Optimal. Leaf size=145 \[ \frac {x}{c^4}-\frac {1}{32 a c^4 (1-a x)^4}+\frac {13}{48 a c^4 (1-a x)^3}-\frac {35}{32 a c^4 (1-a x)^2}+\frac {99}{32 a c^4 (1-a x)}+\frac {1}{64 a c^4 (1+a x)^2}-\frac {11}{64 a c^4 (1+a x)}+\frac {303 \log (1-a x)}{128 a c^4}-\frac {47 \log (1+a x)}{128 a c^4} \]
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Rubi [A]
time = 0.15, antiderivative size = 145, normalized size of antiderivative = 1.00, 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 = {6302, 6292,
6285, 90} \begin {gather*} \frac {99}{32 a c^4 (1-a x)}-\frac {11}{64 a c^4 (a x+1)}-\frac {35}{32 a c^4 (1-a x)^2}+\frac {1}{64 a c^4 (a x+1)^2}+\frac {13}{48 a c^4 (1-a x)^3}-\frac {1}{32 a c^4 (1-a x)^4}+\frac {303 \log (1-a x)}{128 a c^4}-\frac {47 \log (a x+1)}{128 a c^4}+\frac {x}{c^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6285
Rule 6292
Rule 6302
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx\\ &=-\frac {a^8 \int \frac {e^{2 \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)^5 (1+a x)^3} \, dx}{c^4}\\ &=-\frac {a^8 \int \left (-\frac {1}{a^8}-\frac {1}{8 a^8 (-1+a x)^5}-\frac {13}{16 a^8 (-1+a x)^4}-\frac {35}{16 a^8 (-1+a x)^3}-\frac {99}{32 a^8 (-1+a x)^2}-\frac {303}{128 a^8 (-1+a x)}+\frac {1}{32 a^8 (1+a x)^3}-\frac {11}{64 a^8 (1+a x)^2}+\frac {47}{128 a^8 (1+a x)}\right ) \, dx}{c^4}\\ &=\frac {x}{c^4}-\frac {1}{32 a c^4 (1-a x)^4}+\frac {13}{48 a c^4 (1-a x)^3}-\frac {35}{32 a c^4 (1-a x)^2}+\frac {99}{32 a c^4 (1-a x)}+\frac {1}{64 a c^4 (1+a x)^2}-\frac {11}{64 a c^4 (1+a x)}+\frac {303 \log (1-a x)}{128 a c^4}-\frac {47 \log (1+a x)}{128 a c^4}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 98, normalized size = 0.68 \begin {gather*} \frac {\frac {2 \left (400-275 a x-1258 a^2 x^2+866 a^3 x^3+1254 a^4 x^4-819 a^5 x^5-384 a^6 x^6+192 a^7 x^7\right )}{(-1+a x)^4 (1+a x)^2}+909 \log (1-a x)-141 \log (1+a x)}{384 a c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 108, normalized size = 0.74
method | result | size |
default | \(\frac {a^{8} \left (\frac {x}{a^{8}}+\frac {1}{64 a^{9} \left (a x +1\right )^{2}}-\frac {11}{64 a^{9} \left (a x +1\right )}-\frac {47 \ln \left (a x +1\right )}{128 a^{9}}-\frac {1}{32 a^{9} \left (a x -1\right )^{4}}-\frac {13}{48 a^{9} \left (a x -1\right )^{3}}-\frac {35}{32 a^{9} \left (a x -1\right )^{2}}-\frac {99}{32 a^{9} \left (a x -1\right )}+\frac {303 \ln \left (a x -1\right )}{128 a^{9}}\right )}{c^{4}}\) | \(108\) |
risch | \(\frac {x}{c^{4}}+\frac {-\frac {209 a^{4} c^{4} x^{5}}{64}+\frac {81 a^{3} c^{4} x^{4}}{32}+\frac {529 a^{2} c^{4} x^{3}}{96}-\frac {437 c^{4} a \,x^{2}}{96}-\frac {467 c^{4} x}{192}+\frac {25 c^{4}}{12 a}}{c^{8} \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right )^{2}}+\frac {303 \ln \left (-a x +1\right )}{128 a \,c^{4}}-\frac {47 \ln \left (a x +1\right )}{128 a \,c^{4}}\) | \(115\) |
norman | \(\frac {\frac {a^{7} x^{8}}{c}+\frac {175 x}{64 c}-\frac {111 a \,x^{2}}{64 c}-\frac {199 a^{2} x^{3}}{24 c}+\frac {115 a^{3} x^{4}}{24 c}+\frac {545 a^{4} x^{5}}{64 c}-\frac {803 a^{5} x^{6}}{192 c}-\frac {37 a^{6} x^{7}}{12 c}}{\left (a x -1\right )^{4} \left (a x +1\right )^{3} c^{3}}+\frac {303 \ln \left (a x -1\right )}{128 c^{4} a}-\frac {47 \ln \left (a x +1\right )}{128 a \,c^{4}}\) | \(129\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 145, normalized size = 1.00 \begin {gather*} -\frac {627 \, a^{5} x^{5} - 486 \, a^{4} x^{4} - 1058 \, a^{3} x^{3} + 874 \, a^{2} x^{2} + 467 \, a x - 400}{192 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac {x}{c^{4}} - \frac {47 \, \log \left (a x + 1\right )}{128 \, a c^{4}} + \frac {303 \, \log \left (a x - 1\right )}{128 \, a c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 233, normalized size = 1.61 \begin {gather*} \frac {384 \, a^{7} x^{7} - 768 \, a^{6} x^{6} - 1638 \, a^{5} x^{5} + 2508 \, a^{4} x^{4} + 1732 \, a^{3} x^{3} - 2516 \, a^{2} x^{2} - 550 \, a x - 141 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 909 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 800}{384 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.73, size = 156, normalized size = 1.08 \begin {gather*} a^{8} \left (\frac {- 627 a^{5} x^{5} + 486 a^{4} x^{4} + 1058 a^{3} x^{3} - 874 a^{2} x^{2} - 467 a x + 400}{192 a^{15} c^{4} x^{6} - 384 a^{14} c^{4} x^{5} - 192 a^{13} c^{4} x^{4} + 768 a^{12} c^{4} x^{3} - 192 a^{11} c^{4} x^{2} - 384 a^{10} c^{4} x + 192 a^{9} c^{4}} + \frac {x}{a^{8} c^{4}} + \frac {\frac {303 \log {\left (x - \frac {1}{a} \right )}}{128} - \frac {47 \log {\left (x + \frac {1}{a} \right )}}{128}}{a^{9} c^{4}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 96, normalized size = 0.66 \begin {gather*} \frac {x}{c^{4}} - \frac {47 \, \log \left ({\left | a x + 1 \right |}\right )}{128 \, a c^{4}} + \frac {303 \, \log \left ({\left | a x - 1 \right |}\right )}{128 \, a c^{4}} - \frac {627 \, a^{5} x^{5} - 486 \, a^{4} x^{4} - 1058 \, a^{3} x^{3} + 874 \, a^{2} x^{2} + 467 \, a x - 400}{192 \, {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{4} a c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.45, size = 142, normalized size = 0.98 \begin {gather*} \frac {x}{c^4}+\frac {\frac {467\,x}{192}+\frac {437\,a\,x^2}{96}-\frac {25}{12\,a}-\frac {529\,a^2\,x^3}{96}-\frac {81\,a^3\,x^4}{32}+\frac {209\,a^4\,x^5}{64}}{-a^6\,c^4\,x^6+2\,a^5\,c^4\,x^5+a^4\,c^4\,x^4-4\,a^3\,c^4\,x^3+a^2\,c^4\,x^2+2\,a\,c^4\,x-c^4}+\frac {303\,\ln \left (a\,x-1\right )}{128\,a\,c^4}-\frac {47\,\ln \left (a\,x+1\right )}{128\,a\,c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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