Optimal. Leaf size=151 \[ -\frac {(a+b \text {ArcCos}(c x))^3}{x}-6 i b c (a+b \text {ArcCos}(c x))^2 \text {ArcTan}\left (e^{i \text {ArcCos}(c x)}\right )+6 i b^2 c (a+b \text {ArcCos}(c x)) \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(c x)}\right )-6 i b^2 c (a+b \text {ArcCos}(c x)) \text {PolyLog}\left (2,i e^{i \text {ArcCos}(c x)}\right )-6 b^3 c \text {PolyLog}\left (3,-i e^{i \text {ArcCos}(c x)}\right )+6 b^3 c \text {PolyLog}\left (3,i e^{i \text {ArcCos}(c x)}\right ) \]
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Rubi [A]
time = 0.17, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4724, 4804,
4266, 2611, 2320, 6724} \begin {gather*} -6 i b c \text {ArcTan}\left (e^{i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))^2+6 i b^2 c \text {Li}_2\left (-i e^{i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))-6 i b^2 c \text {Li}_2\left (i e^{i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))-\frac {(a+b \text {ArcCos}(c x))^3}{x}-6 b^3 c \text {Li}_3\left (-i e^{i \text {ArcCos}(c x)}\right )+6 b^3 c \text {Li}_3\left (i e^{i \text {ArcCos}(c x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 4266
Rule 4724
Rule 4804
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \cos ^{-1}(c x)\right )^3}{x^2} \, dx &=-\frac {\left (a+b \cos ^{-1}(c x)\right )^3}{x}-(3 b c) \int \frac {\left (a+b \cos ^{-1}(c x)\right )^2}{x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {\left (a+b \cos ^{-1}(c x)\right )^3}{x}+(3 b c) \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {\left (a+b \cos ^{-1}(c x)\right )^3}{x}-6 i b c \left (a+b \cos ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )-\left (6 b^2 c\right ) \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )+\left (6 b^2 c\right ) \text {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {\left (a+b \cos ^{-1}(c x)\right )^3}{x}-6 i b c \left (a+b \cos ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )+6 i b^2 c \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \cos ^{-1}(c x)}\right )-6 i b^2 c \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \cos ^{-1}(c x)}\right )-\left (6 i b^3 c\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )+\left (6 i b^3 c\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {\left (a+b \cos ^{-1}(c x)\right )^3}{x}-6 i b c \left (a+b \cos ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )+6 i b^2 c \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \cos ^{-1}(c x)}\right )-6 i b^2 c \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \cos ^{-1}(c x)}\right )-\left (6 b^3 c\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )+\left (6 b^3 c\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )\\ &=-\frac {\left (a+b \cos ^{-1}(c x)\right )^3}{x}-6 i b c \left (a+b \cos ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )+6 i b^2 c \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \cos ^{-1}(c x)}\right )-6 i b^2 c \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \cos ^{-1}(c x)}\right )-6 b^3 c \text {Li}_3\left (-i e^{i \cos ^{-1}(c x)}\right )+6 b^3 c \text {Li}_3\left (i e^{i \cos ^{-1}(c 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. \(308\) vs. \(2(151)=302\).
time = 0.20, size = 308, normalized size = 2.04 \begin {gather*} -\frac {a^3}{x}-\frac {3 a^2 b \text {ArcCos}(c x)}{x}-3 a^2 b c \log (x)+3 a^2 b c \log \left (1+\sqrt {1-c^2 x^2}\right )+3 a b^2 c \left (-\frac {\text {ArcCos}(c x)^2}{c x}+2 \left (\text {ArcCos}(c x) \left (\log \left (1-i e^{i \text {ArcCos}(c x)}\right )-\log \left (1+i e^{i \text {ArcCos}(c x)}\right )\right )+i \left (\text {PolyLog}\left (2,-i e^{i \text {ArcCos}(c x)}\right )-\text {PolyLog}\left (2,i e^{i \text {ArcCos}(c x)}\right )\right )\right )\right )+b^3 c \left (-\frac {\text {ArcCos}(c x)^3}{c x}+3 \left (\text {ArcCos}(c x)^2 \left (\log \left (1-i e^{i \text {ArcCos}(c x)}\right )-\log \left (1+i e^{i \text {ArcCos}(c x)}\right )\right )+2 i \text {ArcCos}(c x) \left (\text {PolyLog}\left (2,-i e^{i \text {ArcCos}(c x)}\right )-\text {PolyLog}\left (2,i e^{i \text {ArcCos}(c x)}\right )\right )-2 \left (\text {PolyLog}\left (3,-i e^{i \text {ArcCos}(c x)}\right )-\text {PolyLog}\left (3,i e^{i \text {ArcCos}(c x)}\right )\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arccos \left (c x \right )\right )^{3}}{x^{2}}\, dx\]
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^{2}}\, 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^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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