Optimal. Leaf size=140 \[ \frac {x \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {7 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \sqrt {1-\frac {1}{a x}}} \]
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Rubi [A] time = 0.11, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6182, 6179, 89, 78, 63, 208} \[ \frac {x \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {7 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 89
Rule 208
Rule 6179
Rule 6182
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx &=\frac {\sqrt {c-\frac {c}{a x}} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} \, dx}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^2 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \frac {-\frac {7}{2 a}+\frac {x}{a^2}}{x \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\left (7 \sqrt {c-\frac {c}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \sqrt {1-\frac {1}{a x}}}\\ &=\frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\left (7 \sqrt {c-\frac {c}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {7 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 67, normalized size = 0.48 \[ \frac {\sqrt {c-\frac {c}{a x}} \left (a x-7 \sqrt {\frac {1}{a x}+1} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )+9\right )}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 299, normalized size = 2.14 \[ \left [\frac {7 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + 9 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, \frac {7 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + 9 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 146, normalized size = 1.04 \[ \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-7 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) x a +18 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}-7 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{2 \left (a x -1\right )^{2} \sqrt {a}\, \sqrt {\left (a x +1\right ) x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {c-\frac {c}{a\,x}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
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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