Integrand size = 23, antiderivative size = 139 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=-\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {1}{2} b c^2 d \arcsin (c x)-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {i c^2 d (a+b \arcsin (c x))^2}{2 b}-c^2 d (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )+\frac {1}{2} i b c^2 d \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4775, 283, 222, 4721, 3798, 2221, 2317, 2438} \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {i c^2 d (a+b \arcsin (c x))^2}{2 b}-c^2 d \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{2} i b c^2 d \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )-\frac {1}{2} b c^2 d \arcsin (c x)-\frac {b c d \sqrt {1-c^2 x^2}}{2 x} \]
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Rule 222
Rule 283
Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4721
Rule 4775
Rubi steps \begin{align*} \text {integral}& = -\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} (b c d) \int \frac {\sqrt {1-c^2 x^2}}{x^2} \, dx-\left (c^2 d\right ) \int \frac {a+b \arcsin (c x)}{x} \, dx \\ & = -\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}-\left (c^2 d\right ) \text {Subst}(\int (a+b x) \cot (x) \, dx,x,\arcsin (c x))-\frac {1}{2} \left (b c^3 d\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {1}{2} b c^2 d \arcsin (c x)-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {i c^2 d (a+b \arcsin (c x))^2}{2 b}+\left (2 i c^2 d\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\arcsin (c x)\right ) \\ & = -\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {1}{2} b c^2 d \arcsin (c x)-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {i c^2 d (a+b \arcsin (c x))^2}{2 b}-c^2 d (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )+\left (b c^2 d\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right ) \\ & = -\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {1}{2} b c^2 d \arcsin (c x)-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {i c^2 d (a+b \arcsin (c x))^2}{2 b}-c^2 d (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )-\frac {1}{2} \left (i b c^2 d\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right ) \\ & = -\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {1}{2} b c^2 d \arcsin (c x)-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {i c^2 d (a+b \arcsin (c x))^2}{2 b}-c^2 d (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )+\frac {1}{2} i b c^2 d \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.81 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=-\frac {a d}{2 x^2}-\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {b d \arcsin (c x)}{2 x^2}-b c^2 d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-a c^2 d \log (x)+\frac {1}{2} i b c^2 d \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right ) \]
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Time = 0.13 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.24
method | result | size |
parts | \(-\frac {d a}{2 x^{2}}-d a \,c^{2} \ln \left (x \right )-d b \,c^{2} \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\) | \(172\) |
derivativedivides | \(c^{2} \left (-d a \left (\frac {1}{2 c^{2} x^{2}}+\ln \left (c x \right )\right )-d b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )\) | \(174\) |
default | \(c^{2} \left (-d a \left (\frac {1}{2 c^{2} x^{2}}+\ln \left (c x \right )\right )-d b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )\) | \(174\) |
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\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=- d \left (\int \left (- \frac {a}{x^{3}}\right )\, dx + \int \frac {a c^{2}}{x}\, dx + \int \left (- \frac {b \operatorname {asin}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {b c^{2} \operatorname {asin}{\left (c x \right )}}{x}\, dx\right ) \]
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\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right )}{x^3} \,d x \]
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