Optimal. Leaf size=82 \[ \frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Si}\left (3 \cos ^{-1}(a x)\right )}{8 a^3}+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {x}{a^2 \cos ^{-1}(a x)}+\frac {3 x^3}{2 \cos ^{-1}(a x)} \]
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Rubi [A] time = 0.24, 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 = {4634, 4720, 4636, 4406, 3299, 4624} \[ \frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Si}\left (3 \cos ^{-1}(a x)\right )}{8 a^3}+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {x}{a^2 \cos ^{-1}(a x)}+\frac {3 x^3}{2 \cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 4406
Rule 4624
Rule 4634
Rule 4636
Rule 4720
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 {\operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^3}+\frac {9 \operatorname {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 \operatorname {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 \operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}+\frac {9 \operatorname {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 = 0.14, size = 65, normalized size = 0.79 \[ \frac {\frac {4 a x \left (a x \sqrt {1-a^2 x^2}+\left (3 a^2 x^2-2\right ) \cos ^{-1}(a x)\right )}{\cos ^{-1}(a x)^2}+\text {Si}\left (\cos ^{-1}(a x)\right )+9 \text {Si}\left (3 \cos ^{-1}(a x)\right )}{8 a^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\arccos \left (a x\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 72, normalized size = 0.88 \[ \frac {3 \, x^{3}}{2 \, \arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{2 \, a \arccos \left (a x\right )^{2}} - \frac {x}{a^{2} \arccos \left (a x\right )} + \frac {9 \, \operatorname {Si}\left (3 \, \arccos \left (a x\right )\right )}{8 \, a^{3}} + \frac {\operatorname {Si}\left (\arccos \left (a x\right )\right )}{8 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 82, normalized size = 1.00 \[ \frac {\frac {\sin \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}+\frac {3 \cos \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )}+\frac {9 \Si \left (3 \arccos \left (a x \right )\right )}{8}+\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arccos \left (a x \right )^{2}}+\frac {a x}{8 \arccos \left (a x \right )}+\frac {\Si \left (\arccos \left (a x \right )\right )}{8}}{a^{3}} \]
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 \[ \frac {\sqrt {a x + 1} \sqrt {-a x + 1} a x^{2} - \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2} \int \frac {9 \, a^{2} x^{2} - 2}{\arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}\,{d x} + {\left (3 \, a^{2} x^{3} - 2 \, x\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}{2 \, a^{2} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2}} \]
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 \[ \int \frac {x^2}{{\mathrm {acos}\left (a\,x\right )}^3} \,d x \]
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 \[ \int \frac {x^{2}}{\operatorname {acos}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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