Optimal. Leaf size=118 \[ \frac {2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\sqrt {c-a c x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{a x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x}} \]
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Rubi [A]
time = 0.11, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6311, 6316, 96,
95, 212} \begin {gather*} \frac {2 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{\sqrt {c-a c x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{a x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 212
Rule 6311
Rule 6316
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx &=\frac {\left (\sqrt {1-\frac {1}{a x}} \sqrt {x}\right ) \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a x}} \sqrt {x}} \, dx}{\sqrt {c-a c x}}\\ &=-\frac {\sqrt {1-\frac {1}{a x}} \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{3/2} \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{\sqrt {\frac {1}{x}} \sqrt {c-a c x}}\\ &=\frac {2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\sqrt {c-a c x}}-\frac {\left (2 \sqrt {1-\frac {1}{a x}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a \sqrt {\frac {1}{x}} \sqrt {c-a c x}}\\ &=\frac {2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\sqrt {c-a c x}}-\frac {\left (4 \sqrt {1-\frac {1}{a x}}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {2 x^2}{a}} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{a \sqrt {\frac {1}{x}} \sqrt {c-a c x}}\\ &=\frac {2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\sqrt {c-a c x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{a x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 99, normalized size = 0.84 \begin {gather*} \frac {2 \sqrt {1-\frac {1}{a x}} x \left (\sqrt {a} \sqrt {1+\frac {1}{a x}}-\sqrt {2} \sqrt {\frac {1}{x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{\sqrt {a} \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 83, normalized size = 0.70
method | result | size |
default | \(\frac {2 \sqrt {-c \left (a x -1\right )}\, \left (\sqrt {c}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-\sqrt {-c \left (a x +1\right )}\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x +1\right )}\, c a}\) | \(83\) |
risch | \(\frac {2 a x -2}{a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {-a c x -c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {-c \left (a x +1\right )}\, \left (a x -1\right )}{a \sqrt {c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}}\) | \(115\) |
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 = 0.36, size = 239, normalized size = 2.03 \begin {gather*} \left [\frac {\sqrt {2} {\left (a c x - c\right )} \sqrt {-\frac {1}{c}} \log \left (-\frac {a^{2} x^{2} - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {-\frac {1}{c}} + 2 \, a x - 3}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c x - a c}, -\frac {2 \, {\left (\sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} - \frac {\sqrt {2} {\left (a c x - c\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x - 1\right )} \sqrt {c}}\right )}{\sqrt {c}}\right )}}{a^{2} c x - 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 {1}{\sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \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 = 0.42, size = 94, normalized size = 0.80 \begin {gather*} \frac {2 \, c {\left (\frac {\sqrt {2} {\left (\sqrt {c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {c}}\right ) - \sqrt {-c}\right )}}{c} - \frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right ) - \sqrt {-a c x - c}}{c}\right )}}{a {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )} \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 = -0.01 \begin {gather*} \int \frac {1}{\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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