Optimal. Leaf size=128 \[ -\frac {a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {\sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {2} \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 128, 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 {a x \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{3/2}}{\left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {\sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {2} \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}} \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)}}{(c-a c x)^{3/2}} \, dx &=\frac {\left (\left (1-\frac {1}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{3/2} x^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{\sqrt {x} \left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac {a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \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 )}{2 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac {a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \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 )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac {a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {\sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {2} \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 116, normalized size = 0.91 \begin {gather*} \frac {\sqrt {1-\frac {1}{a x}} x \left (2 \sqrt {a} \sqrt {1+\frac {1}{a x}}+\sqrt {2} \sqrt {\frac {1}{x}} (-1+a x) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{2 \sqrt {a} c (-1+a x) \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 118, normalized size = 0.92
method | result | size |
default | \(-\frac {\sqrt {-c \left (a x -1\right )}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x -\sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +2 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x -1\right ) \sqrt {-c \left (a x +1\right )}\, c^{\frac {5}{2}} a}\) | \(118\) |
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 = 0.39, size = 281, normalized size = 2.20 \begin {gather*} \left [-\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{4 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, -\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\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}} \left (- c \left (a x - 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 60, normalized size = 0.47 \begin {gather*} \frac {\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {2 \, \sqrt {-a c x - c}}{a c x - c}}{2 \, a {\left | c \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}{{\left (c-a\,c\,x\right )}^{3/2}\,\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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