Optimal. Leaf size=113 \[ \frac {c \sqrt {c-\frac {c}{a x}}}{a}-\left (c-\frac {c}{a x}\right )^{3/2} x+\frac {7 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {8 \sqrt {2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
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Rubi [A]
time = 0.13, antiderivative size = 113, 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, 100, 159, 162, 65, 214} \begin {gather*} \frac {7 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {8 \sqrt {2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}+\frac {c \sqrt {c-\frac {c}{a x}}}{a}-x \left (c-\frac {c}{a x}\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 65
Rule 100
Rule 159
Rule 162
Rule 214
Rule 382
Rule 528
Rule 6268
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx &=\int \frac {\left (c-\frac {c}{a x}\right )^{3/2} (1-a x)}{1+a x} \, dx\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{5/2} x}{1+a x} \, dx}{c}\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{5/2}}{a+\frac {1}{x}} \, dx}{c}\\ &=\frac {a \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{5/2}}{x^2 (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\left (c-\frac {c}{a x}\right )^{3/2} x-\frac {\text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}} \left (\frac {7 c^2}{2}-\frac {c^2 x}{2 a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {c \sqrt {c-\frac {c}{a x}}}{a}-\left (c-\frac {c}{a x}\right )^{3/2} x-\frac {2 \text {Subst}\left (\int \frac {\frac {7 c^3}{4}-\frac {9 c^3 x}{4 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {c \sqrt {c-\frac {c}{a x}}}{a}-\left (c-\frac {c}{a x}\right )^{3/2} x-\frac {\left (7 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a}+\frac {\left (8 c^2\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {c \sqrt {c-\frac {c}{a x}}}{a}-\left (c-\frac {c}{a x}\right )^{3/2} x+(7 c) \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )-(16 c) \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=\frac {c \sqrt {c-\frac {c}{a x}}}{a}-\left (c-\frac {c}{a x}\right )^{3/2} x+\frac {7 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {8 \sqrt {2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 96, normalized size = 0.85 \begin {gather*} \frac {c \sqrt {c-\frac {c}{a x}} (2-a x)+7 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )-8 \sqrt {2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \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. \(228\) vs.
\(2(94)=188\).
time = 0.40, size = 229, normalized size = 2.03
method | result | size |
risch | \(-\frac {\left (a^{2} x^{2}-3 a x +2\right ) c \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a \left (a x -1\right )}-\frac {\left (-\frac {7 a \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c x a}\right )}{2 \sqrt {a^{2} c}}-\frac {4 \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{\sqrt {c}}\right ) c \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {a c x \left (a x -1\right )}}{a \left (a x -1\right )}\) | \(195\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c \left (8 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}\, x^{2}-10 a^{\frac {3}{2}} \sqrt {a \,x^{2}-x}\, \sqrt {\frac {1}{a}}\, x^{2}-8 \sqrt {a}\, \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{2}+4 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}+5 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a \,x^{2}-12 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a \,x^{2}\right )}{2 x \,a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}}\) | \(229\) |
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 = 231, normalized size = 2.04 \begin {gather*} \left [\frac {8 \, \sqrt {2} c^{\frac {3}{2}} \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 ) + 7 \, c^{\frac {3}{2}} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (a c x - 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{2 \, a}, \frac {8 \, \sqrt {2} \sqrt {-c} c \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) - 7 \, \sqrt {-c} c \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (a c x - 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{a}\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 \left (- \frac {2 c \sqrt {c - \frac {c}{a x}}}{a x + 1}\right )\, dx - \int \frac {c \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2} + a x}\, dx - \int \frac {a c x \sqrt {c - \frac {c}{a x}}}{a x + 1}\, 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 {{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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