Optimal. Leaf size=116 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a c^{5/2}}+\frac {x}{c^2 \sqrt {c-\frac {c}{a x}}}-\frac {2}{a c^2 \sqrt {c-\frac {c}{a x}}} \]
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Rubi [A] time = 0.24, antiderivative size = 116, 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, 103, 152, 156, 63, 208} \[ \frac {x}{c^2 \sqrt {c-\frac {c}{a x}}}-\frac {2}{a c^2 \sqrt {c-\frac {c}{a x}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a c^{5/2}} \]
Antiderivative was successfully verified.
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Rule 25
Rule 63
Rule 103
Rule 152
Rule 156
Rule 208
Rule 375
Rule 514
Rule 6133
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx &=-\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 \operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c^2}-\frac {\operatorname {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 {\operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^3}+\frac {\operatorname {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] time = 0.06, size = 70, normalized size = 0.60 \[ \frac {-\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a-\frac {1}{x}}{2 a}\right )-\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1-\frac {1}{a x}\right )+a x}{a c^2 \sqrt {c-\frac {c}{a x}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.60, size = 287, normalized size = 2.47 \[ \left [\frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 2 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 4 \, {\left (a^{2} x^{2} - 2 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}, -\frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + 2 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - 2 \, {\left (a^{2} x^{2} - 2 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 166, normalized size = 1.43 \[ -\frac {1}{2} \, a c {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a c x - c}{a x}}}{2 \, \sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{3}} + \frac {2 \, \arctan \left (\frac {\sqrt {\frac {a c x - c}{a x}}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{3}} + \frac {2 \, {\left (c - \frac {2 \, {\left (a c x - c\right )}}{a x}\right )}}{{\left (c \sqrt {\frac {a c x - c}{a x}} - \frac {{\left (a c x - c\right )} \sqrt {\frac {a c x - c}{a x}}}{a x}\right )} a^{2} c^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 370, normalized size = 3.19 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (8 \sqrt {\frac {1}{a}}\, a^{\frac {7}{2}} \sqrt {\left (a x -1\right ) x}\, x^{2}+2 \sqrt {\frac {1}{a}}\, \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{2} a^{3}-4 \sqrt {\frac {1}{a}}\, a^{\frac {5}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}-\ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) a^{\frac {5}{2}} \sqrt {2}\, x^{2}-16 \sqrt {\frac {1}{a}}\, a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, x -4 \sqrt {\frac {1}{a}}\, \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x \,a^{2}+2 \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) a^{\frac {3}{2}} \sqrt {2}\, x +8 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+2 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-\sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}\right )}{4 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, c^{3} \left (a x -1\right )^{2} \sqrt {\frac {1}{a}}} \]
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 \[ \int \frac {a x - 1}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {a\,x-1}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\left (a\,x+1\right )} \,d x \]
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 \[ \int \frac {a x - 1}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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