Optimal. Leaf size=85 \[ -\frac {(1-4 a x) e^{\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )^2}-\frac {4 (1-2 a x) e^{\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )}+\frac {8 e^{\coth ^{-1}(a x)}}{15 a c^3} \]
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Rubi [A] time = 0.09, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6185, 6183} \[ -\frac {(1-4 a x) e^{\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )^2}-\frac {4 (1-2 a x) e^{\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )}+\frac {8 e^{\coth ^{-1}(a x)}}{15 a c^3} \]
Antiderivative was successfully verified.
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Rule 6183
Rule 6185
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx &=-\frac {e^{\coth ^{-1}(a x)} (1-4 a x)}{15 a c^3 \left (1-a^2 x^2\right )^2}+\frac {4 \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{5 c}\\ &=-\frac {e^{\coth ^{-1}(a x)} (1-4 a x)}{15 a c^3 \left (1-a^2 x^2\right )^2}-\frac {4 e^{\coth ^{-1}(a x)} (1-2 a x)}{15 a c^3 \left (1-a^2 x^2\right )}+\frac {8 \int \frac {e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{15 c^2}\\ &=\frac {8 e^{\coth ^{-1}(a x)}}{15 a c^3}-\frac {e^{\coth ^{-1}(a x)} (1-4 a x)}{15 a c^3 \left (1-a^2 x^2\right )^2}-\frac {4 e^{\coth ^{-1}(a x)} (1-2 a x)}{15 a c^3 \left (1-a^2 x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 66, normalized size = 0.78 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (8 a^4 x^4-8 a^3 x^3-12 a^2 x^2+12 a x+3\right )}{15 c^3 (a x-1)^3 (a x+1)^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 86, normalized size = 1.01 \[ \frac {{\left (8 \, a^{4} x^{4} - 8 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 12 \, a x + 3\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 117, normalized size = 1.38 \[ -\frac {\frac {{\left (a x + 1\right )}^{2} {\left (\frac {20 \, {\left (a x - 1\right )}}{a x + 1} - \frac {90 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3\right )}}{{\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}} + \frac {5 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} - 60 \, \sqrt {\frac {a x - 1}{a x + 1}}}{240 \, a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 65, normalized size = 0.76 \[ \frac {8 x^{4} a^{4}-8 x^{3} a^{3}-12 a^{2} x^{2}+12 a x +3}{15 \left (a^{2} x^{2}-1\right )^{2} c^{3} \sqrt {\frac {a x -1}{a x +1}}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 99, normalized size = 1.16 \[ -\frac {1}{240} \, a {\left (\frac {5 \, {\left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 12 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{2} c^{3}} + \frac {\frac {20 \, {\left (a x - 1\right )}}{a x + 1} - \frac {90 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 60, normalized size = 0.71 \[ \frac {8\,a^4\,x^4-8\,a^3\,x^3-12\,a^2\,x^2+12\,a\,x+3}{15\,a\,c^3\,{\left (a\,x+1\right )}^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \]
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 \[ - \frac {\int \frac {1}{a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 3 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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