Integrand size = 14, antiderivative size = 92 \[ \int \frac {(a+b \arccos (c x))^2}{x} \, dx=-\frac {i (a+b \arccos (c x))^3}{3 b}+(a+b \arccos (c x))^2 \log \left (1+e^{2 i \arccos (c x)}\right )-i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4722, 3800, 2221, 2611, 2320, 6724} \[ \int \frac {(a+b \arccos (c x))^2}{x} \, dx=-i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {i (a+b \arccos (c x))^3}{3 b}+\log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2+\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right ) \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 4722
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int (a+b x)^2 \tan (x) \, dx,x,\arccos (c x)\right ) \\ & = -\frac {i (a+b \arccos (c x))^3}{3 b}+2 i \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^2}{1+e^{2 i x}} \, dx,x,\arccos (c x)\right ) \\ & = -\frac {i (a+b \arccos (c x))^3}{3 b}+(a+b \arccos (c x))^2 \log \left (1+e^{2 i \arccos (c x)}\right )-(2 b) \text {Subst}\left (\int (a+b x) \log \left (1+e^{2 i x}\right ) \, dx,x,\arccos (c x)\right ) \\ & = -\frac {i (a+b \arccos (c x))^3}{3 b}+(a+b \arccos (c x))^2 \log \left (1+e^{2 i \arccos (c x)}\right )-i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+\left (i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \, dx,x,\arccos (c x)\right ) \\ & = -\frac {i (a+b \arccos (c x))^3}{3 b}+(a+b \arccos (c x))^2 \log \left (1+e^{2 i \arccos (c x)}\right )-i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+\frac {1}{2} b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 i \arccos (c x)}\right ) \\ & = -\frac {i (a+b \arccos (c x))^3}{3 b}+(a+b \arccos (c x))^2 \log \left (1+e^{2 i \arccos (c x)}\right )-i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b \arccos (c x))^2}{x} \, dx=-i a b \arccos (c x)^2-\frac {1}{3} i b^2 \arccos (c x)^3+2 a b \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+b^2 \arccos (c x)^2 \log \left (1+e^{2 i \arccos (c x)}\right )+a^2 \log (c x)-i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right ) \]
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Time = 1.14 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.01
method | result | size |
parts | \(a^{2} \ln \left (x \right )+b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}+\arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )-i a b \arccos \left (c x \right )^{2}-i a b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 a b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\) | \(185\) |
derivativedivides | \(a^{2} \ln \left (c x \right )+b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}+\arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )-i a b \arccos \left (c x \right )^{2}+2 a b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i a b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\) | \(187\) |
default | \(a^{2} \ln \left (c x \right )+b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}+\arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )-i a b \arccos \left (c x \right )^{2}+2 a b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i a b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\) | \(187\) |
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\[ \int \frac {(a+b \arccos (c x))^2}{x} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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\[ \int \frac {(a+b \arccos (c x))^2}{x} \, dx=\int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{x}\, dx \]
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\[ \int \frac {(a+b \arccos (c x))^2}{x} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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\[ \int \frac {(a+b \arccos (c x))^2}{x} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arccos (c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{x} \,d x \]
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