Optimal. Leaf size=97 \[ \frac{2 \text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{3 a^2}-\frac{2 x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)}+\frac{x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{1}{6 a^2 \cos ^{-1}(a x)^2}+\frac{x^2}{3 \cos ^{-1}(a x)^2} \]
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Rubi [A] time = 0.166116, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4634, 4720, 4632, 3302, 4642} \[ \frac{2 \text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{3 a^2}-\frac{2 x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)}+\frac{x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{1}{6 a^2 \cos ^{-1}(a x)^2}+\frac{x^2}{3 \cos ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 4634
Rule 4720
Rule 4632
Rule 3302
Rule 4642
Rubi steps
\begin{align*} \int \frac{x}{\cos ^{-1}(a x)^4} \, dx &=\frac{x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3} \, dx}{3 a}+\frac{1}{3} (2 a) \int \frac{x^2}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3} \, dx\\ &=\frac{x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{1}{6 a^2 \cos ^{-1}(a x)^2}+\frac{x^2}{3 \cos ^{-1}(a x)^2}-\frac{2}{3} \int \frac{x}{\cos ^{-1}(a x)^2} \, dx\\ &=\frac{x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{1}{6 a^2 \cos ^{-1}(a x)^2}+\frac{x^2}{3 \cos ^{-1}(a x)^2}-\frac{2 x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^2}\\ &=\frac{x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{1}{6 a^2 \cos ^{-1}(a x)^2}+\frac{x^2}{3 \cos ^{-1}(a x)^2}-\frac{2 x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)}+\frac{2 \text{Ci}\left (2 \cos ^{-1}(a x)\right )}{3 a^2}\\ \end{align*}
Mathematica [A] time = 0.103208, size = 86, normalized size = 0.89 \[ \frac{2 a x \sqrt{1-a^2 x^2}-4 a x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2+\left (2 a^2 x^2-1\right ) \cos ^{-1}(a x)+4 \cos ^{-1}(a x)^3 \text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{6 a^2 \cos ^{-1}(a x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 60, normalized size = 0.6 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{\sin \left ( 2\,\arccos \left ( ax \right ) \right ) }{6\, \left ( \arccos \left ( ax \right ) \right ) ^{3}}}+{\frac{\cos \left ( 2\,\arccos \left ( ax \right ) \right ) }{6\, \left ( \arccos \left ( ax \right ) \right ) ^{2}}}-{\frac{\sin \left ( 2\,\arccos \left ( ax \right ) \right ) }{3\,\arccos \left ( ax \right ) }}+{\frac{2\,{\it Ci} \left ( 2\,\arccos \left ( ax \right ) \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{4 \, a^{2} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{3} \int \frac{{\left (2 \, a^{2} x^{2} - 1\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{{\left (a^{3} x^{2} - a\right )} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )}\,{d x} - 2 \,{\left (2 \, a x \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{2} - a x\right )} \sqrt{a x + 1} \sqrt{-a x + 1} +{\left (2 \, a^{2} x^{2} - 1\right )} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )}{6 \, a^{2} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{\arccos \left (a x\right )^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\operatorname{acos}^{4}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15565, size = 112, normalized size = 1.15 \begin{align*} \frac{x^{2}}{3 \, \arccos \left (a x\right )^{2}} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} x}{3 \, a \arccos \left (a x\right )} + \frac{2 \, \operatorname{Ci}\left (2 \, \arccos \left (a x\right )\right )}{3 \, a^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1} x}{3 \, a \arccos \left (a x\right )^{3}} - \frac{1}{6 \, a^{2} \arccos \left (a x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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