Optimal. Leaf size=102 \[ -\frac {3}{2} i a^2 \text {ArcCos}(a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^2}{2 x}-\frac {\text {ArcCos}(a x)^3}{2 x^2}+3 a^2 \text {ArcCos}(a x) \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )-\frac {3}{2} i a^2 \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(a x)}\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4724, 4772,
4722, 3800, 2221, 2317, 2438} \begin {gather*} -\frac {3}{2} i a^2 \text {Li}_2\left (-e^{2 i \text {ArcCos}(a x)}\right )+\frac {3 a \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^2}{2 x}-\frac {3}{2} i a^2 \text {ArcCos}(a x)^2+3 a^2 \text {ArcCos}(a x) \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )-\frac {\text {ArcCos}(a x)^3}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4722
Rule 4724
Rule 4772
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a x)^3}{x^3} \, dx &=-\frac {\cos ^{-1}(a x)^3}{2 x^2}-\frac {1}{2} (3 a) \int \frac {\cos ^{-1}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \int \frac {\cos ^{-1}(a x)}{x} \, dx\\ &=\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{2 x^2}-\left (3 a^2\right ) \text {Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {3}{2} i a^2 \cos ^{-1}(a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{2 x^2}+\left (6 i a^2\right ) \text {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {3}{2} i a^2 \cos ^{-1}(a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{2 x^2}+3 a^2 \cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-\left (3 a^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {3}{2} i a^2 \cos ^{-1}(a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{2 x^2}+3 a^2 \cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )+\frac {1}{2} \left (3 i a^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}(a x)}\right )\\ &=-\frac {3}{2} i a^2 \cos ^{-1}(a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{2 x^2}+3 a^2 \cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-\frac {3}{2} i a^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 92, normalized size = 0.90 \begin {gather*} \frac {1}{2} \left (\frac {3 a \left (-i a x+\sqrt {1-a^2 x^2}\right ) \text {ArcCos}(a x)^2}{x}-\frac {\text {ArcCos}(a x)^3}{x^2}+6 a^2 \text {ArcCos}(a x) \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )-3 i a^2 \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(a x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 117, normalized size = 1.15
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\arccos \left (a x \right )^{2} \left (-3 i a^{2} x^{2}-3 a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{2 a^{2} x^{2}}-3 i \arccos \left (a x \right )^{2}+3 \arccos \left (a x \right ) \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \polylog \left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\) | \(117\) |
default | \(a^{2} \left (-\frac {\arccos \left (a x \right )^{2} \left (-3 i a^{2} x^{2}-3 a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{2 a^{2} x^{2}}-3 i \arccos \left (a x \right )^{2}+3 \arccos \left (a x \right ) \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \polylog \left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\) | \(117\) |
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^{3}}\, 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^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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