Optimal. Leaf size=128 \[ -\frac {18 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^3}+\frac {23 a \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^2}-\frac {12 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )}+\frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^5 \left (a-\frac {1}{x}\right )^4} \]
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Rubi [A] time = 0.25, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {18 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^3}+\frac {23 a \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^2}-\frac {12 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )} \]
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)^5} \, dx &=-\frac {\int \frac {e^{-\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^5 x^5} \, dx}{a^5 c^5}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{\left (1-\frac {x}{a}\right )^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^5 c^5}\\ &=\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{c^5 \left (a-\frac {1}{x}\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {\frac {2}{a^2}-\frac {3 x}{a^3}}{\left (1-\frac {x}{a}\right )^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^5}\\ &=\frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^5 \left (a-\frac {1}{x}\right )^4}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{c^5 \left (a-\frac {1}{x}\right )^2}-\frac {18 \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right )^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{7 a^2 c^5}\\ &=\frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {18 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^3}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{c^5 \left (a-\frac {1}{x}\right )^2}-\frac {36 \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 )}{35 a^2 c^5}\\ &=\frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {18 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^3}+\frac {23 a \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^2}-\frac {12 \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 )}{35 a^2 c^5}\\ &=\frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {18 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^3}+\frac {23 a \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^2}-\frac {12 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 51, normalized size = 0.40 \[ -\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a^3 x^3-8 a^2 x^2+13 a x-12\right )}{35 c^5 (a x-1)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 95, normalized size = 0.74 \[ -\frac {{\left (2 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 5 \, a^{2} x^{2} + a x - 12\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} + 6 \, a^{3} c^{5} x^{2} - 4 \, a^{2} c^{5} x + a c^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 85, normalized size = 0.66 \[ \frac {4 \, {\left (35 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{3} x^{3} - 21 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} + 7 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}}{35 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{7} a c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 58, normalized size = 0.45 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (2 x^{3} a^{3}-8 a^{2} x^{2}+13 a x -12\right ) \left (a x +1\right )}{35 \left (a x -1\right )^{4} c^{5} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 71, normalized size = 0.55 \[ -\frac {\frac {21 \, {\left (a x - 1\right )}}{a x + 1} - \frac {35 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {35 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{280 \, a c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 71, normalized size = 0.55 \[ \frac {\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {3\,\left (a\,x-1\right )}{5\,\left (a\,x+1\right )}+\frac {1}{7}}{8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]
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 \[ - \frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{5} x^{5} - 5 a^{4} x^{4} + 10 a^{3} x^{3} - 10 a^{2} x^{2} + 5 a x - 1}\, dx}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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