Optimal. Leaf size=83 \[ \frac {\text {Si}\left (2 \cos ^{-1}(a x)\right )}{2 a^4}+\frac {\text {Si}\left (4 \cos ^{-1}(a x)\right )}{a^4}-\frac {3 x^2}{2 a^2 \cos ^{-1}(a x)}+\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac {2 x^4}{\cos ^{-1}(a x)} \]
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Rubi [A] time = 0.30, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 12, 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, 12} \[ \frac {\text {Si}\left (2 \cos ^{-1}(a x)\right )}{2 a^4}+\frac {\text {Si}\left (4 \cos ^{-1}(a x)\right )}{a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {3 x^2}{2 a^2 \cos ^{-1}(a x)}+\frac {2 x^4}{\cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 4406
Rule 4634
Rule 4636
Rule 4720
Rubi steps
\begin {align*} \int \frac {x^3}{\cos ^{-1}(a x)^3} \, dx &=\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx}{2 a}+(2 a) \int \frac {x^4}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {3 x^2}{2 a^2 \cos ^{-1}(a x)}+\frac {2 x^4}{\cos ^{-1}(a x)}-8 \int \frac {x^3}{\cos ^{-1}(a x)} \, dx+\frac {3 \int \frac {x}{\cos ^{-1}(a x)} \, dx}{a^2}\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {3 x^2}{2 a^2 \cos ^{-1}(a x)}+\frac {2 x^4}{\cos ^{-1}(a x)}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}+\frac {8 \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {3 x^2}{2 a^2 \cos ^{-1}(a x)}+\frac {2 x^4}{\cos ^{-1}(a x)}-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}+\frac {8 \operatorname {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 {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {3 x^2}{2 a^2 \cos ^{-1}(a x)}+\frac {2 x^4}{\cos ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^4}+\frac {2 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {3 x^2}{2 a^2 \cos ^{-1}(a x)}+\frac {2 x^4}{\cos ^{-1}(a x)}+\frac {\text {Si}\left (2 \cos ^{-1}(a x)\right )}{2 a^4}+\frac {\text {Si}\left (4 \cos ^{-1}(a x)\right )}{a^4}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 70, normalized size = 0.84 \[ \frac {\frac {a^2 x^2 \left (a x \sqrt {1-a^2 x^2}+\left (4 a^2 x^2-3\right ) \cos ^{-1}(a x)\right )}{\cos ^{-1}(a x)^2}+\text {Si}\left (2 \cos ^{-1}(a x)\right )+2 \text {Si}\left (4 \cos ^{-1}(a x)\right )}{2 a^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{\arccos \left (a x\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 75, normalized size = 0.90 \[ \frac {2 \, x^{4}}{\arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{2 \, a \arccos \left (a x\right )^{2}} - \frac {3 \, x^{2}}{2 \, a^{2} \arccos \left (a x\right )} + \frac {\operatorname {Si}\left (4 \, \arccos \left (a x\right )\right )}{a^{4}} + \frac {\operatorname {Si}\left (2 \, \arccos \left (a x\right )\right )}{2 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 82, normalized size = 0.99 \[ \frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\frac {\Si \left (2 \arccos \left (a x \right )\right )}{2}+\frac {\sin \left (4 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )^{2}}+\frac {\cos \left (4 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\Si \left (4 \arccos \left (a x \right )\right )}{a^{4}} \]
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 \[ \frac {\sqrt {a x + 1} \sqrt {-a x + 1} a x^{3} - 2 \, \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2} \int \frac {{\left (8 \, a^{2} x^{2} - 3\right )} x}{\arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}\,{d x} + {\left (4 \, a^{2} x^{4} - 3 \, x^{2}\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}{2 \, a^{2} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2}} \]
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 \[ \int \frac {x^3}{{\mathrm {acos}\left (a\,x\right )}^3} \,d x \]
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 \[ \int \frac {x^{3}}{\operatorname {acos}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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