Optimal. Leaf size=140 \[ \frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {7 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {1-\frac {1}{a x}}} \]
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Rubi [A]
time = 0.09, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6317, 6314, 91,
79, 65, 214} \begin {gather*} \frac {x \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {7 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \sqrt {1-\frac {1}{a x}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 79
Rule 91
Rule 214
Rule 6314
Rule 6317
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx &=\frac {\sqrt {c-\frac {c}{a x}} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} \, dx}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^2 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {-\frac {7}{2 a}+\frac {x}{a^2}}{x \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\left (7 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \sqrt {1-\frac {1}{a x}}}\\ &=\frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\left (7 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {7 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 67, normalized size = 0.48 \begin {gather*} \frac {\sqrt {c-\frac {c}{a x}} \left (9+a x-7 \sqrt {1+\frac {1}{a x}} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{a \sqrt {1-\frac {1}{a^2 x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 146, normalized size = 1.04
method | result | size |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 a^{\frac {3}{2}} x \sqrt {x \left (a x +1\right )}-7 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x +18 \sqrt {x \left (a x +1\right )}\, \sqrt {a}-7 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{2 \left (a x -1\right )^{2} \sqrt {a}\, \sqrt {x \left (a x +1\right )}}\) | \(146\) |
risch | \(\frac {x \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}+\frac {\left (-\frac {7 \ln \left (\frac {\frac {1}{2} a c +c \,a^{2} x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 \sqrt {a^{2} c}}+\frac {8 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-\left (x +\frac {1}{a}\right ) a c}}{a^{2} c \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c a x \left (a x +1\right )}}{a x -1}\) | \(180\) |
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 = 0.38, size = 299, normalized size = 2.14 \begin {gather*} \left [\frac {7 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + 9 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, \frac {7 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + 9 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \sqrt {c-\frac {c}{a\,x}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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