Optimal. Leaf size=95 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a c^{3/2}}-\frac {x \sqrt {c-\frac {c}{a x}}}{c^2} \]
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Rubi [A] time = 0.16, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6133, 25, 514, 375, 103, 21, 83, 63, 208} \[ -\frac {x \sqrt {c-\frac {c}{a x}}}{c^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 25
Rule 63
Rule 83
Rule 103
Rule 208
Rule 375
Rule 514
Rule 6133
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx &=\int \frac {1-a x}{\left (c-\frac {c}{a x}\right )^{3/2} (1+a x)} \, dx\\ &=-\frac {a \int \frac {x}{\sqrt {c-\frac {c}{a x}} (1+a x)} \, dx}{c}\\ &=-\frac {a \int \frac {1}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}} \, dx}{c}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {1}{x^2 (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x}{c^2}-\frac {\operatorname {Subst}\left (\int \frac {\frac {c}{2}-\frac {c x}{2 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x}{c^2}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{2 c^2}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x}{c^2}-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c}+\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a c}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x}{c^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^2}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^2}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x}{c^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 95, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a c^{3/2}}-\frac {x \sqrt {c-\frac {c}{a x}}}{c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 231, normalized size = 2.43 \[ \left [-\frac {2 \, a x \sqrt {\frac {a c x - c}{a x}} - \sqrt {2} \sqrt {c} \log \left (\frac {\frac {2 \, \sqrt {2} a x \sqrt {\frac {a c x - c}{a x}}}{\sqrt {c}} - 3 \, a x + 1}{a x + 1}\right ) - \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{2 \, a c^{2}}, \frac {\sqrt {2} c \sqrt {-\frac {1}{c}} \arctan \left (\frac {\sqrt {2} a x \sqrt {-\frac {1}{c}} \sqrt {\frac {a c x - c}{a x}}}{a x - 1}\right ) - a x \sqrt {\frac {a c x - c}{a x}} - \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right )}{a c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 129, normalized size = 1.36 \[ a c {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a c x - c}{a x}}}{2 \, \sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{2}} - \frac {\arctan \left (\frac {\sqrt {\frac {a c x - c}{a x}}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{2}} - \frac {\sqrt {\frac {a c x - c}{a x}}}{a^{2} {\left (c - \frac {a c x - c}{a x}\right )} c^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 134, normalized size = 1.41 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-2 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+\ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+\sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}\right )}{2 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, c^{2} \sqrt {\frac {1}{a}}} \]
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 \[ -\int \frac {a^{2} x^{2} - 1}{{\left (a x + 1\right )}^{2} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ -\int \frac {a^2\,x^2-1}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,{\left (a\,x+1\right )}^2} \,d x \]
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 \[ - \int \frac {a x}{a c x \sqrt {c - \frac {c}{a x}} - \frac {c \sqrt {c - \frac {c}{a x}}}{a x}}\, dx - \int \left (- \frac {1}{a c x \sqrt {c - \frac {c}{a x}} - \frac {c \sqrt {c - \frac {c}{a x}}}{a x}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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