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 = 0.04, 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 = {6262, 675, 214}
\begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 675
Rule 6262
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) \text {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 = 0.03, size = 44, normalized size = 0.86 \begin {gather*} \frac {\sqrt {2-2 a x} \tanh ^{-1}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )}{a c \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.32, size = 68, normalized size = 1.33
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\left (-a x +1\right ) \sqrt {\left (a x +1\right ) c}\, c^{\frac {3}{2}} a}\) | \(68\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 133, normalized size = 2.61 \begin {gather*} \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 ] \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 62, normalized size = 1.22 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {1-a^2\,x^2}}{{\left (c-a\,c\,x\right )}^{3/2}\,\left (a\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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