Optimal. Leaf size=192 \[ -\frac {a^2 \text {ArcCos}(a x)}{x}+\frac {a \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^2}{2 x^2}-\frac {\text {ArcCos}(a x)^3}{3 x^3}-i a^3 \text {ArcCos}(a x)^2 \text {ArcTan}\left (e^{i \text {ArcCos}(a x)}\right )+a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+i a^3 \text {ArcCos}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )-i a^3 \text {ArcCos}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcCos}(a x)}\right )-a^3 \text {PolyLog}\left (3,-i e^{i \text {ArcCos}(a x)}\right )+a^3 \text {PolyLog}\left (3,i e^{i \text {ArcCos}(a x)}\right ) \]
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Rubi [A]
time = 0.20, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4724, 4790,
4804, 4266, 2611, 2320, 6724, 272, 65, 214} \begin {gather*} -i a^3 \text {ArcCos}(a x)^2 \text {ArcTan}\left (e^{i \text {ArcCos}(a x)}\right )+i a^3 \text {ArcCos}(a x) \text {Li}_2\left (-i e^{i \text {ArcCos}(a x)}\right )-i a^3 \text {ArcCos}(a x) \text {Li}_2\left (i e^{i \text {ArcCos}(a x)}\right )-a^3 \text {Li}_3\left (-i e^{i \text {ArcCos}(a x)}\right )+a^3 \text {Li}_3\left (i e^{i \text {ArcCos}(a x)}\right )+\frac {a \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^2}{2 x^2}-\frac {a^2 \text {ArcCos}(a x)}{x}+a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {\text {ArcCos}(a x)^3}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 2320
Rule 2611
Rule 4266
Rule 4724
Rule 4790
Rule 4804
Rule 6724
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a x)^3}{x^4} \, dx &=-\frac {\cos ^{-1}(a x)^3}{3 x^3}-a \int \frac {\cos ^{-1}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac {\cos ^{-1}(a x)^3}{3 x^3}+a^2 \int \frac {\cos ^{-1}(a x)}{x^2} \, dx-\frac {1}{2} a^3 \int \frac {\cos ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a^2 \cos ^{-1}(a x)}{x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac {\cos ^{-1}(a x)^3}{3 x^3}+\frac {1}{2} a^3 \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\cos ^{-1}(a x)\right )-a^3 \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a^2 \cos ^{-1}(a x)}{x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac {\cos ^{-1}(a x)^3}{3 x^3}-i a^3 \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-\frac {1}{2} a^3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )-a^3 \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+a^3 \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {a^2 \cos ^{-1}(a x)}{x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac {\cos ^{-1}(a x)^3}{3 x^3}-i a^3 \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+i a^3 \cos ^{-1}(a x) \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-i a^3 \cos ^{-1}(a x) \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )+a \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )-\left (i a^3\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+\left (i a^3\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {a^2 \cos ^{-1}(a x)}{x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac {\cos ^{-1}(a x)^3}{3 x^3}-i a^3 \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+i a^3 \cos ^{-1}(a x) \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-i a^3 \cos ^{-1}(a x) \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-a^3 \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )+a^3 \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )\\ &=-\frac {a^2 \cos ^{-1}(a x)}{x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac {\cos ^{-1}(a x)^3}{3 x^3}-i a^3 \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+i a^3 \cos ^{-1}(a x) \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-i a^3 \cos ^{-1}(a x) \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-a^3 \text {Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+a^3 \text {Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(501\) vs. \(2(192)=384\).
time = 1.08, size = 501, normalized size = 2.61 \begin {gather*} \frac {1}{2} a^3 \left (\text {ArcCos}(a x)^2 \log \left (1-i e^{i \text {ArcCos}(a x)}\right )+\pi \text {ArcCos}(a x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcCos}(a x)} \left (1-i e^{i \text {ArcCos}(a x)}\right )\right )-\text {ArcCos}(a x)^2 \log \left (1+i e^{i \text {ArcCos}(a x)}\right )-\text {ArcCos}(a x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \text {ArcCos}(a x)} \left (-i+e^{i \text {ArcCos}(a x)}\right )\right )+\pi \text {ArcCos}(a x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcCos}(a x)} \left (-i+e^{i \text {ArcCos}(a x)}\right )\right )+\text {ArcCos}(a x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \text {ArcCos}(a x)} \left ((1+i)+(1-i) e^{i \text {ArcCos}(a x)}\right )\right )-\pi \text {ArcCos}(a x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcCos}(a x))\right )\right )-2 \log \left (\cos \left (\frac {1}{2} \text {ArcCos}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcCos}(a x)\right )\right )+\text {ArcCos}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcCos}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcCos}(a x)\right )\right )+2 \log \left (\cos \left (\frac {1}{2} \text {ArcCos}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcCos}(a x)\right )\right )-\text {ArcCos}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcCos}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcCos}(a x)\right )\right )-\pi \text {ArcCos}(a x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcCos}(a x))\right )\right )+2 i \text {ArcCos}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )-2 i \text {ArcCos}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcCos}(a x)}\right )-2 \text {PolyLog}\left (3,-i e^{i \text {ArcCos}(a x)}\right )+2 \text {PolyLog}\left (3,i e^{i \text {ArcCos}(a x)}\right )\right )-\frac {\text {ArcCos}(a x) \left (12 a^2 x^2+4 \text {ArcCos}(a x)^2-3 \text {ArcCos}(a x) \sin (2 \text {ArcCos}(a x))\right )}{12 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 256, normalized size = 1.33
method | result | size |
derivativedivides | \(a^{3} \left (-\frac {\arccos \left (a x \right ) \left (-3 a x \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}+2 \arccos \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}+\frac {\arccos \left (a x \right )^{2} \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{2}-i \arccos \left (a x \right ) \polylog \left (2, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+\polylog \left (3, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-\frac {\arccos \left (a x \right )^{2} \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{2}+i \arccos \left (a x \right ) \polylog \left (2, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-\polylog \left (3, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-2 i \arctan \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )\) | \(256\) |
default | \(a^{3} \left (-\frac {\arccos \left (a x \right ) \left (-3 a x \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}+2 \arccos \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}+\frac {\arccos \left (a x \right )^{2} \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{2}-i \arccos \left (a x \right ) \polylog \left (2, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+\polylog \left (3, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-\frac {\arccos \left (a x \right )^{2} \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{2}+i \arccos \left (a x \right ) \polylog \left (2, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-\polylog \left (3, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-2 i \arctan \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )\) | \(256\) |
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}^{3}{\left (a x \right )}}{x^{4}}\, 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 )}^3}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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