Optimal. Leaf size=119 \[ -\frac {1}{5} i \text {ArcCos}(a x)^5+\text {ArcCos}(a x)^4 \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )-2 i \text {ArcCos}(a x)^3 \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(a x)}\right )+3 \text {ArcCos}(a x)^2 \text {PolyLog}\left (3,-e^{2 i \text {ArcCos}(a x)}\right )+3 i \text {ArcCos}(a x) \text {PolyLog}\left (4,-e^{2 i \text {ArcCos}(a x)}\right )-\frac {3}{2} \text {PolyLog}\left (5,-e^{2 i \text {ArcCos}(a x)}\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4722, 3800,
2221, 2611, 6744, 2320, 6724} \begin {gather*} -2 i \text {ArcCos}(a x)^3 \text {Li}_2\left (-e^{2 i \text {ArcCos}(a x)}\right )+3 \text {ArcCos}(a x)^2 \text {Li}_3\left (-e^{2 i \text {ArcCos}(a x)}\right )+3 i \text {ArcCos}(a x) \text {Li}_4\left (-e^{2 i \text {ArcCos}(a x)}\right )-\frac {3}{2} \text {Li}_5\left (-e^{2 i \text {ArcCos}(a x)}\right )-\frac {1}{5} i \text {ArcCos}(a x)^5+\text {ArcCos}(a x)^4 \log \left (1+e^{2 i \text {ArcCos}(a 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 {\cos ^{-1}(a x)^4}{x} \, dx &=-\text {Subst}\left (\int x^4 \tan (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {1}{5} i \cos ^{-1}(a x)^5+2 i \text {Subst}\left (\int \frac {e^{2 i x} x^4}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {1}{5} i \cos ^{-1}(a x)^5+\cos ^{-1}(a x)^4 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-4 \text {Subst}\left (\int x^3 \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {1}{5} i \cos ^{-1}(a x)^5+\cos ^{-1}(a x)^4 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-2 i \cos ^{-1}(a x)^3 \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )+6 i \text {Subst}\left (\int x^2 \text {Li}_2\left (-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {1}{5} i \cos ^{-1}(a x)^5+\cos ^{-1}(a x)^4 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-2 i \cos ^{-1}(a x)^3 \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )+3 \cos ^{-1}(a x)^2 \text {Li}_3\left (-e^{2 i \cos ^{-1}(a x)}\right )-6 \text {Subst}\left (\int x \text {Li}_3\left (-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {1}{5} i \cos ^{-1}(a x)^5+\cos ^{-1}(a x)^4 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-2 i \cos ^{-1}(a x)^3 \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )+3 \cos ^{-1}(a x)^2 \text {Li}_3\left (-e^{2 i \cos ^{-1}(a x)}\right )+3 i \cos ^{-1}(a x) \text {Li}_4\left (-e^{2 i \cos ^{-1}(a x)}\right )-3 i \text {Subst}\left (\int \text {Li}_4\left (-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {1}{5} i \cos ^{-1}(a x)^5+\cos ^{-1}(a x)^4 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-2 i \cos ^{-1}(a x)^3 \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )+3 \cos ^{-1}(a x)^2 \text {Li}_3\left (-e^{2 i \cos ^{-1}(a x)}\right )+3 i \cos ^{-1}(a x) \text {Li}_4\left (-e^{2 i \cos ^{-1}(a x)}\right )-\frac {3}{2} \text {Subst}\left (\int \frac {\text {Li}_4(-x)}{x} \, dx,x,e^{2 i \cos ^{-1}(a x)}\right )\\ &=-\frac {1}{5} i \cos ^{-1}(a x)^5+\cos ^{-1}(a x)^4 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-2 i \cos ^{-1}(a x)^3 \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )+3 \cos ^{-1}(a x)^2 \text {Li}_3\left (-e^{2 i \cos ^{-1}(a x)}\right )+3 i \cos ^{-1}(a x) \text {Li}_4\left (-e^{2 i \cos ^{-1}(a x)}\right )-\frac {3}{2} \text {Li}_5\left (-e^{2 i \cos ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 119, normalized size = 1.00 \begin {gather*} -\frac {1}{5} i \text {ArcCos}(a x)^5+\text {ArcCos}(a x)^4 \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )-2 i \text {ArcCos}(a x)^3 \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(a x)}\right )+3 \text {ArcCos}(a x)^2 \text {PolyLog}\left (3,-e^{2 i \text {ArcCos}(a x)}\right )+3 i \text {ArcCos}(a x) \text {PolyLog}\left (4,-e^{2 i \text {ArcCos}(a x)}\right )-\frac {3}{2} \text {PolyLog}\left (5,-e^{2 i \text {ArcCos}(a x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 168, normalized size = 1.41
method | result | size |
derivativedivides | \(-\frac {i \arccos \left (a x \right )^{5}}{5}+\arccos \left (a x \right )^{4} \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-2 i \arccos \left (a x \right )^{3} \polylog \left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+3 \arccos \left (a x \right )^{2} \polylog \left (3, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+3 i \arccos \left (a x \right ) \polylog \left (4, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-\frac {3 \polylog \left (5, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(168\) |
default | \(-\frac {i \arccos \left (a x \right )^{5}}{5}+\arccos \left (a x \right )^{4} \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-2 i \arccos \left (a x \right )^{3} \polylog \left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+3 \arccos \left (a x \right )^{2} \polylog \left (3, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+3 i \arccos \left (a x \right ) \polylog \left (4, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-\frac {3 \polylog \left (5, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(168\) |
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 {\operatorname {acos}^{4}{\left (a x \right )}}{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 {{\mathrm {acos}\left (a\,x\right )}^4}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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