Optimal. Leaf size=57 \[ \frac{a+b \cos ^{-1}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{b x}{2 c d^2 \sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 0.0461148, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {4678, 191} \[ \frac{a+b \cos ^{-1}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{b x}{2 c d^2 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 4678
Rule 191
Rubi steps
\begin{align*} \int \frac{x \left (a+b \cos ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac{a+b \cos ^{-1}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{b \int \frac{1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 c d^2}\\ &=\frac{b x}{2 c d^2 \sqrt{1-c^2 x^2}}+\frac{a+b \cos ^{-1}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.114275, size = 49, normalized size = 0.86 \[ \frac{a+b c x \sqrt{1-c^2 x^2}+b \cos ^{-1}(c x)}{2 c^2 d^2-2 c^4 d^2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 98, normalized size = 1.7 \begin{align*}{\frac{1}{{c}^{2}} \left ( -{\frac{a}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b}{{d}^{2}} \left ( -{\frac{\arccos \left ( cx \right ) }{2\,{c}^{2}{x}^{2}-2}}-{\frac{1}{4\,cx-4}\sqrt{- \left ( cx-1 \right ) ^{2}-2\,cx+2}}-{\frac{1}{4\,cx+4}\sqrt{- \left ( cx+1 \right ) ^{2}+2\,cx+2}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.47584, size = 225, normalized size = 3.95 \begin{align*} -\frac{1}{4} \,{\left (\frac{{\left (\frac{\sqrt{-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{6} d^{4} + \sqrt{c^{6} d^{4}} c^{4} d^{2} x} - \frac{\sqrt{-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{6} d^{4} - \sqrt{c^{6} d^{4}} c^{4} d^{2} x}\right )} c^{5} d^{2}}{\sqrt{c^{6} d^{4}}} + \frac{2 \, \arccos \left (c x\right )}{c^{4} d^{2} x^{2} - c^{2} d^{2}}\right )} b - \frac{a}{2 \,{\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21101, size = 115, normalized size = 2.02 \begin{align*} -\frac{a c^{2} x^{2} + \sqrt{-c^{2} x^{2} + 1} b c x + b \arccos \left (c x\right )}{2 \,{\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\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*} \frac{\int \frac{a x}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac{b x \operatorname{acos}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26603, size = 135, normalized size = 2.37 \begin{align*} -\frac{b x^{2} \arccos \left (c x\right )}{2 \,{\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac{a x^{2}}{2 \,{\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac{\sqrt{-c^{2} x^{2} + 1} b x}{2 \,{\left (c^{2} x^{2} - 1\right )} c d^{2}} + \frac{b \arccos \left (c x\right )}{2 \, c^{2} d^{2}} + \frac{a}{2 \, c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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