Optimal. Leaf size=57 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a c^{5/2}}-\frac {1}{a c^2 \sqrt {c-a c x}} \]
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Rubi [A] time = 0.10, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6167, 6130, 21, 51, 63, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a c^{5/2}}-\frac {1}{a c^2 \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 51
Rule 63
Rule 206
Rule 6130
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx\\ &=-\int \frac {1-a x}{(1+a x) (c-a c x)^{5/2}} \, dx\\ &=-\frac {\int \frac {1}{(1+a x) (c-a c x)^{3/2}} \, dx}{c}\\ &=-\frac {1}{a c^2 \sqrt {c-a c x}}-\frac {\int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx}{2 c^2}\\ &=-\frac {1}{a c^2 \sqrt {c-a c x}}+\frac {\operatorname {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a c^3}\\ &=-\frac {1}{a c^2 \sqrt {c-a c x}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a c^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 37, normalized size = 0.65 \[ -\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2} (1-a x)\right )}{a c^2 \sqrt {c-a c x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.71, size = 146, normalized size = 2.56 \[ \left [\frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 4 \, \sqrt {-a c x + c}}{4 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}, -\frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 2 \, \sqrt {-a c x + c}}{2 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 54, normalized size = 0.95 \[ -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{2 \, a \sqrt {-c} c^{2}} - \frac {1}{\sqrt {-a c x + c} a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 50, normalized size = 0.88 \[ -\frac {2 \left (\frac {1}{2 c \sqrt {-a c x +c}}-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4 c^{\frac {3}{2}}}\right )}{c a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 71, normalized size = 1.25 \[ -\frac {\frac {\sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {3}{2}}} + \frac {4}{\sqrt {-a c x + c} c}}{4 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 47, normalized size = 0.82 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{2\,a\,c^{5/2}}-\frac {1}{a\,c^2\,\sqrt {c-a\,c\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.83, size = 61, normalized size = 1.07 \[ - \frac {1}{a c^{2} \sqrt {- a c x + c}} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{2 a c^{2} \sqrt {- c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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