Optimal. Leaf size=91 \[ -\frac {(3-4 a x) e^{3 \coth ^{-1}(a x)}}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac {12 (3-2 a x) e^{3 \coth ^{-1}(a x)}}{35 a c^3 \left (1-a^2 x^2\right )}-\frac {8 e^{3 \coth ^{-1}(a x)}}{35 a c^3} \]
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Rubi [A] time = 0.10, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6185, 6183} \[ -\frac {(3-4 a x) e^{3 \coth ^{-1}(a x)}}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac {12 (3-2 a x) e^{3 \coth ^{-1}(a x)}}{35 a c^3 \left (1-a^2 x^2\right )}-\frac {8 e^{3 \coth ^{-1}(a x)}}{35 a c^3} \]
Antiderivative was successfully verified.
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Rule 6183
Rule 6185
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx &=-\frac {e^{3 \coth ^{-1}(a x)} (3-4 a x)}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac {12 \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{7 c}\\ &=-\frac {e^{3 \coth ^{-1}(a x)} (3-4 a x)}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac {12 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{35 a c^3 \left (1-a^2 x^2\right )}-\frac {24 \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{35 c^2}\\ &=-\frac {8 e^{3 \coth ^{-1}(a x)}}{35 a c^3}-\frac {e^{3 \coth ^{-1}(a x)} (3-4 a x)}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac {12 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{35 a c^3 \left (1-a^2 x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 66, normalized size = 0.73 \[ -\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (8 a^4 x^4-24 a^3 x^3+20 a^2 x^2+4 a x-13\right )}{35 c^3 (a x-1)^4 (a x+1)} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 96, normalized size = 1.05 \[ -\frac {{\left (8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 104, normalized size = 1.14 \[ -\frac {\frac {{\left (a x + 1\right )}^{3} {\left (\frac {28 \, {\left (a x - 1\right )}}{a x + 1} - \frac {70 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {140 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5\right )}}{{\left (a x - 1\right )}^{3} \sqrt {\frac {a x - 1}{a x + 1}}} + 35 \, \sqrt {\frac {a x - 1}{a x + 1}}}{560 \, a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 65, normalized size = 0.71 \[ -\frac {8 x^{4} a^{4}-24 x^{3} a^{3}+20 a^{2} x^{2}+4 a x -13}{35 \left (a^{2} x^{2}-1\right )^{2} c^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 97, normalized size = 1.07 \[ -\frac {1}{560} \, a {\left (\frac {35 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac {\frac {28 \, {\left (a x - 1\right )}}{a x + 1} - \frac {70 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {140 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 60, normalized size = 0.66 \[ -\frac {8\,a^4\,x^4-24\,a^3\,x^3+20\,a^2\,x^2+4\,a\,x-13}{35\,a\,c^3\,{\left (a\,x+1\right )}^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]
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 \[ - \frac {\int \frac {1}{\frac {a^{7} x^{7} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {3 a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {3 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {3 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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