Optimal. Leaf size=127 \[ -\frac {i (a+b \text {ArcCos}(c x))^4}{4 b}+(a+b \text {ArcCos}(c x))^3 \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )-\frac {3}{2} i b (a+b \text {ArcCos}(c x))^2 \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(c x)}\right )+\frac {3}{2} b^2 (a+b \text {ArcCos}(c x)) \text {PolyLog}\left (3,-e^{2 i \text {ArcCos}(c x)}\right )+\frac {3}{4} i b^3 \text {PolyLog}\left (4,-e^{2 i \text {ArcCos}(c x)}\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4722, 3800,
2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {3}{2} b^2 \text {Li}_3\left (-e^{2 i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))-\frac {3}{2} i b \text {Li}_2\left (-e^{2 i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))^2-\frac {i (a+b \text {ArcCos}(c x))^4}{4 b}+\log \left (1+e^{2 i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))^3+\frac {3}{4} i b^3 \text {Li}_4\left (-e^{2 i \text {ArcCos}(c x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 4722
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {\left (a+b \cos ^{-1}(c x)\right )^3}{x} \, dx &=-\text {Subst}\left (\int (a+b x)^3 \tan (x) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^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,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^4}{4 b}+\left (a+b \cos ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-(3 b) \text {Subst}\left (\int (a+b x)^2 \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^4}{4 b}+\left (a+b \cos ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-\frac {3}{2} i b \left (a+b \cos ^{-1}(c x)\right )^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\left (3 i b^2\right ) \text {Subst}\left (\int (a+b x) \text {Li}_2\left (-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^4}{4 b}+\left (a+b \cos ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-\frac {3}{2} i b \left (a+b \cos ^{-1}(c x)\right )^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_3\left (-e^{2 i \cos ^{-1}(c x)}\right )-\frac {1}{2} \left (3 b^3\right ) \text {Subst}\left (\int \text {Li}_3\left (-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^4}{4 b}+\left (a+b \cos ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-\frac {3}{2} i b \left (a+b \cos ^{-1}(c x)\right )^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_3\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac {1}{4} \left (3 i b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 i \cos ^{-1}(c x)}\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^4}{4 b}+\left (a+b \cos ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-\frac {3}{2} i b \left (a+b \cos ^{-1}(c x)\right )^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_3\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac {3}{4} i b^3 \text {Li}_4\left (-e^{2 i \cos ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 204, normalized size = 1.61 \begin {gather*} \frac {1}{4} \left (-6 i a^2 b \text {ArcCos}(c x)^2-4 i a b^2 \text {ArcCos}(c x)^3-i b^3 \text {ArcCos}(c x)^4+12 a^2 b \text {ArcCos}(c x) \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )+12 a b^2 \text {ArcCos}(c x)^2 \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )+4 b^3 \text {ArcCos}(c x)^3 \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )+4 a^3 \log (c x)-6 i b (a+b \text {ArcCos}(c x))^2 \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(c x)}\right )+6 b^2 (a+b \text {ArcCos}(c x)) \text {PolyLog}\left (3,-e^{2 i \text {ArcCos}(c x)}\right )+3 i b^3 \text {PolyLog}\left (4,-e^{2 i \text {ArcCos}(c x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 352 vs. \(2 (160 ) = 320\).
time = 0.16, size = 353, normalized size = 2.78
method | result | size |
derivativedivides | \(a^{3} \ln \left (c x \right )-\frac {i b^{3} \arccos \left (c x \right )^{4}}{4}+b^{3} \arccos \left (c x \right )^{3} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i b^{3} \arccos \left (c x \right )^{2} \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 b^{3} \arccos \left (c x \right ) \polylog \left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 i b^{3} \polylog \left (4, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{4}-i a \,b^{2} \arccos \left (c x \right )^{3}+3 a \,b^{2} \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-3 i a \,b^{2} \arccos \left (c x \right ) \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 a \,b^{2} \polylog \left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\frac {3 i a^{2} b \arccos \left (c x \right )^{2}}{2}+3 a^{2} b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i a^{2} b \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(353\) |
default | \(a^{3} \ln \left (c x \right )-\frac {i b^{3} \arccos \left (c x \right )^{4}}{4}+b^{3} \arccos \left (c x \right )^{3} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i b^{3} \arccos \left (c x \right )^{2} \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 b^{3} \arccos \left (c x \right ) \polylog \left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 i b^{3} \polylog \left (4, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{4}-i a \,b^{2} \arccos \left (c x \right )^{3}+3 a \,b^{2} \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-3 i a \,b^{2} \arccos \left (c x \right ) \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 a \,b^{2} \polylog \left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\frac {3 i a^{2} b \arccos \left (c x \right )^{2}}{2}+3 a^{2} b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i a^{2} b \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(353\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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