Integrand size = 20, antiderivative size = 50 \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\arcsin (a x)} \, dx=\frac {5 c^2 \operatorname {CosIntegral}(\arcsin (a x))}{8 a}+\frac {5 c^2 \operatorname {CosIntegral}(3 \arcsin (a x))}{16 a}+\frac {c^2 \operatorname {CosIntegral}(5 \arcsin (a x))}{16 a} \]
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Time = 0.06 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4753, 3393, 3383} \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\arcsin (a x)} \, dx=\frac {5 c^2 \operatorname {CosIntegral}(\arcsin (a x))}{8 a}+\frac {5 c^2 \operatorname {CosIntegral}(3 \arcsin (a x))}{16 a}+\frac {c^2 \operatorname {CosIntegral}(5 \arcsin (a x))}{16 a} \]
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Rule 3383
Rule 3393
Rule 4753
Rubi steps \begin{align*} \text {integral}& = \frac {c^2 \text {Subst}\left (\int \frac {\cos ^5(x)}{x} \, dx,x,\arcsin (a x)\right )}{a} \\ & = \frac {c^2 \text {Subst}\left (\int \left (\frac {5 \cos (x)}{8 x}+\frac {5 \cos (3 x)}{16 x}+\frac {\cos (5 x)}{16 x}\right ) \, dx,x,\arcsin (a x)\right )}{a} \\ & = \frac {c^2 \text {Subst}\left (\int \frac {\cos (5 x)}{x} \, dx,x,\arcsin (a x)\right )}{16 a}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arcsin (a x)\right )}{16 a}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arcsin (a x)\right )}{8 a} \\ & = \frac {5 c^2 \operatorname {CosIntegral}(\arcsin (a x))}{8 a}+\frac {5 c^2 \operatorname {CosIntegral}(3 \arcsin (a x))}{16 a}+\frac {c^2 \operatorname {CosIntegral}(5 \arcsin (a x))}{16 a} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.68 \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\arcsin (a x)} \, dx=\frac {c^2 (10 \operatorname {CosIntegral}(\arcsin (a x))+5 \operatorname {CosIntegral}(3 \arcsin (a x))+\operatorname {CosIntegral}(5 \arcsin (a x)))}{16 a} \]
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Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {c^{2} \left (10 \,\operatorname {Ci}\left (\arcsin \left (a x \right )\right )+5 \,\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )+\operatorname {Ci}\left (5 \arcsin \left (a x \right )\right )\right )}{16 a}\) | \(33\) |
default | \(\frac {c^{2} \left (10 \,\operatorname {Ci}\left (\arcsin \left (a x \right )\right )+5 \,\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )+\operatorname {Ci}\left (5 \arcsin \left (a x \right )\right )\right )}{16 a}\) | \(33\) |
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\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\arcsin (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} - c\right )}^{2}}{\arcsin \left (a x\right )} \,d x } \]
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\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\arcsin (a x)} \, dx=c^{2} \left (\int \left (- \frac {2 a^{2} x^{2}}{\operatorname {asin}{\left (a x \right )}}\right )\, dx + \int \frac {a^{4} x^{4}}{\operatorname {asin}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {asin}{\left (a x \right )}}\, dx\right ) \]
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\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\arcsin (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} - c\right )}^{2}}{\arcsin \left (a x\right )} \,d x } \]
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none
Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\arcsin (a x)} \, dx=\frac {c^{2} \operatorname {Ci}\left (5 \, \arcsin \left (a x\right )\right )}{16 \, a} + \frac {5 \, c^{2} \operatorname {Ci}\left (3 \, \arcsin \left (a x\right )\right )}{16 \, a} + \frac {5 \, c^{2} \operatorname {Ci}\left (\arcsin \left (a x\right )\right )}{8 \, a} \]
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Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\arcsin (a x)} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^2}{\mathrm {asin}\left (a\,x\right )} \,d x \]
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