Optimal. Leaf size=119 \[ \frac {5}{64 a c^4 (1-a x)}-\frac {5}{32 a c^4 (a x+1)}+\frac {1}{64 a c^4 (1-a x)^2}-\frac {3}{32 a c^4 (a x+1)^2}-\frac {1}{16 a c^4 (a x+1)^3}-\frac {1}{32 a c^4 (a x+1)^4}+\frac {15 \tanh ^{-1}(a x)}{64 a c^4} \]
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Rubi [A] time = 0.09, antiderivative size = 119, 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} \[ \frac {5}{64 a c^4 (1-a x)}-\frac {5}{32 a c^4 (a x+1)}+\frac {1}{64 a c^4 (1-a x)^2}-\frac {3}{32 a c^4 (a x+1)^2}-\frac {1}{16 a c^4 (a x+1)^3}-\frac {1}{32 a c^4 (a x+1)^4}+\frac {15 \tanh ^{-1}(a x)}{64 a c^4} \]
Antiderivative was successfully verified.
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Rule 44
Rule 207
Rule 6140
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=\frac {\int \frac {1}{(1-a x)^3 (1+a x)^5} \, dx}{c^4}\\ &=\frac {\int \left (-\frac {1}{32 (-1+a x)^3}+\frac {5}{64 (-1+a x)^2}+\frac {1}{8 (1+a x)^5}+\frac {3}{16 (1+a x)^4}+\frac {3}{16 (1+a x)^3}+\frac {5}{32 (1+a x)^2}-\frac {15}{64 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^4}\\ &=\frac {1}{64 a c^4 (1-a x)^2}+\frac {5}{64 a c^4 (1-a x)}-\frac {1}{32 a c^4 (1+a x)^4}-\frac {1}{16 a c^4 (1+a x)^3}-\frac {3}{32 a c^4 (1+a x)^2}-\frac {5}{32 a c^4 (1+a x)}-\frac {15 \int \frac {1}{-1+a^2 x^2} \, dx}{64 c^4}\\ &=\frac {1}{64 a c^4 (1-a x)^2}+\frac {5}{64 a c^4 (1-a x)}-\frac {1}{32 a c^4 (1+a x)^4}-\frac {1}{16 a c^4 (1+a x)^3}-\frac {3}{32 a c^4 (1+a x)^2}-\frac {5}{32 a c^4 (1+a x)}+\frac {15 \tanh ^{-1}(a x)}{64 a c^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 80, normalized size = 0.67 \[ \frac {-15 a^5 x^5-30 a^4 x^4+10 a^3 x^3+50 a^2 x^2+17 a x+15 (a x-1)^2 (a x+1)^4 \tanh ^{-1}(a x)-16}{64 a (a x-1)^2 (a c x+c)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 217, normalized size = 1.82 \[ -\frac {30 \, a^{5} x^{5} + 60 \, a^{4} x^{4} - 20 \, a^{3} x^{3} - 100 \, a^{2} x^{2} - 34 \, a x - 15 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 15 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 32}{128 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 122, normalized size = 1.03 \[ -\frac {15 \, \log \left ({\left | -\frac {2}{a x + 1} + 1 \right |}\right )}{128 \, a c^{4}} + \frac {\frac {24}{a x + 1} - 11}{256 \, a c^{4} {\left (\frac {2}{a x + 1} - 1\right )}^{2}} - \frac {\frac {5 \, a^{11} c^{12}}{a x + 1} + \frac {3 \, a^{11} c^{12}}{{\left (a x + 1\right )}^{2}} + \frac {2 \, a^{11} c^{12}}{{\left (a x + 1\right )}^{3}} + \frac {a^{11} c^{12}}{{\left (a x + 1\right )}^{4}}}{32 \, a^{12} c^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 120, normalized size = 1.01 \[ \frac {1}{64 c^{4} a \left (a x -1\right )^{2}}-\frac {5}{64 c^{4} a \left (a x -1\right )}-\frac {15 \ln \left (a x -1\right )}{128 c^{4} a}-\frac {1}{32 a \,c^{4} \left (a x +1\right )^{4}}-\frac {1}{16 a \,c^{4} \left (a x +1\right )^{3}}-\frac {3}{32 a \,c^{4} \left (a x +1\right )^{2}}-\frac {5}{32 a \,c^{4} \left (a x +1\right )}+\frac {15 \ln \left (a x +1\right )}{128 a \,c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 140, normalized size = 1.18 \[ -\frac {15 \, a^{5} x^{5} + 30 \, a^{4} x^{4} - 10 \, a^{3} x^{3} - 50 \, a^{2} x^{2} - 17 \, a x + 16}{64 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac {15 \, \log \left (a x + 1\right )}{128 \, a c^{4}} - \frac {15 \, \log \left (a x - 1\right )}{128 \, a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 120, normalized size = 1.01 \[ \frac {\frac {17\,x}{64}+\frac {25\,a\,x^2}{32}-\frac {1}{4\,a}+\frac {5\,a^2\,x^3}{32}-\frac {15\,a^3\,x^4}{32}-\frac {15\,a^4\,x^5}{64}}{a^6\,c^4\,x^6+2\,a^5\,c^4\,x^5-a^4\,c^4\,x^4-4\,a^3\,c^4\,x^3-a^2\,c^4\,x^2+2\,a\,c^4\,x+c^4}+\frac {15\,\mathrm {atanh}\left (a\,x\right )}{64\,a\,c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 143, normalized size = 1.20 \[ - \frac {15 a^{5} x^{5} + 30 a^{4} x^{4} - 10 a^{3} x^{3} - 50 a^{2} x^{2} - 17 a x + 16}{64 a^{7} c^{4} x^{6} + 128 a^{6} c^{4} x^{5} - 64 a^{5} c^{4} x^{4} - 256 a^{4} c^{4} x^{3} - 64 a^{3} c^{4} x^{2} + 128 a^{2} c^{4} x + 64 a c^{4}} - \frac {\frac {15 \log {\left (x - \frac {1}{a} \right )}}{128} - \frac {15 \log {\left (x + \frac {1}{a} \right )}}{128}}{a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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