Optimal. Leaf size=56 \[ \frac {x^3 \sqrt {1-a^2 x^2}}{a \text {ArcCos}(a x)}-\frac {\text {CosIntegral}(2 \text {ArcCos}(a x))}{2 a^4}-\frac {\text {CosIntegral}(4 \text {ArcCos}(a x))}{2 a^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4728, 3383}
\begin {gather*} -\frac {\text {CosIntegral}(2 \text {ArcCos}(a x))}{2 a^4}-\frac {\text {CosIntegral}(4 \text {ArcCos}(a x))}{2 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{a \text {ArcCos}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3383
Rule 4728
Rubi steps
\begin {align*} \int \frac {x^3}{\cos ^{-1}(a x)^2} \, dx &=\frac {x^3 \sqrt {1-a^2 x^2}}{a \cos ^{-1}(a x)}+\frac {\text {Subst}\left (\int \left (-\frac {\cos (2 x)}{2 x}-\frac {\cos (4 x)}{2 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^4}-\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^4}\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac {\text {Ci}\left (2 \cos ^{-1}(a x)\right )}{2 a^4}-\frac {\text {Ci}\left (4 \cos ^{-1}(a x)\right )}{2 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 50, normalized size = 0.89 \begin {gather*} -\frac {-\frac {2 a^3 x^3 \sqrt {1-a^2 x^2}}{\text {ArcCos}(a x)}+\text {CosIntegral}(2 \text {ArcCos}(a x))+\text {CosIntegral}(4 \text {ArcCos}(a x))}{2 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 54, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}-\frac {\cosineIntegral \left (2 \arccos \left (a x \right )\right )}{2}+\frac {\sin \left (4 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )}-\frac {\cosineIntegral \left (4 \arccos \left (a x \right )\right )}{2}}{a^{4}}\) | \(54\) |
default | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}-\frac {\cosineIntegral \left (2 \arccos \left (a x \right )\right )}{2}+\frac {\sin \left (4 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )}-\frac {\cosineIntegral \left (4 \arccos \left (a x \right )\right )}{2}}{a^{4}}\) | \(54\) |
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 {x^{3}}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 50, normalized size = 0.89 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{a \arccos \left (a x\right )} - \frac {\operatorname {Ci}\left (4 \, \arccos \left (a x\right )\right )}{2 \, a^{4}} - \frac {\operatorname {Ci}\left (2 \, \arccos \left (a x\right )\right )}{2 \, a^{4}} \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 = -0.02 \begin {gather*} \int \frac {x^3}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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