Optimal. Leaf size=113 \[ -\frac {7 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {8 \sqrt {2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}-\frac {c \sqrt {c-\frac {c}{a x}}}{a}+x \left (c-\frac {c}{a x}\right )^{3/2} \]
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Rubi [A] time = 0.24, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6167, 6133, 25, 514, 375, 98, 154, 156, 63, 208} \[ -\frac {7 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {8 \sqrt {2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}-\frac {c \sqrt {c-\frac {c}{a x}}}{a}+x \left (c-\frac {c}{a x}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 25
Rule 63
Rule 98
Rule 154
Rule 156
Rule 208
Rule 375
Rule 514
Rule 6133
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx &=-\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 \operatorname {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 {\operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )+(16 c) \operatorname {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 = 0.09, size = 95, normalized size = 0.84 \[ \frac {-7 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )+8 \sqrt {2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )+c (a x-2) \sqrt {c-\frac {c}{a x}}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 235, normalized size = 2.08 \[ \left [\frac {8 \, \sqrt {2} c^{\frac {3}{2}} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 7 \, c^{\frac {3}{2}} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (a c x - 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{2 \, a}, -\frac {8 \, \sqrt {2} \sqrt {-c} c \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) - 7 \, \sqrt {-c} c \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (a c x - 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{a}\right ] \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 229, normalized size = 2.03 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c \left (-10 \sqrt {a \,x^{2}-x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}\, x^{2}+8 a^{\frac {3}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, x^{2}+4 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}+5 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{2} a -8 \sqrt {a}\, \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{2}-12 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{2} a \right )}{2 x \,a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}} \]
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 \[ \int \frac {{\left (a x - 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}{a x + 1}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
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 \[ \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \left (a x - 1\right )}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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