Optimal. Leaf size=76 \[ \frac {4 \sqrt {c-a c x}}{a}+\frac {2 (c-a c x)^{3/2}}{3 a c}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
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Rubi [A]
time = 0.05, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6265, 21, 52,
65, 212} \begin {gather*} \frac {2 (c-a c x)^{3/2}}{3 a c}+\frac {4 \sqrt {c-a c x}}{a}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 52
Rule 65
Rule 212
Rule 6265
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \sqrt {c-a c x} \, dx &=\int \frac {(1-a x) \sqrt {c-a c x}}{1+a x} \, dx\\ &=\frac {\int \frac {(c-a c x)^{3/2}}{1+a x} \, dx}{c}\\ &=\frac {2 (c-a c x)^{3/2}}{3 a c}+2 \int \frac {\sqrt {c-a c x}}{1+a x} \, dx\\ &=\frac {4 \sqrt {c-a c x}}{a}+\frac {2 (c-a c x)^{3/2}}{3 a c}+(4 c) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=\frac {4 \sqrt {c-a c x}}{a}+\frac {2 (c-a c x)^{3/2}}{3 a c}-\frac {8 \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a}\\ &=\frac {4 \sqrt {c-a c x}}{a}+\frac {2 (c-a c x)^{3/2}}{3 a c}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 61, normalized size = 0.80 \begin {gather*} -\frac {2 (-7+a x) \sqrt {c-a c x}+12 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 59, normalized size = 0.78
method | result | size |
risch | \(\frac {2 \left (a x -7\right ) \left (a x -1\right ) c}{3 a \sqrt {-c \left (a x -1\right )}}-\frac {4 \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {c}}{a}\) | \(57\) |
derivativedivides | \(\frac {\frac {2 \left (-c x a +c \right )^{\frac {3}{2}}}{3}+4 c \sqrt {-c x a +c}-4 c^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{a c}\) | \(59\) |
default | \(\frac {\frac {2 \left (-c x a +c \right )^{\frac {3}{2}}}{3}+4 c \sqrt {-c x a +c}-4 c^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{a c}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 79, normalized size = 1.04 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {2} c^{\frac {3}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + {\left (-a c x + c\right )}^{\frac {3}{2}} + 6 \, \sqrt {-a c x + c} c\right )}}{3 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 120, normalized size = 1.58 \begin {gather*} \left [\frac {2 \, {\left (3 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) - \sqrt {-a c x + c} {\left (a x - 7\right )}\right )}}{3 \, a}, \frac {2 \, {\left (6 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - \sqrt {-a c x + c} {\left (a x - 7\right )}\right )}}{3 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.64, size = 75, normalized size = 0.99 \begin {gather*} - \frac {2 \left (- \frac {2 \sqrt {2} c^{2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} - 2 c \sqrt {- a c x + c} - \frac {\left (- a c x + c\right )^{\frac {3}{2}}}{3}\right )}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 77, normalized size = 1.01 \begin {gather*} \frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} + \frac {2 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} a^{2} c^{2} + 6 \, \sqrt {-a c x + c} a^{2} c^{3}\right )}}{3 \, a^{3} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.00, size = 61, normalized size = 0.80 \begin {gather*} \frac {4\,\sqrt {c-a\,c\,x}}{a}+\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a\,c}-\frac {4\,\sqrt {2}\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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