Optimal. Leaf size=57 \[ \frac{1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{b x^2}{10 c^3}+\frac{b \log \left (1-c^2 x^2\right )}{10 c^5}+\frac{b x^4}{20 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0426034, antiderivative size = 57, 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, 43} \[ \frac{1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{b x^2}{10 c^3}+\frac{b \log \left (1-c^2 x^2\right )}{10 c^5}+\frac{b x^4}{20 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5916
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^4 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{5} (b c) \int \frac{x^5}{1-c^2 x^2} \, dx\\ &=\frac{1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{10} (b c) \operatorname{Subst}\left (\int \frac{x^2}{1-c^2 x} \, dx,x,x^2\right )\\ &=\frac{1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{10} (b c) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}-\frac{x}{c^2}-\frac{1}{c^4 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{b x^2}{10 c^3}+\frac{b x^4}{20 c}+\frac{1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{b \log \left (1-c^2 x^2\right )}{10 c^5}\\ \end{align*}
Mathematica [A] time = 0.0085123, size = 62, normalized size = 1.09 \[ \frac{a x^5}{5}+\frac{b x^2}{10 c^3}+\frac{b \log \left (1-c^2 x^2\right )}{10 c^5}+\frac{b x^4}{20 c}+\frac{1}{5} b x^5 \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 60, normalized size = 1.1 \begin{align*}{\frac{a{x}^{5}}{5}}+{\frac{b{x}^{5}{\it Artanh} \left ( cx \right ) }{5}}+{\frac{b{x}^{4}}{20\,c}}+{\frac{b{x}^{2}}{10\,{c}^{3}}}+{\frac{b\ln \left ( cx-1 \right ) }{10\,{c}^{5}}}+{\frac{b\ln \left ( cx+1 \right ) }{10\,{c}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.966233, size = 74, normalized size = 1.3 \begin{align*} \frac{1}{5} \, a x^{5} + \frac{1}{20} \,{\left (4 \, x^{5} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.05552, size = 153, normalized size = 2.68 \begin{align*} \frac{2 \, b c^{5} x^{5} \log \left (-\frac{c x + 1}{c x - 1}\right ) + 4 \, a c^{5} x^{5} + b c^{4} x^{4} + 2 \, b c^{2} x^{2} + 2 \, b \log \left (c^{2} x^{2} - 1\right )}{20 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.61565, size = 68, normalized size = 1.19 \begin{align*} \begin{cases} \frac{a x^{5}}{5} + \frac{b x^{5} \operatorname{atanh}{\left (c x \right )}}{5} + \frac{b x^{4}}{20 c} + \frac{b x^{2}}{10 c^{3}} + \frac{b \log{\left (x - \frac{1}{c} \right )}}{5 c^{5}} + \frac{b \operatorname{atanh}{\left (c x \right )}}{5 c^{5}} & \text{for}\: c \neq 0 \\\frac{a x^{5}}{5} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15635, size = 84, normalized size = 1.47 \begin{align*} \frac{1}{10} \, b x^{5} \log \left (-\frac{c x + 1}{c x - 1}\right ) + \frac{1}{5} \, a x^{5} + \frac{b x^{4}}{20 \, c} + \frac{b x^{2}}{10 \, c^{3}} + \frac{b \log \left (c^{2} x^{2} - 1\right )}{10 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]