Integrand size = 19, antiderivative size = 132 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=-\frac {b e x \sqrt {1-c^2 x^2}}{4 c}+\frac {1}{2} e x^2 (a+b \arccos (c x))+\frac {b e \arcsin (c x)}{4 c^2}+\frac {1}{2} i b d \arcsin (c x)^2-b d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+d (a+b \arccos (c x)) \log (x)+b d \arcsin (c x) \log (x)+\frac {1}{2} i b d \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {14, 4816, 12, 6874, 327, 222, 2363, 4721, 3798, 2221, 2317, 2438} \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=d \log (x) (a+b \arccos (c x))+\frac {1}{2} e x^2 (a+b \arccos (c x))+\frac {b e \arcsin (c x)}{4 c^2}+\frac {1}{2} i b d \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+\frac {1}{2} i b d \arcsin (c x)^2-b d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+b d \log (x) \arcsin (c x)-\frac {b e x \sqrt {1-c^2 x^2}}{4 c} \]
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Rule 12
Rule 14
Rule 222
Rule 327
Rule 2221
Rule 2317
Rule 2363
Rule 2438
Rule 3798
Rule 4721
Rule 4816
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} e x^2 (a+b \arccos (c x))+d (a+b \arccos (c x)) \log (x)+(b c) \int \frac {e x^2+2 d \log (x)}{2 \sqrt {1-c^2 x^2}} \, dx \\ & = \frac {1}{2} e x^2 (a+b \arccos (c x))+d (a+b \arccos (c x)) \log (x)+\frac {1}{2} (b c) \int \frac {e x^2+2 d \log (x)}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {1}{2} e x^2 (a+b \arccos (c x))+d (a+b \arccos (c x)) \log (x)+\frac {1}{2} (b c) \int \left (\frac {e x^2}{\sqrt {1-c^2 x^2}}+\frac {2 d \log (x)}{\sqrt {1-c^2 x^2}}\right ) \, dx \\ & = \frac {1}{2} e x^2 (a+b \arccos (c x))+d (a+b \arccos (c x)) \log (x)+(b c d) \int \frac {\log (x)}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{2} (b c e) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {b e x \sqrt {1-c^2 x^2}}{4 c}+\frac {1}{2} e x^2 (a+b \arccos (c x))+d (a+b \arccos (c x)) \log (x)+b d \arcsin (c x) \log (x)-(b d) \int \frac {\arcsin (c x)}{x} \, dx+\frac {(b e) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 c} \\ & = -\frac {b e x \sqrt {1-c^2 x^2}}{4 c}+\frac {1}{2} e x^2 (a+b \arccos (c x))+\frac {b e \arcsin (c x)}{4 c^2}+d (a+b \arccos (c x)) \log (x)+b d \arcsin (c x) \log (x)-(b d) \text {Subst}(\int x \cot (x) \, dx,x,\arcsin (c x)) \\ & = -\frac {b e x \sqrt {1-c^2 x^2}}{4 c}+\frac {1}{2} e x^2 (a+b \arccos (c x))+\frac {b e \arcsin (c x)}{4 c^2}+\frac {1}{2} i b d \arcsin (c x)^2+d (a+b \arccos (c x)) \log (x)+b d \arcsin (c x) \log (x)+(2 i b d) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\arcsin (c x)\right ) \\ & = -\frac {b e x \sqrt {1-c^2 x^2}}{4 c}+\frac {1}{2} e x^2 (a+b \arccos (c x))+\frac {b e \arcsin (c x)}{4 c^2}+\frac {1}{2} i b d \arcsin (c x)^2-b d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+d (a+b \arccos (c x)) \log (x)+b d \arcsin (c x) \log (x)+(b d) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right ) \\ & = -\frac {b e x \sqrt {1-c^2 x^2}}{4 c}+\frac {1}{2} e x^2 (a+b \arccos (c x))+\frac {b e \arcsin (c x)}{4 c^2}+\frac {1}{2} i b d \arcsin (c x)^2-b d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+d (a+b \arccos (c x)) \log (x)+b d \arcsin (c x) \log (x)-\frac {1}{2} (i b d) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right ) \\ & = -\frac {b e x \sqrt {1-c^2 x^2}}{4 c}+\frac {1}{2} e x^2 (a+b \arccos (c x))+\frac {b e \arcsin (c x)}{4 c^2}+\frac {1}{2} i b d \arcsin (c x)^2-b d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+d (a+b \arccos (c x)) \log (x)+b d \arcsin (c x) \log (x)+\frac {1}{2} i b d \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=\frac {1}{2} a e x^2-\frac {b e x \sqrt {1-c^2 x^2}}{4 c}+\frac {1}{2} b e x^2 \arccos (c x)-\frac {1}{2} i b d \arccos (c x)^2+\frac {b e \arcsin (c x)}{4 c^2}+b d \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+a d \log (x)-\frac {1}{2} i b d \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) \]
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Time = 4.93 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.97
method | result | size |
parts | \(\frac {a e \,x^{2}}{2}+a d \ln \left (x \right )-\frac {i b d \arccos \left (c x \right )^{2}}{2}-\frac {b e x \sqrt {-c^{2} x^{2}+1}}{4 c}+\frac {b \arccos \left (c x \right ) e \,x^{2}}{2}-\frac {b \arccos \left (c x \right ) e}{4 c^{2}}+b d \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i b d \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(128\) |
derivativedivides | \(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {i b d \arccos \left (c x \right )^{2}}{2}-\frac {b e x \sqrt {-c^{2} x^{2}+1}}{4 c}+\frac {b \arccos \left (c x \right ) e \,x^{2}}{2}-\frac {b \arccos \left (c x \right ) e}{4 c^{2}}+b d \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i b d \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(130\) |
default | \(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {i b d \arccos \left (c x \right )^{2}}{2}-\frac {b e x \sqrt {-c^{2} x^{2}+1}}{4 c}+\frac {b \arccos \left (c x \right ) e \,x^{2}}{2}-\frac {b \arccos \left (c x \right ) e}{4 c^{2}}+b d \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i b d \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(130\) |
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\[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arccos \left (c x\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=\int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arccos \left (c x\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arccos \left (c x\right ) + a\right )}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x} \,d x \]
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