Optimal. Leaf size=55 \[ \frac {(3-2 a x) e^{3 \coth ^{-1}(a x)}}{5 a c^2 \left (1-a^2 x^2\right )}-\frac {2 e^{3 \coth ^{-1}(a x)}}{15 a c^2} \]
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Rubi [A] time = 0.07, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6185, 6183} \[ \frac {(3-2 a x) e^{3 \coth ^{-1}(a x)}}{5 a c^2 \left (1-a^2 x^2\right )}-\frac {2 e^{3 \coth ^{-1}(a x)}}{15 a c^2} \]
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 )^2} \, dx &=\frac {e^{3 \coth ^{-1}(a x)} (3-2 a x)}{5 a c^2 \left (1-a^2 x^2\right )}-\frac {2 \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{5 c}\\ &=-\frac {2 e^{3 \coth ^{-1}(a x)}}{15 a c^2}+\frac {e^{3 \coth ^{-1}(a x)} (3-2 a x)}{5 a c^2 \left (1-a^2 x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 43, normalized size = 0.78 \[ -\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a^2 x^2-6 a x+7\right )}{15 c^2 (a x-1)^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 77, normalized size = 1.40 \[ -\frac {{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 7\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 69, normalized size = 1.25 \[ \frac {{\left (a x + 1\right )}^{2} {\left (\frac {10 \, {\left (a x - 1\right )}}{a x + 1} - \frac {15 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3\right )}}{60 \, {\left (a x - 1\right )}^{2} a c^{2} \sqrt {\frac {a x - 1}{a x + 1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 49, normalized size = 0.89 \[ -\frac {2 a^{2} x^{2}-6 a x +7}{15 \left (a^{2} x^{2}-1\right ) c^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 55, normalized size = 1.00 \[ \frac {\frac {10 \, {\left (a x - 1\right )}}{a x + 1} - \frac {15 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3}{60 \, a c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 55, normalized size = 1.00 \[ -\frac {\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}+\frac {1}{5}}{4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \]
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 \[ \frac {\int \frac {1}{\frac {a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {2 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {2 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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