Optimal. Leaf size=91 \[ -\frac {12 (2 a x+3) e^{-3 \coth ^{-1}(a x)}}{35 a c^3 \left (1-a^2 x^2\right )}+\frac {(4 a x+3) e^{-3 \coth ^{-1}(a x)}}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac {8 e^{-3 \coth ^{-1}(a x)}}{35 a c^3} \]
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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} \[ -\frac {12 (2 a x+3) e^{-3 \coth ^{-1}(a x)}}{35 a c^3 \left (1-a^2 x^2\right )}+\frac {(4 a x+3) e^{-3 \coth ^{-1}(a x)}}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac {8 e^{-3 \coth ^{-1}(a x)}}{35 a c^3} \]
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 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (8 a^4 x^4+24 a^3 x^3+20 a^2 x^2-4 a x-13\right )}{35 c^3 (a x-1) (a x+1)^4} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.54, size = 86, normalized size = 0.95 \[ \frac {{\left (8 \, a^{4} x^{4} + 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} - 4 \, a x - 13\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (a^{2} c x^{2} - c\right )}^{3}}\,{d x} \]
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 \[ \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (8 x^{4} a^{4}+24 x^{3} a^{3}+20 a^{2} x^{2}-4 a x -13\right )}{35 \left (a^{2} x^{2}-1\right )^{2} c^{3} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 103, normalized size = 1.13 \[ -\frac {1}{560} \, a {\left (\frac {5 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 28 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 70 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 140 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}} - \frac {35}{a^{2} c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 116, normalized size = 1.27 \[ \frac {1}{16\,a\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}+\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{4\,a\,c^3}-\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{8\,a\,c^3}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{20\,a\,c^3}-\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{112\,a\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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