Optimal. Leaf size=57 \[ \frac{\left (3 a^2 c+d\right ) \log \left (1-a^2 x^2\right )}{6 a^3}+c x \tanh ^{-1}(a x)+\frac{d x^2}{6 a}+\frac{1}{3} d x^3 \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.066788, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5976, 1593, 444, 43} \[ \frac{\left (3 a^2 c+d\right ) \log \left (1-a^2 x^2\right )}{6 a^3}+c x \tanh ^{-1}(a x)+\frac{d x^2}{6 a}+\frac{1}{3} d x^3 \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 5976
Rule 1593
Rule 444
Rule 43
Rubi steps
\begin{align*} \int \left (c+d x^2\right ) \tanh ^{-1}(a x) \, dx &=c x \tanh ^{-1}(a x)+\frac{1}{3} d x^3 \tanh ^{-1}(a x)-a \int \frac{c x+\frac{d x^3}{3}}{1-a^2 x^2} \, dx\\ &=c x \tanh ^{-1}(a x)+\frac{1}{3} d x^3 \tanh ^{-1}(a x)-a \int \frac{x \left (c+\frac{d x^2}{3}\right )}{1-a^2 x^2} \, dx\\ &=c x \tanh ^{-1}(a x)+\frac{1}{3} d x^3 \tanh ^{-1}(a x)-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{c+\frac{d x}{3}}{1-a^2 x} \, dx,x,x^2\right )\\ &=c x \tanh ^{-1}(a x)+\frac{1}{3} d x^3 \tanh ^{-1}(a x)-\frac{1}{2} a \operatorname{Subst}\left (\int \left (-\frac{d}{3 a^2}+\frac{-3 a^2 c-d}{3 a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{d x^2}{6 a}+c x \tanh ^{-1}(a x)+\frac{1}{3} d x^3 \tanh ^{-1}(a x)+\frac{\left (3 a^2 c+d\right ) \log \left (1-a^2 x^2\right )}{6 a^3}\\ \end{align*}
Mathematica [A] time = 0.0113418, size = 69, normalized size = 1.21 \[ \frac{c \log \left (1-a^2 x^2\right )}{2 a}+\frac{d \log \left (1-a^2 x^2\right )}{6 a^3}+c x \tanh ^{-1}(a x)+\frac{d x^2}{6 a}+\frac{1}{3} d x^3 \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 76, normalized size = 1.3 \begin{align*}{\frac{d{x}^{3}{\it Artanh} \left ( ax \right ) }{3}}+cx{\it Artanh} \left ( ax \right ) +{\frac{d{x}^{2}}{6\,a}}+{\frac{\ln \left ( ax-1 \right ) c}{2\,a}}+{\frac{\ln \left ( ax-1 \right ) d}{6\,{a}^{3}}}+{\frac{\ln \left ( ax+1 \right ) c}{2\,a}}+{\frac{\ln \left ( ax+1 \right ) d}{6\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953481, size = 88, normalized size = 1.54 \begin{align*} \frac{1}{6} \, a{\left (\frac{d x^{2}}{a^{2}} + \frac{{\left (3 \, a^{2} c + d\right )} \log \left (a x + 1\right )}{a^{4}} + \frac{{\left (3 \, a^{2} c + d\right )} \log \left (a x - 1\right )}{a^{4}}\right )} + \frac{1}{3} \,{\left (d x^{3} + 3 \, c x\right )} \operatorname{artanh}\left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96813, size = 143, normalized size = 2.51 \begin{align*} \frac{a^{2} d x^{2} +{\left (3 \, a^{2} c + d\right )} \log \left (a^{2} x^{2} - 1\right ) +{\left (a^{3} d x^{3} + 3 \, a^{3} c x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{6 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.51762, size = 73, normalized size = 1.28 \begin{align*} \begin{cases} c x \operatorname{atanh}{\left (a x \right )} + \frac{d x^{3} \operatorname{atanh}{\left (a x \right )}}{3} + \frac{c \log{\left (x - \frac{1}{a} \right )}}{a} + \frac{c \operatorname{atanh}{\left (a x \right )}}{a} + \frac{d x^{2}}{6 a} + \frac{d \log{\left (x - \frac{1}{a} \right )}}{3 a^{3}} + \frac{d \operatorname{atanh}{\left (a x \right )}}{3 a^{3}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20345, size = 82, normalized size = 1.44 \begin{align*} \frac{d x^{2}}{6 \, a} + \frac{1}{6} \,{\left (d x^{3} + 3 \, c x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) + \frac{{\left (3 \, a^{2} c + d\right )} \log \left ({\left | a^{2} x^{2} - 1 \right |}\right )}{6 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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