Optimal. Leaf size=54 \[ -\frac{a+b \tanh ^{-1}(c x)}{3 x^3}-\frac{1}{6} b c^3 \log \left (1-c^2 x^2\right )+\frac{1}{3} b c^3 \log (x)-\frac{b c}{6 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0360336, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5916, 266, 44} \[ -\frac{a+b \tanh ^{-1}(c x)}{3 x^3}-\frac{1}{6} b c^3 \log \left (1-c^2 x^2\right )+\frac{1}{3} b c^3 \log (x)-\frac{b c}{6 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5916
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}(c x)}{x^4} \, dx &=-\frac{a+b \tanh ^{-1}(c x)}{3 x^3}+\frac{1}{3} (b c) \int \frac{1}{x^3 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{a+b \tanh ^{-1}(c x)}{3 x^3}+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{a+b \tanh ^{-1}(c x)}{3 x^3}+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}+\frac{c^2}{x}-\frac{c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac{b c}{6 x^2}-\frac{a+b \tanh ^{-1}(c x)}{3 x^3}+\frac{1}{3} b c^3 \log (x)-\frac{1}{6} b c^3 \log \left (1-c^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.008483, size = 59, normalized size = 1.09 \[ -\frac{a}{3 x^3}-\frac{1}{6} b c^3 \log \left (1-c^2 x^2\right )+\frac{1}{3} b c^3 \log (x)-\frac{b c}{6 x^2}-\frac{b \tanh ^{-1}(c x)}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 59, normalized size = 1.1 \begin{align*} -{\frac{a}{3\,{x}^{3}}}-{\frac{b{\it Artanh} \left ( cx \right ) }{3\,{x}^{3}}}-{\frac{{c}^{3}b\ln \left ( cx-1 \right ) }{6}}-{\frac{bc}{6\,{x}^{2}}}+{\frac{{c}^{3}b\ln \left ( cx \right ) }{3}}-{\frac{{c}^{3}b\ln \left ( cx+1 \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.967296, size = 66, normalized size = 1.22 \begin{align*} -\frac{1}{6} \,{\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac{1}{x^{2}}\right )} c + \frac{2 \, \operatorname{artanh}\left (c x\right )}{x^{3}}\right )} b - \frac{a}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.92786, size = 140, normalized size = 2.59 \begin{align*} -\frac{b c^{3} x^{3} \log \left (c^{2} x^{2} - 1\right ) - 2 \, b c^{3} x^{3} \log \left (x\right ) + b c x + b \log \left (-\frac{c x + 1}{c x - 1}\right ) + 2 \, a}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.5614, size = 70, normalized size = 1.3 \begin{align*} \begin{cases} - \frac{a}{3 x^{3}} + \frac{b c^{3} \log{\left (x \right )}}{3} - \frac{b c^{3} \log{\left (x - \frac{1}{c} \right )}}{3} - \frac{b c^{3} \operatorname{atanh}{\left (c x \right )}}{3} - \frac{b c}{6 x^{2}} - \frac{b \operatorname{atanh}{\left (c x \right )}}{3 x^{3}} & \text{for}\: c \neq 0 \\- \frac{a}{3 x^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20603, size = 80, normalized size = 1.48 \begin{align*} -\frac{1}{6} \, b c^{3} \log \left (c^{2} x^{2} - 1\right ) + \frac{1}{3} \, b c^{3} \log \left (x\right ) - \frac{b \log \left (-\frac{c x + 1}{c x - 1}\right )}{6 \, x^{3}} - \frac{b c x + 2 \, a}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]