Optimal. Leaf size=39 \[ \frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )-\frac{1}{2} b c^2 \tanh ^{-1}\left (\frac{x}{c}\right )+\frac{b c x}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0202438, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6097, 193, 321, 207} \[ \frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )-\frac{1}{2} b c^2 \tanh ^{-1}\left (\frac{x}{c}\right )+\frac{b c x}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6097
Rule 193
Rule 321
Rule 207
Rubi steps
\begin{align*} \int x \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{2} (b c) \int \frac{1}{1-\frac{c^2}{x^2}} \, dx\\ &=\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{2} (b c) \int \frac{x^2}{-c^2+x^2} \, dx\\ &=\frac{b c x}{2}+\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{2} \left (b c^3\right ) \int \frac{1}{-c^2+x^2} \, dx\\ &=\frac{b c x}{2}+\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )-\frac{1}{2} b c^2 \tanh ^{-1}\left (\frac{x}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.007661, size = 56, normalized size = 1.44 \[ \frac{a x^2}{2}+\frac{1}{4} b c^2 \log (x-c)-\frac{1}{4} b c^2 \log (c+x)+\frac{1}{2} b x^2 \tanh ^{-1}\left (\frac{c}{x}\right )+\frac{b c x}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 53, normalized size = 1.4 \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{b{x}^{2}}{2}{\it Artanh} \left ({\frac{c}{x}} \right ) }+{\frac{xbc}{2}}+{\frac{{c}^{2}b}{4}\ln \left ({\frac{c}{x}}-1 \right ) }-{\frac{{c}^{2}b}{4}\ln \left ( 1+{\frac{c}{x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.969734, size = 59, normalized size = 1.51 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \operatorname{artanh}\left (\frac{c}{x}\right ) -{\left (c \log \left (c + x\right ) - c \log \left (-c + x\right ) - 2 \, x\right )} c\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.68662, size = 90, normalized size = 2.31 \begin{align*} \frac{1}{2} \, b c x + \frac{1}{2} \, a x^{2} - \frac{1}{4} \,{\left (b c^{2} - b x^{2}\right )} \log \left (-\frac{c + x}{c - x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.486917, size = 36, normalized size = 0.92 \begin{align*} \frac{a x^{2}}{2} - \frac{b c^{2} \operatorname{atanh}{\left (\frac{c}{x} \right )}}{2} + \frac{b c x}{2} + \frac{b x^{2} \operatorname{atanh}{\left (\frac{c}{x} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14094, size = 72, normalized size = 1.85 \begin{align*} -\frac{1}{4} \, b c^{2} \log \left (c + x\right ) + \frac{1}{4} \, b c^{2} \log \left (c - x\right ) + \frac{1}{4} \, b x^{2} \log \left (-\frac{c + x}{c - x}\right ) + \frac{1}{2} \, b c x + \frac{1}{2} \, a x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]