Optimal. Leaf size=82 \[ \frac {x^2 \sqrt {1-a^2 x^2}}{2 a \text {ArcCos}(a x)^2}-\frac {x}{a^2 \text {ArcCos}(a x)}+\frac {3 x^3}{2 \text {ArcCos}(a x)}+\frac {\text {Si}(\text {ArcCos}(a x))}{8 a^3}+\frac {9 \text {Si}(3 \text {ArcCos}(a x))}{8 a^3} \]
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Rubi [A]
time = 0.17, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 10, 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, 4720} \begin {gather*} \frac {\text {Si}(\text {ArcCos}(a x))}{8 a^3}+\frac {9 \text {Si}(3 \text {ArcCos}(a x))}{8 a^3}+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \text {ArcCos}(a x)^2}-\frac {x}{a^2 \text {ArcCos}(a x)}+\frac {3 x^3}{2 \text {ArcCos}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 4491
Rule 4720
Rule 4730
Rule 4732
Rule 4808
Rubi steps
\begin {align*} \int \frac {x^2}{\cos ^{-1}(a x)^3} \, dx &=\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {\int \frac {x}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx}{a}+\frac {1}{2} (3 a) \int \frac {x^3}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx\\ &=\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {x}{a^2 \cos ^{-1}(a x)}+\frac {3 x^3}{2 \cos ^{-1}(a x)}-\frac {9}{2} \int \frac {x^2}{\cos ^{-1}(a x)} \, dx+\frac {\int \frac {1}{\cos ^{-1}(a x)} \, dx}{a^2}\\ &=\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {x}{a^2 \cos ^{-1}(a x)}+\frac {3 x^3}{2 \cos ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^3}+\frac {9 \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^3}\\ &=\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {x}{a^2 \cos ^{-1}(a x)}+\frac {3 x^3}{2 \cos ^{-1}(a x)}-\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{a^3}+\frac {9 \text {Subst}\left (\int \left (\frac {\sin (x)}{4 x}+\frac {\sin (3 x)}{4 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{2 a^3}\\ &=\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {x}{a^2 \cos ^{-1}(a x)}+\frac {3 x^3}{2 \cos ^{-1}(a x)}-\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{a^3}+\frac {9 \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}\\ &=\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {x}{a^2 \cos ^{-1}(a x)}+\frac {3 x^3}{2 \cos ^{-1}(a x)}+\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Si}\left (3 \cos ^{-1}(a x)\right )}{8 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 65, normalized size = 0.79 \begin {gather*} \frac {\frac {4 a x \left (a x \sqrt {1-a^2 x^2}+\left (-2+3 a^2 x^2\right ) \text {ArcCos}(a x)\right )}{\text {ArcCos}(a x)^2}+\text {Si}(\text {ArcCos}(a x))+9 \text {Si}(3 \text {ArcCos}(a x))}{8 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 82, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}+\frac {3 \cos \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )}+\frac {9 \sinIntegral \left (3 \arccos \left (a x \right )\right )}{8}+\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arccos \left (a x \right )^{2}}+\frac {a x}{8 \arccos \left (a x \right )}+\frac {\sinIntegral \left (\arccos \left (a x \right )\right )}{8}}{a^{3}}\) | \(82\) |
default | \(\frac {\frac {\sin \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}+\frac {3 \cos \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )}+\frac {9 \sinIntegral \left (3 \arccos \left (a x \right )\right )}{8}+\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arccos \left (a x \right )^{2}}+\frac {a x}{8 \arccos \left (a x \right )}+\frac {\sinIntegral \left (\arccos \left (a x \right )\right )}{8}}{a^{3}}\) | \(82\) |
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^{2}}{\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 = 0.45, size = 72, normalized size = 0.88 \begin {gather*} \frac {3 \, x^{3}}{2 \, \arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{2 \, a \arccos \left (a x\right )^{2}} - \frac {x}{a^{2} \arccos \left (a x\right )} + \frac {9 \, \operatorname {Si}\left (3 \, \arccos \left (a x\right )\right )}{8 \, a^{3}} + \frac {\operatorname {Si}\left (\arccos \left (a x\right )\right )}{8 \, a^{3}} \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^2}{{\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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