Optimal. Leaf size=48 \[ -\frac{a+b \tanh ^{-1}(c x)}{4 x^4}-\frac{b c^3}{4 x}+\frac{1}{4} b c^4 \tanh ^{-1}(c x)-\frac{b c}{12 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0266312, antiderivative size = 48, 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, 325, 206} \[ -\frac{a+b \tanh ^{-1}(c x)}{4 x^4}-\frac{b c^3}{4 x}+\frac{1}{4} b c^4 \tanh ^{-1}(c x)-\frac{b c}{12 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5916
Rule 325
Rule 206
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}(c x)}{x^5} \, dx &=-\frac{a+b \tanh ^{-1}(c x)}{4 x^4}+\frac{1}{4} (b c) \int \frac{1}{x^4 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{b c}{12 x^3}-\frac{a+b \tanh ^{-1}(c x)}{4 x^4}+\frac{1}{4} \left (b c^3\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{b c}{12 x^3}-\frac{b c^3}{4 x}-\frac{a+b \tanh ^{-1}(c x)}{4 x^4}+\frac{1}{4} \left (b c^5\right ) \int \frac{1}{1-c^2 x^2} \, dx\\ &=-\frac{b c}{12 x^3}-\frac{b c^3}{4 x}+\frac{1}{4} b c^4 \tanh ^{-1}(c x)-\frac{a+b \tanh ^{-1}(c x)}{4 x^4}\\ \end{align*}
Mathematica [A] time = 0.0082598, size = 70, normalized size = 1.46 \[ -\frac{a}{4 x^4}-\frac{b c^3}{4 x}-\frac{1}{8} b c^4 \log (1-c x)+\frac{1}{8} b c^4 \log (c x+1)-\frac{b c}{12 x^3}-\frac{b \tanh ^{-1}(c x)}{4 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 58, normalized size = 1.2 \begin{align*} -{\frac{a}{4\,{x}^{4}}}-{\frac{b{\it Artanh} \left ( cx \right ) }{4\,{x}^{4}}}-{\frac{{c}^{4}b\ln \left ( cx-1 \right ) }{8}}-{\frac{bc}{12\,{x}^{3}}}-{\frac{b{c}^{3}}{4\,x}}+{\frac{{c}^{4}b\ln \left ( cx+1 \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.987132, size = 81, normalized size = 1.69 \begin{align*} \frac{1}{24} \,{\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac{2 \,{\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac{6 \, \operatorname{artanh}\left (c x\right )}{x^{4}}\right )} b - \frac{a}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.92559, size = 117, normalized size = 2.44 \begin{align*} -\frac{6 \, b c^{3} x^{3} + 2 \, b c x - 3 \,{\left (b c^{4} x^{4} - b\right )} \log \left (-\frac{c x + 1}{c x - 1}\right ) + 6 \, a}{24 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.16585, size = 46, normalized size = 0.96 \begin{align*} - \frac{a}{4 x^{4}} + \frac{b c^{4} \operatorname{atanh}{\left (c x \right )}}{4} - \frac{b c^{3}}{4 x} - \frac{b c}{12 x^{3}} - \frac{b \operatorname{atanh}{\left (c x \right )}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21371, size = 92, normalized size = 1.92 \begin{align*} \frac{1}{8} \, b c^{4} \log \left (c x + 1\right ) - \frac{1}{8} \, b c^{4} \log \left (c x - 1\right ) - \frac{b \log \left (-\frac{c x + 1}{c x - 1}\right )}{8 \, x^{4}} - \frac{3 \, b c^{3} x^{3} + b c x + 3 \, a}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]