Optimal. Leaf size=119 \[ \frac {2}{a c^2 \sqrt {c-\frac {c}{a x}}}-\frac {x}{c^2 \sqrt {c-\frac {c}{a x}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a c^{5/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6268, 25, 528,
382, 105, 157, 162, 65, 214} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a c^{5/2}}-\frac {x}{c^2 \sqrt {c-\frac {c}{a x}}}+\frac {2}{a c^2 \sqrt {c-\frac {c}{a x}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 65
Rule 105
Rule 157
Rule 162
Rule 214
Rule 382
Rule 528
Rule 6268
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx &=\int \frac {1-a x}{\left (c-\frac {c}{a x}\right )^{5/2} (1+a x)} \, dx\\ &=-\frac {a \int \frac {x}{\left (c-\frac {c}{a x}\right )^{3/2} (1+a x)} \, dx}{c}\\ &=-\frac {a \int \frac {1}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}} \, dx}{c}\\ &=\frac {a \text {Subst}\left (\int \frac {1}{x^2 (a+x) \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {x}{c^2 \sqrt {c-\frac {c}{a x}}}-\frac {\text {Subst}\left (\int \frac {-\frac {c}{2}-\frac {3 c x}{2 a}}{x (a+x) \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=\frac {2}{a c^2 \sqrt {c-\frac {c}{a x}}}-\frac {x}{c^2 \sqrt {c-\frac {c}{a x}}}+\frac {\text {Subst}\left (\int \frac {\frac {c^2}{2}+\frac {c^2 x}{a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{c^4}\\ &=\frac {2}{a c^2 \sqrt {c-\frac {c}{a x}}}-\frac {x}{c^2 \sqrt {c-\frac {c}{a x}}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c^2}+\frac {\text {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c^2}\\ &=\frac {2}{a c^2 \sqrt {c-\frac {c}{a x}}}-\frac {x}{c^2 \sqrt {c-\frac {c}{a x}}}-\frac {\text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^3}-\frac {\text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^3}\\ &=\frac {2}{a c^2 \sqrt {c-\frac {c}{a x}}}-\frac {x}{c^2 \sqrt {c-\frac {c}{a x}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a c^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 67, normalized size = 0.56 \begin {gather*} \frac {-a x+\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a-\frac {1}{x}}{2 a}\right )+\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1-\frac {1}{a x}\right )}{a c^2 \sqrt {c-\frac {c}{a x}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(369\) vs.
\(2(100)=200\).
time = 0.48, size = 370, normalized size = 3.11
method | result | size |
risch | \(-\frac {a x -1}{a \,c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}}-\frac {\left (\frac {\ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c x a}\right )}{2 a^{3} \sqrt {a^{2} c}}-\frac {\sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{a^{5} c \left (x -\frac {1}{a}\right )}-\frac {\sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{4 a^{4} \sqrt {c}}\right ) a^{2} \sqrt {a c x \left (a x -1\right )}}{c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x}\) | \(229\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (8 a^{\frac {7}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}\, x^{2}-a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{2}-4 a^{\frac {5}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} \sqrt {\frac {1}{a}}-16 a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}\, x +2 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{3} x^{2}+2 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x +8 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-4 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x -\sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}+2 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}\right )}{4 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, c^{3} \left (a x -1\right )^{2} \sqrt {\frac {1}{a}}}\) | \(370\) |
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 = 284, normalized size = 2.39 \begin {gather*} \left [\frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + 2 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 4 \, {\left (a^{2} x^{2} - 2 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}, \frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + 2 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - 2 \, {\left (a^{2} x^{2} - 2 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} c^{3} x - a c^{3}\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 {a x}{a c^{2} x \sqrt {c - \frac {c}{a x}} - c^{2} \sqrt {c - \frac {c}{a x}} - \frac {c^{2} \sqrt {c - \frac {c}{a x}}}{a x} + \frac {c^{2} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}}}\, dx - \int \left (- \frac {1}{a c^{2} x \sqrt {c - \frac {c}{a x}} - c^{2} \sqrt {c - \frac {c}{a x}} - \frac {c^{2} \sqrt {c - \frac {c}{a x}}}{a x} + \frac {c^{2} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}}}\right )\, dx \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 \frac {a^2\,x^2-1}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,{\left (a\,x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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