Optimal. Leaf size=143 \[ \frac {\text {Ci}\left (2 \cos ^{-1}(a x)\right )}{3 a^4}+\frac {4 \text {Ci}\left (4 \cos ^{-1}(a x)\right )}{3 a^4}-\frac {x^2}{2 a^2 \cos ^{-1}(a x)^2}-\frac {8 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)}+\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}+\frac {x \sqrt {1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}+\frac {2 x^4}{3 \cos ^{-1}(a x)^2} \]
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Rubi [A] time = 0.29, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4634, 4720, 4632, 3302} \[ \frac {\text {CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{3 a^4}+\frac {4 \text {CosIntegral}\left (4 \cos ^{-1}(a x)\right )}{3 a^4}-\frac {8 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)}+\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac {x^2}{2 a^2 \cos ^{-1}(a x)^2}+\frac {x \sqrt {1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}+\frac {2 x^4}{3 \cos ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 4632
Rule 4634
Rule 4720
Rubi steps
\begin {align*} \int \frac {x^3}{\cos ^{-1}(a x)^4} \, dx &=\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac {\int \frac {x^2}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3} \, dx}{a}+\frac {1}{3} (4 a) \int \frac {x^4}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3} \, dx\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac {x^2}{2 a^2 \cos ^{-1}(a x)^2}+\frac {2 x^4}{3 \cos ^{-1}(a x)^2}-\frac {8}{3} \int \frac {x^3}{\cos ^{-1}(a x)^2} \, dx+\frac {\int \frac {x}{\cos ^{-1}(a x)^2} \, dx}{a^2}\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac {x^2}{2 a^2 \cos ^{-1}(a x)^2}+\frac {2 x^4}{3 \cos ^{-1}(a x)^2}+\frac {x \sqrt {1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}-\frac {8 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}-\frac {8 \operatorname {Subst}\left (\int \left (-\frac {\cos (2 x)}{2 x}-\frac {\cos (4 x)}{2 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{3 a^4}\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac {x^2}{2 a^2 \cos ^{-1}(a x)^2}+\frac {2 x^4}{3 \cos ^{-1}(a x)^2}+\frac {x \sqrt {1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}-\frac {8 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)}-\frac {\text {Ci}\left (2 \cos ^{-1}(a x)\right )}{a^4}+\frac {4 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^4}+\frac {4 \operatorname {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^4}\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac {x^2}{2 a^2 \cos ^{-1}(a x)^2}+\frac {2 x^4}{3 \cos ^{-1}(a x)^2}+\frac {x \sqrt {1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}-\frac {8 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)}+\frac {\text {Ci}\left (2 \cos ^{-1}(a x)\right )}{3 a^4}+\frac {4 \text {Ci}\left (4 \cos ^{-1}(a x)\right )}{3 a^4}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 107, normalized size = 0.75 \[ \frac {\frac {a x \left (2 a^2 x^2 \sqrt {1-a^2 x^2}+a x \left (4 a^2 x^2-3\right ) \cos ^{-1}(a x)-2 \sqrt {1-a^2 x^2} \left (8 a^2 x^2-3\right ) \cos ^{-1}(a x)^2\right )}{\cos ^{-1}(a x)^3}+2 \text {Ci}\left (2 \cos ^{-1}(a x)\right )+8 \text {Ci}\left (4 \cos ^{-1}(a x)\right )}{6 a^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{\arccos \left (a x\right )^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 125, normalized size = 0.87 \[ \frac {2 \, x^{4}}{3 \, \arccos \left (a x\right )^{2}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} x^{3}}{3 \, a \arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{3 \, a \arccos \left (a x\right )^{3}} - \frac {x^{2}}{2 \, a^{2} \arccos \left (a x\right )^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{3} \arccos \left (a x\right )} + \frac {4 \, \operatorname {Ci}\left (4 \, \arccos \left (a x\right )\right )}{3 \, a^{4}} + \frac {\operatorname {Ci}\left (2 \, \arccos \left (a x\right )\right )}{3 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 114, normalized size = 0.80 \[ \frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{12 \arccos \left (a x \right )^{3}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{12 \arccos \left (a x \right )^{2}}-\frac {\sin \left (2 \arccos \left (a x \right )\right )}{6 \arccos \left (a x \right )}+\frac {\Ci \left (2 \arccos \left (a x \right )\right )}{3}+\frac {\sin \left (4 \arccos \left (a x \right )\right )}{24 \arccos \left (a x \right )^{3}}+\frac {\cos \left (4 \arccos \left (a x \right )\right )}{12 \arccos \left (a x \right )^{2}}-\frac {\sin \left (4 \arccos \left (a x \right )\right )}{3 \arccos \left (a x \right )}+\frac {4 \Ci \left (4 \arccos \left (a x \right )\right )}{3}}{a^{4}} \]
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 \[ \frac {2 \, a^{3} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{3} \int \frac {{\left (32 \, a^{4} x^{4} - 30 \, a^{2} x^{2} + 3\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{{\left (a^{5} x^{2} - a^{3}\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}\,{d x} + 2 \, {\left (a^{2} x^{3} - {\left (8 \, a^{2} x^{3} - 3 \, x\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2}\right )} \sqrt {a x + 1} \sqrt {-a x + 1} + {\left (4 \, a^{3} x^{4} - 3 \, a x^{2}\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}{6 \, a^{3} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{3}} \]
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 \[ \int \frac {x^3}{{\mathrm {acos}\left (a\,x\right )}^4} \,d x \]
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 \[ \int \frac {x^{3}}{\operatorname {acos}^{4}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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