Optimal. Leaf size=98 \[ \frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{16 a^5}+\frac {27 \text {Si}\left (3 \cos ^{-1}(a x)\right )}{32 a^5}+\frac {25 \text {Si}\left (5 \cos ^{-1}(a x)\right )}{32 a^5}-\frac {2 x^3}{a^2 \cos ^{-1}(a x)}+\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac {5 x^5}{2 \cos ^{-1}(a x)} \]
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Rubi [A] time = 0.34, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4634, 4720, 4636, 4406, 3299} \[ \frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{16 a^5}+\frac {27 \text {Si}\left (3 \cos ^{-1}(a x)\right )}{32 a^5}+\frac {25 \text {Si}\left (5 \cos ^{-1}(a x)\right )}{32 a^5}+\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {2 x^3}{a^2 \cos ^{-1}(a x)}+\frac {5 x^5}{2 \cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 4406
Rule 4634
Rule 4636
Rule 4720
Rubi steps
\begin {align*} \int \frac {x^4}{\cos ^{-1}(a x)^3} \, dx &=\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {2 \int \frac {x^3}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx}{a}+\frac {1}{2} (5 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2} \, dx\\ &=\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {2 x^3}{a^2 \cos ^{-1}(a x)}+\frac {5 x^5}{2 \cos ^{-1}(a x)}-\frac {25}{2} \int \frac {x^4}{\cos ^{-1}(a x)} \, dx+\frac {6 \int \frac {x^2}{\cos ^{-1}(a x)} \, dx}{a^2}\\ &=\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {2 x^3}{a^2 \cos ^{-1}(a x)}+\frac {5 x^5}{2 \cos ^{-1}(a x)}-\frac {6 \operatorname {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {\cos ^4(x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^5}\\ &=\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {2 x^3}{a^2 \cos ^{-1}(a x)}+\frac {5 x^5}{2 \cos ^{-1}(a x)}-\frac {6 \operatorname {Subst}\left (\int \left (\frac {\sin (x)}{4 x}+\frac {\sin (3 x)}{4 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^5}+\frac {25 \operatorname {Subst}\left (\int \left (\frac {\sin (x)}{8 x}+\frac {3 \sin (3 x)}{16 x}+\frac {\sin (5 x)}{16 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{2 a^5}\\ &=\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {2 x^3}{a^2 \cos ^{-1}(a x)}+\frac {5 x^5}{2 \cos ^{-1}(a x)}+\frac {25 \operatorname {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^5}-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^5}-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^5}+\frac {75 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^5}\\ &=\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac {2 x^3}{a^2 \cos ^{-1}(a x)}+\frac {5 x^5}{2 \cos ^{-1}(a x)}+\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{16 a^5}+\frac {27 \text {Si}\left (3 \cos ^{-1}(a x)\right )}{32 a^5}+\frac {25 \text {Si}\left (5 \cos ^{-1}(a x)\right )}{32 a^5}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 103, normalized size = 1.05 \[ \frac {80 a^5 x^5 \cos ^{-1}(a x)-64 a^3 x^3 \cos ^{-1}(a x)+16 a^4 x^4 \sqrt {1-a^2 x^2}+2 \cos ^{-1}(a x)^2 \text {Si}\left (\cos ^{-1}(a x)\right )+27 \cos ^{-1}(a x)^2 \text {Si}\left (3 \cos ^{-1}(a x)\right )+25 \cos ^{-1}(a x)^2 \text {Si}\left (5 \cos ^{-1}(a x)\right )}{32 a^5 \cos ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4}}{\arccos \left (a x\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 86, normalized size = 0.88 \[ \frac {5 \, x^{5}}{2 \, \arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{4}}{2 \, a \arccos \left (a x\right )^{2}} - \frac {2 \, x^{3}}{a^{2} \arccos \left (a x\right )} + \frac {25 \, \operatorname {Si}\left (5 \, \arccos \left (a x\right )\right )}{32 \, a^{5}} + \frac {27 \, \operatorname {Si}\left (3 \, \arccos \left (a x\right )\right )}{32 \, a^{5}} + \frac {\operatorname {Si}\left (\arccos \left (a x\right )\right )}{16 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 121, normalized size = 1.23 \[ \frac {\frac {3 \sin \left (3 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )^{2}}+\frac {9 \cos \left (3 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )}+\frac {27 \Si \left (3 \arccos \left (a x \right )\right )}{32}+\frac {\sin \left (5 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )^{2}}+\frac {5 \cos \left (5 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )}+\frac {25 \Si \left (5 \arccos \left (a x \right )\right )}{32}+\frac {\sqrt {-a^{2} x^{2}+1}}{16 \arccos \left (a x \right )^{2}}+\frac {a x}{16 \arccos \left (a x \right )}+\frac {\Si \left (\arccos \left (a x \right )\right )}{16}}{a^{5}} \]
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 \[ \frac {\sqrt {a x + 1} \sqrt {-a x + 1} a x^{4} - \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2} \int \frac {{\left (25 \, a^{2} x^{2} - 12\right )} x^{2}}{\arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}\,{d x} + {\left (5 \, a^{2} x^{5} - 4 \, x^{3}\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}{2 \, a^{2} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2}} \]
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 \[ \int \frac {x^4}{{\mathrm {acos}\left (a\,x\right )}^3} \,d x \]
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 \[ \int \frac {x^{4}}{\operatorname {acos}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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