Optimal. Leaf size=122 \[ \frac {5}{64 a c^4 (1-a x)}-\frac {1}{64 a c^4 (a x+1)}+\frac {1}{16 a c^4 (1-a x)^2}+\frac {1}{16 a c^4 (1-a x)^3}+\frac {1}{16 a c^4 (1-a x)^4}+\frac {1}{20 a c^4 (1-a x)^5}+\frac {3 \tanh ^{-1}(a x)}{32 a c^4} \]
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Rubi [A] time = 0.09, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6140, 44, 207} \[ \frac {5}{64 a c^4 (1-a x)}-\frac {1}{64 a c^4 (a x+1)}+\frac {1}{16 a c^4 (1-a x)^2}+\frac {1}{16 a c^4 (1-a x)^3}+\frac {1}{16 a c^4 (1-a x)^4}+\frac {1}{20 a c^4 (1-a x)^5}+\frac {3 \tanh ^{-1}(a x)}{32 a c^4} \]
Antiderivative was successfully verified.
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Rule 44
Rule 207
Rule 6140
Rubi steps
\begin {align*} \int \frac {e^{4 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=\frac {\int \frac {1}{(1-a x)^6 (1+a x)^2} \, dx}{c^4}\\ &=\frac {\int \left (\frac {1}{4 (-1+a x)^6}-\frac {1}{4 (-1+a x)^5}+\frac {3}{16 (-1+a x)^4}-\frac {1}{8 (-1+a x)^3}+\frac {5}{64 (-1+a x)^2}+\frac {1}{64 (1+a x)^2}-\frac {3}{32 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^4}\\ &=\frac {1}{20 a c^4 (1-a x)^5}+\frac {1}{16 a c^4 (1-a x)^4}+\frac {1}{16 a c^4 (1-a x)^3}+\frac {1}{16 a c^4 (1-a x)^2}+\frac {5}{64 a c^4 (1-a x)}-\frac {1}{64 a c^4 (1+a x)}-\frac {3 \int \frac {1}{-1+a^2 x^2} \, dx}{32 c^4}\\ &=\frac {1}{20 a c^4 (1-a x)^5}+\frac {1}{16 a c^4 (1-a x)^4}+\frac {1}{16 a c^4 (1-a x)^3}+\frac {1}{16 a c^4 (1-a x)^2}+\frac {5}{64 a c^4 (1-a x)}-\frac {1}{64 a c^4 (1+a x)}+\frac {3 \tanh ^{-1}(a x)}{32 a c^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 80, normalized size = 0.66 \[ \frac {-15 a^5 x^5+60 a^4 x^4-80 a^3 x^3+20 a^2 x^2+47 a x+15 (a x-1)^5 (a x+1) \tanh ^{-1}(a x)-48}{160 a c^4 (a x-1)^5 (a x+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 191, normalized size = 1.57 \[ -\frac {30 \, a^{5} x^{5} - 120 \, a^{4} x^{4} + 160 \, a^{3} x^{3} - 40 \, a^{2} x^{2} - 94 \, a x - 15 \, {\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (a x + 1\right ) + 15 \, {\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (a x - 1\right ) + 96}{320 \, {\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 91, normalized size = 0.75 \[ \frac {3 \, \log \left ({\left | a x + 1 \right |}\right )}{64 \, a c^{4}} - \frac {3 \, \log \left ({\left | a x - 1 \right |}\right )}{64 \, a c^{4}} - \frac {15 \, a^{5} x^{5} - 60 \, a^{4} x^{4} + 80 \, a^{3} x^{3} - 20 \, a^{2} x^{2} - 47 \, a x + 48}{160 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{5} a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 120, normalized size = 0.98 \[ -\frac {1}{20 c^{4} a \left (a x -1\right )^{5}}+\frac {1}{16 c^{4} a \left (a x -1\right )^{4}}-\frac {1}{16 c^{4} a \left (a x -1\right )^{3}}+\frac {1}{16 c^{4} a \left (a x -1\right )^{2}}-\frac {5}{64 c^{4} a \left (a x -1\right )}-\frac {3 \ln \left (a x -1\right )}{64 c^{4} a}-\frac {1}{64 a \,c^{4} \left (a x +1\right )}+\frac {3 \ln \left (a x +1\right )}{64 a \,c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 130, normalized size = 1.07 \[ -\frac {15 \, a^{5} x^{5} - 60 \, a^{4} x^{4} + 80 \, a^{3} x^{3} - 20 \, a^{2} x^{2} - 47 \, a x + 48}{160 \, {\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} + \frac {3 \, \log \left (a x + 1\right )}{64 \, a c^{4}} - \frac {3 \, \log \left (a x - 1\right )}{64 \, a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 111, normalized size = 0.91 \[ \frac {3\,\mathrm {atanh}\left (a\,x\right )}{32\,a\,c^4}-\frac {\frac {47\,x}{160}+\frac {a\,x^2}{8}-\frac {3}{10\,a}-\frac {a^2\,x^3}{2}+\frac {3\,a^3\,x^4}{8}-\frac {3\,a^4\,x^5}{32}}{-a^6\,c^4\,x^6+4\,a^5\,c^4\,x^5-5\,a^4\,c^4\,x^4+5\,a^2\,c^4\,x^2-4\,a\,c^4\,x+c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 129, normalized size = 1.06 \[ \frac {- 15 a^{5} x^{5} + 60 a^{4} x^{4} - 80 a^{3} x^{3} + 20 a^{2} x^{2} + 47 a x - 48}{160 a^{7} c^{4} x^{6} - 640 a^{6} c^{4} x^{5} + 800 a^{5} c^{4} x^{4} - 800 a^{3} c^{4} x^{2} + 640 a^{2} c^{4} x - 160 a c^{4}} + \frac {- \frac {3 \log {\left (x - \frac {1}{a} \right )}}{64} + \frac {3 \log {\left (x + \frac {1}{a} \right )}}{64}}{a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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