Optimal. Leaf size=66 \[ \frac {8 c \sqrt {1-a^2 x^2}}{3 a \sqrt {c-a c x}}+\frac {2 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{3 a} \]
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Rubi [A]
time = 0.05, antiderivative size = 66, 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, 671, 663}
\begin {gather*} \frac {8 c \sqrt {1-a^2 x^2}}{3 a \sqrt {c-a c x}}+\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a c x}}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 663
Rule 671
Rule 6262
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \sqrt {c-a c x} \, dx &=\frac {\int \frac {(c-a c x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {2 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{3 a}+\frac {4}{3} \int \frac {\sqrt {c-a c x}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {8 c \sqrt {1-a^2 x^2}}{3 a \sqrt {c-a c x}}+\frac {2 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{3 a}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 38, normalized size = 0.58 \begin {gather*} -\frac {2 c (-5+a x) \sqrt {1-a^2 x^2}}{3 a \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 40, normalized size = 0.61
method | result | size |
gosper | \(\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c x a +c}\, \left (a x -5\right )}{3 a \left (a x -1\right )}\) | \(39\) |
default | \(\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (a x -5\right )}{3 \left (a x -1\right ) a}\) | \(40\) |
risch | \(\frac {2 \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c \left (a x -5\right ) \left (a x +1\right )}{3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, a \sqrt {\left (a x +1\right ) c}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 37, normalized size = 0.56 \begin {gather*} -\frac {2 \, {\left (a \sqrt {c} x - 5 \, \sqrt {c}\right )} \sqrt {a x + 1} {\left (a x - 1\right )}}{3 \, {\left (a^{2} x - a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 39, normalized size = 0.59 \begin {gather*} \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x - 5\right )}}{3 \, {\left (a^{2} x - a\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 {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 50, normalized size = 0.76 \begin {gather*} -\frac {8 \, \sqrt {2} {\left | c \right |}}{3 \, a \sqrt {c}} - \frac {2 \, {\left (a c x + c\right )}^{\frac {3}{2}} {\left | c \right |}}{3 \, a c^{2}} + \frac {4 \, \sqrt {a c x + c} {\left | c \right |}}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.00, size = 56, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {c-a\,c\,x}\,\left (\frac {10\,\sqrt {1-a^2\,x^2}}{3\,a^2}-\frac {2\,x\,\sqrt {1-a^2\,x^2}}{3\,a}\right )}{x-\frac {1}{a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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