Optimal. Leaf size=127 \[ -\frac {16 e^{-\coth ^{-1}(a x)}}{35 a c^4}+\frac {e^{-\coth ^{-1}(a x)} (1+6 a x)}{35 a c^4 \left (1-a^2 x^2\right )^3}+\frac {2 e^{-\coth ^{-1}(a x)} (1+4 a x)}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac {8 e^{-\coth ^{-1}(a x)} (1+2 a x)}{35 a c^4 \left (1-a^2 x^2\right )} \]
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Rubi [A]
time = 0.10, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6320, 6318}
\begin {gather*} \frac {8 (2 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )}+\frac {2 (4 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac {(6 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac {16 e^{-\coth ^{-1}(a x)}}{35 a c^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 6318
Rule 6320
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=\frac {e^{-\coth ^{-1}(a x)} (1+6 a x)}{35 a c^4 \left (1-a^2 x^2\right )^3}+\frac {6 \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx}{7 c}\\ &=\frac {e^{-\coth ^{-1}(a x)} (1+6 a x)}{35 a c^4 \left (1-a^2 x^2\right )^3}+\frac {2 e^{-\coth ^{-1}(a x)} (1+4 a x)}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac {24 \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{35 c^2}\\ &=\frac {e^{-\coth ^{-1}(a x)} (1+6 a x)}{35 a c^4 \left (1-a^2 x^2\right )^3}+\frac {2 e^{-\coth ^{-1}(a x)} (1+4 a x)}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac {8 e^{-\coth ^{-1}(a x)} (1+2 a x)}{35 a c^4 \left (1-a^2 x^2\right )}+\frac {16 \int \frac {e^{-\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{35 c^3}\\ &=-\frac {16 e^{-\coth ^{-1}(a x)}}{35 a c^4}+\frac {e^{-\coth ^{-1}(a x)} (1+6 a x)}{35 a c^4 \left (1-a^2 x^2\right )^3}+\frac {2 e^{-\coth ^{-1}(a x)} (1+4 a x)}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac {8 e^{-\coth ^{-1}(a x)} (1+2 a x)}{35 a c^4 \left (1-a^2 x^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 80, normalized size = 0.63 \begin {gather*} -\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (-5+30 a x+30 a^2 x^2-40 a^3 x^3-40 a^4 x^4+16 a^5 x^5+16 a^6 x^6\right )}{35 (-1+a x)^3 (c+a c x)^4} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.22, size = 84, normalized size = 0.66
method | result | size |
gosper | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (16 a^{6} x^{6}+16 a^{5} x^{5}-40 a^{4} x^{4}-40 a^{3} x^{3}+30 a^{2} x^{2}+30 a x -5\right )}{35 \left (a^{2} x^{2}-1\right )^{3} c^{4} a}\) | \(81\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (16 a^{6} x^{6}+16 a^{5} x^{5}-40 a^{4} x^{4}-40 a^{3} x^{3}+30 a^{2} x^{2}+30 a x -5\right )}{35 c^{4} \left (a x +1\right )^{3} \left (a x -1\right )^{3} a}\) | \(84\) |
trager | \(-\frac {\left (16 a^{6} x^{6}+16 a^{5} x^{5}-40 a^{4} x^{4}-40 a^{3} x^{3}+30 a^{2} x^{2}+30 a x -5\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{35 a \,c^{4} \left (a x -1\right )^{3} \left (a x +1\right )^{3}}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 135, normalized size = 1.06 \begin {gather*} \frac {1}{2240} \, a {\left (\frac {5 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 42 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 175 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 700 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4}} + \frac {7 \, {\left (\frac {10 \, {\left (a x - 1\right )}}{a x + 1} - \frac {75 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 1\right )}}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 104, normalized size = 0.82 \begin {gather*} -\frac {{\left (16 \, a^{6} x^{6} + 16 \, a^{5} x^{5} - 40 \, a^{4} x^{4} - 40 \, a^{3} x^{3} + 30 \, a^{2} x^{2} + 30 \, a x - 5\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{7} c^{4} x^{6} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} - a c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{8} x^{8} - 4 a^{6} x^{6} + 6 a^{4} x^{4} - 4 a^{2} x^{2} + 1}\, dx}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 148, normalized size = 1.17 \begin {gather*} \frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{64\,a\,c^4}-\frac {5\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{16\,a\,c^4}-\frac {3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{160\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{448\,a\,c^4}-\frac {\frac {15\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,\left (a\,x-1\right )}{a\,x+1}+\frac {1}{5}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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