Optimal. Leaf size=95 \[ -\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{5 c^4 \left (a-\frac {1}{x}\right )^3}+\frac {8 a \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )^2}-\frac {7 \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )} \]
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Rubi [A] time = 0.23, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6175, 6178, 1639, 793, 659, 651} \[ -\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{5 c^4 \left (a-\frac {1}{x}\right )^3}+\frac {8 a \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )^2}-\frac {7 \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )} \]
Antiderivative was successfully verified.
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Rule 651
Rule 659
Rule 793
Rule 1639
Rule 6175
Rule 6178
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^4} \, dx &=\frac {\int \frac {e^{-\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^4 x^4} \, dx}{a^4 c^4}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (1-\frac {x}{a}\right )^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^4 c^4}\\ &=\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{c^4 \left (a-\frac {1}{x}\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {\frac {2}{a^2}-\frac {x}{a^3}}{\left (1-\frac {x}{a}\right )^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^4}\\ &=-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{5 c^4 \left (a-\frac {1}{x}\right )^3}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{c^4 \left (a-\frac {1}{x}\right )^2}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right )^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{5 a^2 c^4}\\ &=-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{5 c^4 \left (a-\frac {1}{x}\right )^3}+\frac {8 a \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )^2}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{15 a^2 c^4}\\ &=-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{5 c^4 \left (a-\frac {1}{x}\right )^3}+\frac {8 a \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )^2}-\frac {7 \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 43, normalized size = 0.45 \[ -\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a^2 x^2-6 a x+7\right )}{15 c^4 (a x-1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 77, normalized size = 0.81 \[ -\frac {{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 7\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 65, normalized size = 0.68 \[ -\frac {4 \, {\left (10 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} - 5 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x + 1\right )}}{15 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{5} a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 50, normalized size = 0.53 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (2 a^{2} x^{2}-6 a x +7\right ) \left (a x +1\right )}{15 \left (a x -1\right )^{3} c^{4} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 55, normalized size = 0.58 \[ \frac {\frac {10 \, {\left (a x - 1\right )}}{a x + 1} - \frac {15 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3}{60 \, a c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 55, normalized size = 0.58 \[ -\frac {\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}+\frac {1}{5}}{4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \]
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 \[ \frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 4 a^{3} x^{3} + 6 a^{2} x^{2} - 4 a x + 1}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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