Optimal. Leaf size=92 \[ -\frac {i (a+b \text {ArcCos}(c x))^3}{3 b}+(a+b \text {ArcCos}(c x))^2 \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )-i b (a+b \text {ArcCos}(c x)) \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(c x)}\right )+\frac {1}{2} b^2 \text {PolyLog}\left (3,-e^{2 i \text {ArcCos}(c x)}\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4722, 3800,
2221, 2611, 2320, 6724} \begin {gather*} -i b \text {Li}_2\left (-e^{2 i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))-\frac {i (a+b \text {ArcCos}(c x))^3}{3 b}+\log \left (1+e^{2 i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))^2+\frac {1}{2} b^2 \text {Li}_3\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
Rubi steps
\begin {align*} \int \frac {\left (a+b \cos ^{-1}(c x)\right )^2}{x} \, dx &=-\text {Subst}\left (\int (a+b x)^2 \tan (x) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^3}{3 b}+2 i \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^2}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^3}{3 b}+\left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-(2 b) \text {Subst}\left (\int (a+b x) \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 )^3}{3 b}+\left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-i b \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\left (i b^2\right ) \text {Subst}\left (\int \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 )^3}{3 b}+\left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-i b \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac {1}{2} b^2 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \cos ^{-1}(c x)}\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^3}{3 b}+\left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-i b \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac {1}{2} b^2 \text {Li}_3\left (-e^{2 i \cos ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 128, normalized size = 1.39 \begin {gather*} -i a b \text {ArcCos}(c x)^2-\frac {1}{3} i b^2 \text {ArcCos}(c x)^3+2 a b \text {ArcCos}(c x) \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )+b^2 \text {ArcCos}(c x)^2 \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )+a^2 \log (c x)-i b (a+b \text {ArcCos}(c x)) \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(c x)}\right )+\frac {1}{2} b^2 \text {PolyLog}\left (3,-e^{2 i \text {ArcCos}(c x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 194, normalized size = 2.11
method | result | size |
derivativedivides | \(a^{2} \ln \left (c x \right )-\frac {i b^{2} \arccos \left (c x \right )^{3}}{3}+b^{2} \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i b^{2} \arccos \left (c x \right ) \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {b^{2} \polylog \left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-i a b \arccos \left (c x \right )^{2}+2 a b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i a b \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\) | \(194\) |
default | \(a^{2} \ln \left (c x \right )-\frac {i b^{2} \arccos \left (c x \right )^{3}}{3}+b^{2} \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i b^{2} \arccos \left (c x \right ) \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {b^{2} \polylog \left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-i a b \arccos \left (c x \right )^{2}+2 a b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i a b \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\) | \(194\) |
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 )^{2}}{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 )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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