Optimal. Leaf size=63 \[ \frac {x \sqrt {1-a^2 x^2}}{2 a \text {ArcCos}(a x)^2}-\frac {1}{2 a^2 \text {ArcCos}(a x)}+\frac {x^2}{\text {ArcCos}(a x)}+\frac {\text {Si}(2 \text {ArcCos}(a x))}{a^2} \]
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Rubi [A]
time = 0.12, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {4730, 4808,
4732, 4491, 12, 3380, 4738} \begin {gather*} \frac {\text {Si}(2 \text {ArcCos}(a x))}{a^2}+\frac {x \sqrt {1-a^2 x^2}}{2 a \text {ArcCos}(a x)^2}-\frac {1}{2 a^2 \text {ArcCos}(a x)}+\frac {x^2}{\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 4738
Rule 4808
Rubi steps
\begin {align*} \int \frac {x}{\cos ^{-1}(a x)^3} \, dx &=\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx}{2 a}+a \int \frac {x^2}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx\\ &=\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {1}{2 a^2 \cos ^{-1}(a x)}+\frac {x^2}{\cos ^{-1}(a x)}-2 \int \frac {x}{\cos ^{-1}(a x)} \, dx\\ &=\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {1}{2 a^2 \cos ^{-1}(a x)}+\frac {x^2}{\cos ^{-1}(a x)}+\frac {2 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {1}{2 a^2 \cos ^{-1}(a x)}+\frac {x^2}{\cos ^{-1}(a x)}+\frac {2 \text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {1}{2 a^2 \cos ^{-1}(a x)}+\frac {x^2}{\cos ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=\frac {x \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {1}{2 a^2 \cos ^{-1}(a x)}+\frac {x^2}{\cos ^{-1}(a x)}+\frac {\text {Si}\left (2 \cos ^{-1}(a x)\right )}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 63, normalized size = 1.00 \begin {gather*} \frac {x \sqrt {1-a^2 x^2}}{2 a \text {ArcCos}(a x)^2}+\frac {-1+2 a^2 x^2}{2 a^2 \text {ArcCos}(a x)}+\frac {\text {Si}(2 \text {ArcCos}(a x))}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 43, normalized size = 0.68
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )^{2}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{2 \arccos \left (a x \right )}+\sinIntegral \left (2 \arccos \left (a x \right )\right )}{a^{2}}\) | \(43\) |
default | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )^{2}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{2 \arccos \left (a x \right )}+\sinIntegral \left (2 \arccos \left (a x \right )\right )}{a^{2}}\) | \(43\) |
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}{\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 = 0.41, size = 57, normalized size = 0.90 \begin {gather*} \frac {x^{2}}{\arccos \left (a x\right )} + \frac {\operatorname {Si}\left (2 \, \arccos \left (a x\right )\right )}{a^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a \arccos \left (a x\right )^{2}} - \frac {1}{2 \, a^{2} \arccos \left (a x\right )} \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}{{\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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