Optimal. Leaf size=36 \[ \frac {4}{a \sqrt {c-a c x}}+\frac {2 \sqrt {c-a c x}}{a c} \]
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Rubi [A]
time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6265, 21, 45}
\begin {gather*} \frac {2 \sqrt {c-a c x}}{a c}+\frac {4}{a \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 45
Rule 6265
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx &=\int \frac {1+a x}{(1-a x) \sqrt {c-a c x}} \, dx\\ &=c \int \frac {1+a x}{(c-a c x)^{3/2}} \, dx\\ &=c \int \left (\frac {2}{(c-a c x)^{3/2}}-\frac {1}{c \sqrt {c-a c x}}\right ) \, dx\\ &=\frac {4}{a \sqrt {c-a c x}}+\frac {2 \sqrt {c-a c x}}{a c}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 21, normalized size = 0.58 \begin {gather*} \frac {6-2 a x}{a \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.25, size = 31, normalized size = 0.86
method | result | size |
gosper | \(-\frac {2 \left (a x -3\right )}{a \sqrt {-c x a +c}}\) | \(20\) |
trager | \(\frac {2 \left (a x -3\right ) \sqrt {-c x a +c}}{c \left (a x -1\right ) a}\) | \(30\) |
derivativedivides | \(\frac {2 \sqrt {-c x a +c}+\frac {4 c}{\sqrt {-c x a +c}}}{a c}\) | \(31\) |
default | \(\frac {2 \sqrt {-c x a +c}+\frac {4 c}{\sqrt {-c x a +c}}}{a c}\) | \(31\) |
risch | \(-\frac {2 \left (a x -1\right )}{a \sqrt {-c \left (a x -1\right )}}+\frac {4}{a \sqrt {-c \left (a x -1\right )}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 30, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left (\sqrt {-a c x + c} + \frac {2 \, c}{\sqrt {-a c x + c}}\right )}}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 29, normalized size = 0.81 \begin {gather*} \frac {2 \, \sqrt {-a c x + c} {\left (a x - 3\right )}}{a^{2} c x - a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 22.18, size = 48, normalized size = 1.33 \begin {gather*} \begin {cases} \frac {\frac {2}{\sqrt {- a c x + c}} - \frac {2 \left (- \frac {c}{\sqrt {- a c x + c}} - \sqrt {- a c x + c}\right )}{c}}{a} & \text {for}\: a \neq 0 \\\frac {x}{\sqrt {c}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 32, normalized size = 0.89 \begin {gather*} \frac {4}{\sqrt {-a c x + c} a} + \frac {2 \, \sqrt {-a c x + c}}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 20, normalized size = 0.56 \begin {gather*} -\frac {2\,a\,x-6}{a\,\sqrt {c-a\,c\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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