Optimal. Leaf size=172 \[ \frac {5 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{9/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}}-\frac {21}{4 a c^4 \sqrt {c-\frac {c}{a x}}}-\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}} \]
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Rubi [A] time = 0.29, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ \frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {21}{4 a c^4 \sqrt {c-\frac {c}{a x}}}-\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {5 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{9/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}} \]
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 )^{9/2}} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx\\ &=-\int \frac {1-a x}{\left (c-\frac {c}{a x}\right )^{9/2} (1+a x)} \, dx\\ &=\frac {a \int \frac {x}{\left (c-\frac {c}{a x}\right )^{7/2} (1+a x)} \, dx}{c}\\ &=\frac {a \int \frac {1}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}} \, dx}{c}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {1}{x^2 (a+x) \left (c-\frac {c x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {5 c}{2}-\frac {7 c x}{2 a}}{x (a+x) \left (c-\frac {c x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=-\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\operatorname {Subst}\left (\int \frac {\frac {25 c^2}{2}+\frac {15 c^2 x}{a}}{x (a+x) \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 c^4}\\ &=-\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {75 c^3}{2}-\frac {165 c^3 x}{4 a}}{x (a+x) \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{15 c^6}\\ &=-\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {21}{4 a c^4 \sqrt {c-\frac {c}{a x}}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\operatorname {Subst}\left (\int \frac {\frac {75 c^4}{2}+\frac {315 c^4 x}{8 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{15 c^8}\\ &=-\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {21}{4 a c^4 \sqrt {c-\frac {c}{a x}}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 a c^4}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c^4}\\ &=-\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {21}{4 a c^4 \sqrt {c-\frac {c}{a x}}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{4 c^5}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^5}\\ &=-\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {21}{4 a c^4 \sqrt {c-\frac {c}{a x}}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {5 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{9/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 82, normalized size = 0.48 \[ \frac {a x^2 \left (-\, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {a-\frac {1}{x}}{2 a}\right )-5 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};1-\frac {1}{a x}\right )+5 a x\right )}{5 c^4 (a x-1)^2 \sqrt {c-\frac {c}{a x}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.67, size = 431, normalized size = 2.51 \[ \left [\frac {15 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 600 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 4 \, {\left (60 \, a^{4} x^{4} - 497 \, a^{3} x^{3} + 740 \, a^{2} x^{2} - 315 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{240 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}}, -\frac {15 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + 600 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - 2 \, {\left (60 \, a^{4} x^{4} - 497 \, a^{3} x^{3} + 740 \, a^{2} x^{2} - 315 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{120 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 207, normalized size = 1.20 \[ -\frac {1}{120} \, a c {\left (\frac {2 \, {\left (12 \, c^{2} + \frac {50 \, {\left (a c x - c\right )} c}{a x} + \frac {255 \, {\left (a c x - c\right )}^{2}}{a^{2} x^{2}}\right )} x^{2}}{{\left (a c x - c\right )}^{2} c^{5} \sqrt {\frac {a c x - c}{a x}}} + \frac {15 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a c x - c}{a x}}}{2 \, \sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{5}} + \frac {600 \, \arctan \left (\frac {\sqrt {\frac {a c x - c}{a x}}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{5}} - \frac {120 \, \sqrt {\frac {a c x - c}{a x}}}{a^{2} {\left (c - \frac {a c x - c}{a x}\right )} c^{5}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 626, normalized size = 3.64 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-1260 \sqrt {\left (a x -1\right ) x}\, a^{\frac {11}{2}} \sqrt {\frac {1}{a}}\, x^{4}+1020 \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{2}-600 \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^{4} a^{5}+15 a^{\frac {9}{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^{4}+5040 a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, x^{3}-1792 a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x +2400 \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^{3} a^{4}-60 a^{\frac {7}{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^{3}-7560 \sqrt {\frac {1}{a}}\, a^{\frac {7}{2}} \sqrt {\left (a x -1\right ) x}\, x^{2}+820 \sqrt {\frac {1}{a}}\, a^{\frac {5}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}-3600 \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}+90 \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}+5040 \sqrt {\frac {1}{a}}\, a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, x +2400 \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}-60 \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 -1260 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-600 \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}}+15 \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 )}{240 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, c^{5} \left (a x -1\right )^{4} \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 \[ \int \frac {a x - 1}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {a\,x-1}{{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,\left (a\,x+1\right )} \,d x \]
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 \[ \int \frac {a x - 1}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {9}{2}} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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