Optimal. Leaf size=144 \[ \frac {x \sqrt {1-\frac {1}{a x}}}{c \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{3 a c \sqrt {\frac {1}{a x}+1}}+\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c} \]
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Rubi [A] time = 0.10, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6194, 99, 152, 12, 92, 208} \[ \frac {x \sqrt {1-\frac {1}{a x}}}{c \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{3 a c \sqrt {\frac {1}{a x}+1}}+\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 92
Rule 99
Rule 152
Rule 208
Rule 6194
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}}}{x^2 \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{a}+\frac {2 x}{a^2}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a \operatorname {Subst}\left (\int \frac {-\frac {9}{a^2}+\frac {5 x}{a^3}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{3 a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a^2 \operatorname {Subst}\left (\int -\frac {9}{a^3 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{3 a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a c}\\ &=\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{3 a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a^2 c}\\ &=\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{3 a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 69, normalized size = 0.48 \[ \frac {\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a^2 x^2+19 a x+14\right )}{(a x+1)^2}-\frac {9 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{a}}{3 c} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 96, normalized size = 0.67 \[ -\frac {9 \, {\left (a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 9 \, {\left (a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (3 \, a^{2} x^{2} + 19 \, a x + 14\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{2} c x + a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 59, normalized size = 0.41 \[ \frac {3 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{c {\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} \mathrm {sgn}\left (a x + 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 346, normalized size = 2.40 \[ -\frac {\left (9 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}-9 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}+27 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+6 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a -27 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+27 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+5 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-27 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a +9 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )-9 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{3 a \sqrt {a^{2}}\, \left (a x +1\right ) c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 140, normalized size = 0.97 \[ -\frac {1}{3} \, a {\left (\frac {6 \, \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2} c}{a x + 1} - a^{2} c} - \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 12 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c} + \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 114, normalized size = 0.79 \[ \frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c-\frac {a\,c\,\left (a\,x-1\right )}{a\,x+1}}+\frac {4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3\,a\,c}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,6{}\mathrm {i}}{a\,c} \]
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 \[ \frac {a^{2} \left (\int \left (- \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} + a^{2} x^{2} - a x - 1}\right )\, dx + \int \frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} + a^{2} x^{2} - a x - 1}\, dx\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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