Optimal. Leaf size=146 \[ \frac{501}{64 a c^4 (1-a x)}-\frac{1}{64 a c^4 (a x+1)}-\frac{67}{16 a c^4 (1-a x)^2}+\frac{83}{48 a c^4 (1-a x)^3}-\frac{7}{16 a c^4 (1-a x)^4}+\frac{1}{20 a c^4 (1-a x)^5}+\frac{261 \log (1-a x)}{64 a c^4}-\frac{5 \log (a x+1)}{64 a c^4}+\frac{x}{c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.221346, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 6157, 6150, 88} \[ \frac{501}{64 a c^4 (1-a x)}-\frac{1}{64 a c^4 (a x+1)}-\frac{67}{16 a c^4 (1-a x)^2}+\frac{83}{48 a c^4 (1-a x)^3}-\frac{7}{16 a c^4 (1-a x)^4}+\frac{1}{20 a c^4 (1-a x)^5}+\frac{261 \log (1-a x)}{64 a c^4}-\frac{5 \log (a x+1)}{64 a c^4}+\frac{x}{c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6157
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^4} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^4} \, dx\\ &=\frac{a^8 \int \frac{e^{4 \tanh ^{-1}(a x)} x^8}{\left (1-a^2 x^2\right )^4} \, dx}{c^4}\\ &=\frac{a^8 \int \frac{x^8}{(1-a x)^6 (1+a x)^2} \, dx}{c^4}\\ &=\frac{a^8 \int \left (\frac{1}{a^8}+\frac{1}{4 a^8 (-1+a x)^6}+\frac{7}{4 a^8 (-1+a x)^5}+\frac{83}{16 a^8 (-1+a x)^4}+\frac{67}{8 a^8 (-1+a x)^3}+\frac{501}{64 a^8 (-1+a x)^2}+\frac{261}{64 a^8 (-1+a x)}+\frac{1}{64 a^8 (1+a x)^2}-\frac{5}{64 a^8 (1+a x)}\right ) \, dx}{c^4}\\ &=\frac{x}{c^4}+\frac{1}{20 a c^4 (1-a x)^5}-\frac{7}{16 a c^4 (1-a x)^4}+\frac{83}{48 a c^4 (1-a x)^3}-\frac{67}{16 a c^4 (1-a x)^2}+\frac{501}{64 a c^4 (1-a x)}-\frac{1}{64 a c^4 (1+a x)}+\frac{261 \log (1-a x)}{64 a c^4}-\frac{5 \log (1+a x)}{64 a c^4}\\ \end{align*}
Mathematica [A] time = 0.0958636, size = 98, normalized size = 0.67 \[ \frac{\frac{2 \left (480 a^7 x^7-1920 a^6 x^6-1365 a^5 x^5+9300 a^4 x^4-6800 a^3 x^3-4900 a^2 x^2+7541 a x-2384\right )}{(a x-1)^5 (a x+1)}+3915 \log (1-a x)-75 \log (a x+1)}{960 a c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.06, size = 125, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{4}}}-{\frac{1}{64\,a{c}^{4} \left ( ax+1 \right ) }}-{\frac{5\,\ln \left ( ax+1 \right ) }{64\,a{c}^{4}}}-{\frac{1}{20\,a{c}^{4} \left ( ax-1 \right ) ^{5}}}-{\frac{7}{16\,a{c}^{4} \left ( ax-1 \right ) ^{4}}}-{\frac{83}{48\,a{c}^{4} \left ( ax-1 \right ) ^{3}}}-{\frac{67}{16\,a{c}^{4} \left ( ax-1 \right ) ^{2}}}-{\frac{501}{64\,a{c}^{4} \left ( ax-1 \right ) }}+{\frac{261\,\ln \left ( ax-1 \right ) }{64\,a{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.06622, size = 182, normalized size = 1.25 \begin{align*} -\frac{3765 \, a^{5} x^{5} - 9300 \, a^{4} x^{4} + 4400 \, a^{3} x^{3} + 6820 \, a^{2} x^{2} - 8021 \, a x + 2384}{480 \,{\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} + \frac{x}{c^{4}} - \frac{5 \, \log \left (a x + 1\right )}{64 \, a c^{4}} + \frac{261 \, \log \left (a x - 1\right )}{64 \, a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.28138, size = 479, normalized size = 3.28 \begin{align*} \frac{960 \, a^{7} x^{7} - 3840 \, a^{6} x^{6} - 2730 \, a^{5} x^{5} + 18600 \, a^{4} x^{4} - 13600 \, a^{3} x^{3} - 9800 \, a^{2} x^{2} + 15082 \, a x - 75 \,{\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (a x + 1\right ) + 3915 \,{\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (a x - 1\right ) - 4768}{960 \,{\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.37593, size = 144, normalized size = 0.99 \begin{align*} a^{8} \left (- \frac{3765 a^{5} x^{5} - 9300 a^{4} x^{4} + 4400 a^{3} x^{3} + 6820 a^{2} x^{2} - 8021 a x + 2384}{480 a^{15} c^{4} x^{6} - 1920 a^{14} c^{4} x^{5} + 2400 a^{13} c^{4} x^{4} - 2400 a^{11} c^{4} x^{2} + 1920 a^{10} c^{4} x - 480 a^{9} c^{4}} + \frac{x}{a^{8} c^{4}} + \frac{\frac{261 \log{\left (x - \frac{1}{a} \right )}}{64} - \frac{5 \log{\left (x + \frac{1}{a} \right )}}{64}}{a^{9} c^{4}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.109, size = 230, normalized size = 1.58 \begin{align*} \frac{{\left (a x - 1\right )}{\left (\frac{257}{a x - 1} + 128\right )}}{128 \, a c^{4}{\left (\frac{2}{a x - 1} + 1\right )}} - \frac{4 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a c^{4}} - \frac{5 \, \log \left ({\left | -\frac{2}{a x - 1} - 1 \right |}\right )}{64 \, a c^{4}} - \frac{\frac{7515 \, a^{19} c^{16}}{a x - 1} + \frac{4020 \, a^{19} c^{16}}{{\left (a x - 1\right )}^{2}} + \frac{1660 \, a^{19} c^{16}}{{\left (a x - 1\right )}^{3}} + \frac{420 \, a^{19} c^{16}}{{\left (a x - 1\right )}^{4}} + \frac{48 \, a^{19} c^{16}}{{\left (a x - 1\right )}^{5}}}{960 \, a^{20} c^{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]