Optimal. Leaf size=45 \[ \frac{d+e x}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac{e \tanh ^{-1}\left (\frac{c x}{a}\right )}{2 a c^3} \]
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Rubi [A] time = 0.0196568, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {778, 208} \[ \frac{d+e x}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac{e \tanh ^{-1}\left (\frac{c x}{a}\right )}{2 a c^3} \]
Antiderivative was successfully verified.
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Rule 778
Rule 208
Rubi steps
\begin{align*} \int \frac{x (d+e x)}{\left (a^2-c^2 x^2\right )^2} \, dx &=\frac{d+e x}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac{e \int \frac{1}{a^2-c^2 x^2} \, dx}{2 c^2}\\ &=\frac{d+e x}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac{e \tanh ^{-1}\left (\frac{c x}{a}\right )}{2 a c^3}\\ \end{align*}
Mathematica [A] time = 0.0222976, size = 42, normalized size = 0.93 \[ \frac{\frac{c (d+e x)}{a^2-c^2 x^2}-\frac{e \tanh ^{-1}\left (\frac{c x}{a}\right )}{a}}{2 c^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 96, normalized size = 2.1 \begin{align*} -{\frac{e\ln \left ( cx+a \right ) }{4\,a{c}^{3}}}-{\frac{e}{4\,{c}^{3} \left ( cx+a \right ) }}+{\frac{d}{4\,a{c}^{2} \left ( cx+a \right ) }}+{\frac{e\ln \left ( cx-a \right ) }{4\,a{c}^{3}}}-{\frac{e}{4\,{c}^{3} \left ( cx-a \right ) }}-{\frac{d}{4\,a{c}^{2} \left ( cx-a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07953, size = 78, normalized size = 1.73 \begin{align*} -\frac{e x + d}{2 \,{\left (c^{4} x^{2} - a^{2} c^{2}\right )}} - \frac{e \log \left (c x + a\right )}{4 \, a c^{3}} + \frac{e \log \left (c x - a\right )}{4 \, a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54247, size = 162, normalized size = 3.6 \begin{align*} -\frac{2 \, a c e x + 2 \, a c d +{\left (c^{2} e x^{2} - a^{2} e\right )} \log \left (c x + a\right ) -{\left (c^{2} e x^{2} - a^{2} e\right )} \log \left (c x - a\right )}{4 \,{\left (a c^{5} x^{2} - a^{3} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.696135, size = 44, normalized size = 0.98 \begin{align*} - \frac{d + e x}{- 2 a^{2} c^{2} + 2 c^{4} x^{2}} + \frac{e \left (\frac{\log{\left (- \frac{a}{c} + x \right )}}{4} - \frac{\log{\left (\frac{a}{c} + x \right )}}{4}\right )}{a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12998, size = 85, normalized size = 1.89 \begin{align*} -\frac{x e + d}{2 \,{\left (c^{2} x^{2} - a^{2}\right )} c^{2}} - \frac{e \log \left ({\left | c x + a \right |}\right )}{4 \, a c^{3}} + \frac{e \log \left ({\left | c x - a \right |}\right )}{4 \, a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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