Integrand size = 14, antiderivative size = 81 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=-\frac {(a+b \arcsin (c x))^2}{x}-4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )+2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4723, 4803, 4268, 2317, 2438} \[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=-4 b c \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {(a+b \arcsin (c x))^2}{x}+2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \]
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Rule 2317
Rule 2438
Rule 4268
Rule 4723
Rule 4803
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arcsin (c x))^2}{x}+(2 b c) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {(a+b \arcsin (c x))^2}{x}+(2 b c) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x)) \\ & = -\frac {(a+b \arcsin (c x))^2}{x}-4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )-\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )+\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right ) \\ & = -\frac {(a+b \arcsin (c x))^2}{x}-4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )+\left (2 i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )-\left (2 i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (c x)}\right ) \\ & = -\frac {(a+b \arcsin (c x))^2}{x}-4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )+2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.56 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=-\frac {a^2+2 a b \left (\arcsin (c x)+c x \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-i b^2 \left (i \arcsin (c x) \left (\arcsin (c x)+2 c x \left (-\log \left (1-e^{i \arcsin (c x)}\right )+\log \left (1+e^{i \arcsin (c x)}\right )\right )\right )+2 c x \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 c x \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{x} \]
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Time = 0.04 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.04
method | result | size |
parts | \(-\frac {a^{2}}{x}+b^{2} c \left (-\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+2 a b c \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(165\) |
derivativedivides | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+2 a b \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\) | \(168\) |
default | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+2 a b \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\) | \(168\) |
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^2} \,d x \]
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