Optimal. Leaf size=29 \[ -\frac {\text {Si}(2 \text {ArcCos}(a x))}{4 a^4}-\frac {\text {Si}(4 \text {ArcCos}(a x))}{8 a^4} \]
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Rubi [A]
time = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4732, 4491,
3380} \begin {gather*} -\frac {\text {Si}(2 \text {ArcCos}(a x))}{4 a^4}-\frac {\text {Si}(4 \text {ArcCos}(a x))}{8 a^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 4491
Rule 4732
Rubi steps
\begin {align*} \int \frac {x^3}{\cos ^{-1}(a x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac {\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 {\text {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}-\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {\text {Si}\left (2 \cos ^{-1}(a x)\right )}{4 a^4}-\frac {\text {Si}\left (4 \cos ^{-1}(a x)\right )}{8 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 24, normalized size = 0.83 \begin {gather*} -\frac {2 \text {Si}(2 \text {ArcCos}(a x))+\text {Si}(4 \text {ArcCos}(a x))}{8 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 24, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {-\frac {\sinIntegral \left (2 \arccos \left (a x \right )\right )}{4}-\frac {\sinIntegral \left (4 \arccos \left (a x \right )\right )}{8}}{a^{4}}\) | \(24\) |
default | \(\frac {-\frac {\sinIntegral \left (2 \arccos \left (a x \right )\right )}{4}-\frac {\sinIntegral \left (4 \arccos \left (a x \right )\right )}{8}}{a^{4}}\) | \(24\) |
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}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 25, normalized size = 0.86 \begin {gather*} -\frac {\operatorname {Si}\left (4 \, \arccos \left (a x\right )\right )}{8 \, a^{4}} - \frac {\operatorname {Si}\left (2 \, \arccos \left (a x\right )\right )}{4 \, 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 = -0.03 \begin {gather*} \int \frac {x^3}{\mathrm {acos}\left (a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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