Optimal. Leaf size=89 \[ -\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{1-a x}-\frac {2 \sqrt {a} \sqrt {c-\frac {c}{a x}} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {1-a x}} \]
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Rubi [A]
time = 0.19, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6269, 6263,
893, 862, 56, 221} \begin {gather*} -\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-\frac {c}{a x}}}{1-a x}-\frac {2 \sqrt {a} \sqrt {x} \sqrt {c-\frac {c}{a x}} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 221
Rule 862
Rule 893
Rule 6263
Rule 6269
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {e^{-\tanh ^{-1}(a x)} \sqrt {1-a x}}{x^{3/2}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {(1-a x)^{3/2}}{x^{3/2} \sqrt {1-a^2 x^2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{1-a x}-\frac {\left (a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1-a x}}{\sqrt {x} \sqrt {1-a^2 x^2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{1-a x}-\frac {\left (a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{1-a x}-\frac {\left (2 a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{1-a x}-\frac {2 \sqrt {a} \sqrt {c-\frac {c}{a x}} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {1-a x}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 60, normalized size = 0.67 \begin {gather*} -\frac {2 \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+a x}+\sqrt {a} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right )}{\sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.77, size = 91, normalized size = 1.02
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-\arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a x +2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}\right ) \sqrt {-a^{2} x^{2}+1}}{\left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}\, \sqrt {a}}\) | \(91\) |
risch | \(-\frac {2 \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a c x \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}}-\frac {a \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-c x a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {a^{2} c}\, \sqrt {-a^{2} x^{2}+1}}\) | \(164\) |
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 = 0.45, size = 235, normalized size = 2.64 \begin {gather*} \left [\frac {{\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a x - 1\right )}}, \frac {{\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{a x - 1}\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 (-1 + \frac {1}{a x}\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x \left (a x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\sqrt {c-\frac {c}{a\,x}}\,\sqrt {1-a^2\,x^2}}{x\,\left (a\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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