Integrand size = 14, antiderivative size = 127 \[ \int \frac {(a+b \arccos (c x))^3}{x} \, dx=-\frac {i (a+b \arccos (c x))^4}{4 b}+(a+b \arccos (c x))^3 \log \left (1+e^{2 i \arccos (c x)}\right )-\frac {3}{2} i b (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+\frac {3}{2} b^2 (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )+\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \arccos (c x)}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4722, 3800, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(a+b \arccos (c x))^3}{x} \, dx=\frac {3}{2} b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {3}{2} i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2-\frac {i (a+b \arccos (c x))^4}{4 b}+\log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^3+\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \arccos (c x)}\right ) \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 4722
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int (a+b x)^3 \tan (x) \, dx,x,\arccos (c x)\right ) \\ & = -\frac {i (a+b \arccos (c x))^4}{4 b}+2 i \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^3}{1+e^{2 i x}} \, dx,x,\arccos (c x)\right ) \\ & = -\frac {i (a+b \arccos (c x))^4}{4 b}+(a+b \arccos (c x))^3 \log \left (1+e^{2 i \arccos (c x)}\right )-(3 b) \text {Subst}\left (\int (a+b x)^2 \log \left (1+e^{2 i x}\right ) \, dx,x,\arccos (c x)\right ) \\ & = -\frac {i (a+b \arccos (c x))^4}{4 b}+(a+b \arccos (c x))^3 \log \left (1+e^{2 i \arccos (c x)}\right )-\frac {3}{2} i b (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+\left (3 i b^2\right ) \text {Subst}\left (\int (a+b x) \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \, dx,x,\arccos (c x)\right ) \\ & = -\frac {i (a+b \arccos (c x))^4}{4 b}+(a+b \arccos (c x))^3 \log \left (1+e^{2 i \arccos (c x)}\right )-\frac {3}{2} i b (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+\frac {3}{2} b^2 (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )-\frac {1}{2} \left (3 b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{2 i x}\right ) \, dx,x,\arccos (c x)\right ) \\ & = -\frac {i (a+b \arccos (c x))^4}{4 b}+(a+b \arccos (c x))^3 \log \left (1+e^{2 i \arccos (c x)}\right )-\frac {3}{2} i b (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+\frac {3}{2} b^2 (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )+\frac {1}{4} \left (3 i b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{2 i \arccos (c x)}\right ) \\ & = -\frac {i (a+b \arccos (c x))^4}{4 b}+(a+b \arccos (c x))^3 \log \left (1+e^{2 i \arccos (c x)}\right )-\frac {3}{2} i b (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+\frac {3}{2} b^2 (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )+\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \arccos (c x)}\right ) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.61 \[ \int \frac {(a+b \arccos (c x))^3}{x} \, dx=\frac {1}{4} \left (-6 i a^2 b \arccos (c x)^2-4 i a b^2 \arccos (c x)^3-i b^3 \arccos (c x)^4+12 a^2 b \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+12 a b^2 \arccos (c x)^2 \log \left (1+e^{2 i \arccos (c x)}\right )+4 b^3 \arccos (c x)^3 \log \left (1+e^{2 i \arccos (c x)}\right )+4 a^3 \log (c x)-6 i b (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+6 b^2 (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )+3 i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \arccos (c x)}\right )\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (160 ) = 320\).
Time = 1.20 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.56
method | result | size |
parts | \(a^{3} \ln \left (x \right )+b^{3} \left (-\frac {i \arccos \left (c x \right )^{4}}{4}+\arccos \left (c x \right )^{3} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \arccos \left (c x \right )^{2} \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 \arccos \left (c x \right ) \operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 i \operatorname {polylog}\left (4, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{4}\right )+3 a \,b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}+\arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )+3 a^{2} b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\) | \(325\) |
derivativedivides | \(a^{3} \ln \left (c x \right )+b^{3} \left (-\frac {i \arccos \left (c x \right )^{4}}{4}+\arccos \left (c x \right )^{3} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \arccos \left (c x \right )^{2} \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 \arccos \left (c x \right ) \operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 i \operatorname {polylog}\left (4, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{4}\right )+3 a \,b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}+\arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )+3 a^{2} b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\) | \(327\) |
default | \(a^{3} \ln \left (c x \right )+b^{3} \left (-\frac {i \arccos \left (c x \right )^{4}}{4}+\arccos \left (c x \right )^{3} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \arccos \left (c x \right )^{2} \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 \arccos \left (c x \right ) \operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 i \operatorname {polylog}\left (4, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{4}\right )+3 a \,b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}+\arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )+3 a^{2} b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\) | \(327\) |
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\[ \int \frac {(a+b \arccos (c x))^3}{x} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
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\[ \int \frac {(a+b \arccos (c x))^3}{x} \, dx=\int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}}{x}\, dx \]
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\[ \int \frac {(a+b \arccos (c x))^3}{x} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
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\[ \int \frac {(a+b \arccos (c x))^3}{x} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arccos (c x))^3}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3}{x} \,d x \]
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