Optimal. Leaf size=158 \[ \frac {\left (21 a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{3/2} x}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {9 \left (c-\frac {c}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (1-\frac {1}{a x}\right )^{3/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 158, 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,
100, 151, 65, 214} \begin {gather*} \frac {x \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}+\frac {\left (21 a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}-\frac {9 \left (c-\frac {c}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (1-\frac {1}{a x}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 151
Rule 214
Rule 6314
Rule 6317
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx &=\frac {\left (c-\frac {c}{a x}\right )^{3/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{3/2} \, dx}{\left (1-\frac {1}{a x}\right )^{3/2}}\\ &=-\frac {\left (c-\frac {c}{a x}\right )^{3/2} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3}{x^2 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}}\\ &=\frac {\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{3/2} x}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (c-\frac {c}{a x}\right )^{3/2} \text {Subst}\left (\int \frac {\left (\frac {9}{2 a}-\frac {x}{2 a^2}\right ) \left (1-\frac {x}{a}\right )}{x \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}}\\ &=\frac {\left (21 a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{3/2} x}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (9 \left (c-\frac {c}{a x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \left (1-\frac {1}{a x}\right )^{3/2}}\\ &=\frac {\left (21 a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{3/2} x}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (9 \left (c-\frac {c}{a x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}}\\ &=\frac {\left (21 a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{3/2} x}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {9 \left (c-\frac {c}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (1-\frac {1}{a x}\right )^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.03, size = 71, normalized size = 0.45 \begin {gather*} \frac {c \sqrt {c-\frac {c}{a x}} \left (2+10 a x+a^2 x^2+9 a x \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {1}{a x}\right )\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 169, normalized size = 1.07
method | result | size |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, c \left (2 a^{\frac {5}{2}} x^{2} \sqrt {x \left (a x +1\right )}+38 a^{\frac {3}{2}} x \sqrt {x \left (a x +1\right )}-9 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} x^{2}-9 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x +4 \sqrt {x \left (a x +1\right )}\, \sqrt {a}\right )}{2 \left (a x -1\right )^{2} a^{\frac {3}{2}} \sqrt {x \left (a x +1\right )}}\) | \(169\) |
risch | \(\frac {\left (a^{2} x^{2}+3 a x +2\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a \left (a x -1\right )}+\frac {\left (-\frac {9 a \ln \left (\frac {\frac {1}{2} a c +c \,a^{2} x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 \sqrt {a^{2} c}}+\frac {16 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-\left (x +\frac {1}{a}\right ) a c}}{a c \left (x +\frac {1}{a}\right )}\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c a x \left (a x +1\right )}}{a \left (a x -1\right )}\) | \(196\) |
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 = 0.37, size = 315, normalized size = 1.99 \begin {gather*} \left [\frac {9 \, {\left (a c x - c\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} c x^{2} + 19 \, a c x + 2 \, c\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 {9 \, {\left (a c x - c\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} c x^{2} + 19 \, a c x + 2 \, c\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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {\left (c-\frac {c}{a\,x}\right )}^{3/2}\,{\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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