Optimal. Leaf size=83 \[ \frac {x^3 \sqrt {1-a^2 x^2}}{2 a \text {ArcCos}(a x)^2}-\frac {3 x^2}{2 a^2 \text {ArcCos}(a x)}+\frac {2 x^4}{\text {ArcCos}(a x)}+\frac {\text {Si}(2 \text {ArcCos}(a x))}{2 a^4}+\frac {\text {Si}(4 \text {ArcCos}(a x))}{a^4} \]
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Rubi [A]
time = 0.21, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4730, 4808,
4732, 4491, 3380, 12} \begin {gather*} \frac {\text {Si}(2 \text {ArcCos}(a x))}{2 a^4}+\frac {\text {Si}(4 \text {ArcCos}(a x))}{a^4}-\frac {3 x^2}{2 a^2 \text {ArcCos}(a x)}+\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \text {ArcCos}(a x)^2}+\frac {2 x^4}{\text {ArcCos}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3380
Rule 4491
Rule 4730
Rule 4732
Rule 4808
Rubi steps
\begin {align*} \int \frac {x^3}{\cos ^{-1}(a x)^3} \, dx &=\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx}{2 a}+(2 a) \int \frac {x^4}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {3 x^2}{2 a^2 \cos ^{-1}(a x)}+\frac {2 x^4}{\cos ^{-1}(a x)}-8 \int \frac {x^3}{\cos ^{-1}(a x)} \, dx+\frac {3 \int \frac {x}{\cos ^{-1}(a x)} \, dx}{a^2}\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {3 x^2}{2 a^2 \cos ^{-1}(a x)}+\frac {2 x^4}{\cos ^{-1}(a x)}-\frac {3 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}+\frac {8 \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {3 x^2}{2 a^2 \cos ^{-1}(a x)}+\frac {2 x^4}{\cos ^{-1}(a x)}-\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}+\frac {8 \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 x}+\frac {\sin (4 x)}{8 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {3 x^2}{2 a^2 \cos ^{-1}(a x)}+\frac {2 x^4}{\cos ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}-\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^4}+\frac {2 \text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {3 x^2}{2 a^2 \cos ^{-1}(a x)}+\frac {2 x^4}{\cos ^{-1}(a x)}+\frac {\text {Si}\left (2 \cos ^{-1}(a x)\right )}{2 a^4}+\frac {\text {Si}\left (4 \cos ^{-1}(a x)\right )}{a^4}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 70, normalized size = 0.84 \begin {gather*} \frac {\frac {a^2 x^2 \left (a x \sqrt {1-a^2 x^2}+\left (-3+4 a^2 x^2\right ) \text {ArcCos}(a x)\right )}{\text {ArcCos}(a x)^2}+\text {Si}(2 \text {ArcCos}(a x))+2 \text {Si}(4 \text {ArcCos}(a x))}{2 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 82, normalized size = 0.99
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\frac {\sinIntegral \left (2 \arccos \left (a x \right )\right )}{2}+\frac {\sin \left (4 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )^{2}}+\frac {\cos \left (4 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\sinIntegral \left (4 \arccos \left (a x \right )\right )}{a^{4}}\) | \(82\) |
default | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\frac {\sinIntegral \left (2 \arccos \left (a x \right )\right )}{2}+\frac {\sin \left (4 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )^{2}}+\frac {\cos \left (4 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\sinIntegral \left (4 \arccos \left (a x \right )\right )}{a^{4}}\) | \(82\) |
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}^{3}{\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 = 75, normalized size = 0.90 \begin {gather*} \frac {2 \, x^{4}}{\arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{2 \, a \arccos \left (a x\right )^{2}} - \frac {3 \, x^{2}}{2 \, a^{2} \arccos \left (a x\right )} + \frac {\operatorname {Si}\left (4 \, \arccos \left (a x\right )\right )}{a^{4}} + \frac {\operatorname {Si}\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 = -0.01 \begin {gather*} \int \frac {x^3}{{\mathrm {acos}\left (a\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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