Integrand size = 18, antiderivative size = 55 \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=-\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-\frac {3 c \text {Si}(\arcsin (a x))}{4 a}-\frac {3 c \text {Si}(3 \arcsin (a x))}{4 a} \]
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Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4751, 4809, 4491, 3380} \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=-\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-\frac {3 c \text {Si}(\arcsin (a x))}{4 a}-\frac {3 c \text {Si}(3 \arcsin (a x))}{4 a} \]
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Rule 3380
Rule 4491
Rule 4751
Rule 4809
Rubi steps \begin{align*} \text {integral}& = -\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-(3 a c) \int \frac {x \sqrt {1-a^2 x^2}}{\arcsin (a x)} \, dx \\ & = -\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-\frac {(3 c) \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{x} \, dx,x,\arcsin (a x)\right )}{a} \\ & = -\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-\frac {(3 c) \text {Subst}\left (\int \left (\frac {\sin (x)}{4 x}+\frac {\sin (3 x)}{4 x}\right ) \, dx,x,\arcsin (a x)\right )}{a} \\ & = -\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-\frac {(3 c) \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arcsin (a x)\right )}{4 a}-\frac {(3 c) \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arcsin (a x)\right )}{4 a} \\ & = -\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-\frac {3 c \text {Si}(\arcsin (a x))}{4 a}-\frac {3 c \text {Si}(3 \arcsin (a x))}{4 a} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=-\frac {c \left (4 \left (1-a^2 x^2\right )^{3/2}+3 \arcsin (a x) \text {Si}(\arcsin (a x))+3 \arcsin (a x) \text {Si}(3 \arcsin (a x))\right )}{4 a \arcsin (a x)} \]
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Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(-\frac {c \left (3 \,\operatorname {Si}\left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+3 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+\cos \left (3 \arcsin \left (a x \right )\right )+3 \sqrt {-a^{2} x^{2}+1}\right )}{4 a \arcsin \left (a x \right )}\) | \(59\) |
default | \(-\frac {c \left (3 \,\operatorname {Si}\left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+3 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+\cos \left (3 \arcsin \left (a x \right )\right )+3 \sqrt {-a^{2} x^{2}+1}\right )}{4 a \arcsin \left (a x \right )}\) | \(59\) |
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\[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\arcsin \left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=- c \left (\int \frac {a^{2} x^{2}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {asin}^{2}{\left (a x \right )}}\right )\, dx\right ) \]
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\[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\arcsin \left (a x\right )^{2}} \,d x } \]
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none
Time = 0.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=-\frac {3 \, c \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{4 \, a} - \frac {3 \, c \operatorname {Si}\left (\arcsin \left (a x\right )\right )}{4 \, a} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{a \arcsin \left (a x\right )} \]
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Timed out. \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=\int \frac {c-a^2\,c\,x^2}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]
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