Optimal. Leaf size=51 \[ \frac {\sqrt {1-a^2 x^2}}{2 a \text {ArcCos}(a x)^2}+\frac {x}{2 \text {ArcCos}(a x)}+\frac {\text {Si}(\text {ArcCos}(a x))}{2 a} \]
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Rubi [A]
time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4718, 4808,
4720, 3380} \begin {gather*} \frac {\sqrt {1-a^2 x^2}}{2 a \text {ArcCos}(a x)^2}+\frac {\text {Si}(\text {ArcCos}(a x))}{2 a}+\frac {x}{2 \text {ArcCos}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 4718
Rule 4720
Rule 4808
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{-1}(a x)^3} \, dx &=\frac {\sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac {1}{2} a \int \frac {x}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx\\ &=\frac {\sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac {x}{2 \cos ^{-1}(a x)}-\frac {1}{2} \int \frac {1}{\cos ^{-1}(a x)} \, dx\\ &=\frac {\sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac {x}{2 \cos ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a}\\ &=\frac {\sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac {x}{2 \cos ^{-1}(a x)}+\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{2 a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 47, normalized size = 0.92 \begin {gather*} \frac {\sqrt {1-a^2 x^2}+a x \text {ArcCos}(a x)+\text {ArcCos}(a x)^2 \text {Si}(\text {ArcCos}(a x))}{2 a \text {ArcCos}(a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 43, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{2 \arccos \left (a x \right )^{2}}+\frac {a x}{2 \arccos \left (a x \right )}+\frac {\sinIntegral \left (\arccos \left (a x \right )\right )}{2}}{a}\) | \(43\) |
default | \(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{2 \arccos \left (a x \right )^{2}}+\frac {a x}{2 \arccos \left (a x \right )}+\frac {\sinIntegral \left (\arccos \left (a x \right )\right )}{2}}{a}\) | \(43\) |
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 {1}{\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 = 43, normalized size = 0.84 \begin {gather*} \frac {x}{2 \, \arccos \left (a x\right )} + \frac {\operatorname {Si}\left (\arccos \left (a x\right )\right )}{2 \, a} + \frac {\sqrt {-a^{2} x^{2} + 1}}{2 \, a \arccos \left (a x\right )^{2}} \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 {1}{{\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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