \frac {\operatorname {Ci}\left (2 \, \arcsin \left (a x\right )\right )}{2 \, a^{5}} + \frac {3 \, \log \left (\arcsin \left (a x\right )\right )}{8 \, a^{5}} \]
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Timed out. \[ \int \frac {x^4}{\sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=\int \frac {x^4}{\mathrm {asin}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \]
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