Optimal. Leaf size=51 \[ \frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a c^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6127, 661, 208} \[ \frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 661
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=\frac {\int \frac {1}{\sqrt {c-a c x} \sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\left ((2 a) \operatorname {Subst}\left (\int \frac {1}{-2 a^2 c+a^2 c^2 x^2} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\right )\\ &=\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.86 \[ \frac {\sqrt {2-2 a x} \tanh ^{-1}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )}{a c \sqrt {c-a c x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.69, size = 133, normalized size = 2.61 \[ \left [\frac {\sqrt {2} \log \left (-\frac {a^{2} x^{2} + 2 \, a x - \frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{\sqrt {c}} - 3}{a^{2} x^{2} - 2 \, a x + 1}\right )}{2 \, a c^{\frac {3}{2}}}, \frac {\sqrt {2} \sqrt {-\frac {1}{c}} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-\frac {1}{c}}}{a^{2} x^{2} - 1}\right )}{a c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 62, normalized size = 1.22 \[ -\frac {{\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} - \frac {\sqrt {2} \arctan \left (\frac {\sqrt {c}}{\sqrt {-c}}\right )}{a \sqrt {-c}}\right )} {\left | c \right |}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 68, normalized size = 1.33 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\left (-a x +1\right ) \sqrt {c \left (a x +1\right )}\, c^{\frac {3}{2}} a} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (-a c x + c\right )}^{\frac {3}{2}} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {1-a^2\,x^2}}{{\left (c-a\,c\,x\right )}^{3/2}\,\left (a\,x+1\right )} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (- c \left (a x - 1\right )\right )^{\frac {3}{2}} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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