Optimal. Leaf size=91 \[ \frac {4 (2 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )}+\frac {(4 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )^2}-\frac {8 e^{-\coth ^{-1}(a x)}}{15 a c^3} \]
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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} \[ \frac {4 (2 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )}+\frac {(4 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )^2}-\frac {8 e^{-\coth ^{-1}(a x)}}{15 a c^3} \]
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 = 0.21, size = 64, normalized size = 0.70 \[ -\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (8 a^4 x^4+8 a^3 x^3-12 a^2 x^2-12 a x+3\right )}{15 (a x-1)^2 (a c x+c)^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 76, normalized size = 0.84 \[ -\frac {{\left (8 \, a^{4} x^{4} + 8 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 12 \, a x + 3\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
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 \[ \int -\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a^{2} c x^{2} - c\right )}^{3}}\,{d x} \]
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 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (8 x^{4} a^{4}+8 x^{3} a^{3}-12 a^{2} x^{2}-12 a x +3\right )}{15 \left (a^{2} x^{2}-1\right )^{2} c^{3} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 102, normalized size = 1.12 \[ -\frac {1}{240} \, a {\left (\frac {3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 20 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 90 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac {5 \, {\left (\frac {12 \, {\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 109, normalized size = 1.20 \[ \frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{12\,a\,c^3}-\frac {3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8\,a\,c^3}-\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{80\,a\,c^3}-\frac {\frac {4\,\left (a\,x-1\right )}{a\,x+1}-\frac {1}{3}}{16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \]
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 \[ - \frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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