Optimal. Leaf size=215 \[ \frac {a x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{\left (a-\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}-\frac {3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{\left (a-\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}+\frac {7 \sqrt {1-\frac {1}{a x}} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \sqrt {c-\frac {c}{a x}}}-\frac {5 \sqrt {2} \sqrt {1-\frac {1}{a x}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{a \sqrt {c-\frac {c}{a x}}} \]
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Rubi [A] time = 0.15, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6182, 6179, 98, 151, 156, 63, 208, 206} \[ \frac {a x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{\left (a-\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}-\frac {3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{\left (a-\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}+\frac {7 \sqrt {1-\frac {1}{a x}} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \sqrt {c-\frac {c}{a x}}}-\frac {5 \sqrt {2} \sqrt {1-\frac {1}{a x}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{a \sqrt {c-\frac {c}{a x}}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 151
Rule 156
Rule 206
Rule 208
Rule 6179
Rule 6182
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx &=\frac {\sqrt {1-\frac {1}{a x}} \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a x}}} \, dx}{\sqrt {c-\frac {c}{a x}}}\\ &=-\frac {\sqrt {1-\frac {1}{a x}} \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^2 \left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{\sqrt {c-\frac {c}{a x}}}\\ &=\frac {a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} \operatorname {Subst}\left (\int \frac {-\frac {7}{2 a}-\frac {5 x}{2 a^2}}{x \left (1-\frac {x}{a}\right )^2 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {c-\frac {c}{a x}}}\\ &=-\frac {3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}+\frac {a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}-\frac {\left (a \sqrt {1-\frac {1}{a x}}\right ) \operatorname {Subst}\left (\int \frac {\frac {7}{a^2}+\frac {3 x}{a^3}}{x \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 \sqrt {c-\frac {c}{a x}}}\\ &=-\frac {3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}+\frac {a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}-\frac {\left (5 \sqrt {1-\frac {1}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^2 \sqrt {c-\frac {c}{a x}}}-\frac {\left (7 \sqrt {1-\frac {1}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \sqrt {c-\frac {c}{a x}}}\\ &=-\frac {3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}+\frac {a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}-\frac {\left (7 \sqrt {1-\frac {1}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\sqrt {c-\frac {c}{a x}}}-\frac {\left (10 \sqrt {1-\frac {1}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {c-\frac {c}{a x}}}\\ &=-\frac {3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}+\frac {a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}+\frac {7 \sqrt {1-\frac {1}{a x}} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {c-\frac {c}{a x}}}-\frac {5 \sqrt {2} \sqrt {1-\frac {1}{a x}} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a \sqrt {c-\frac {c}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 115, normalized size = 0.53 \[ \frac {\sqrt {1-\frac {1}{a x}} \left (a x \sqrt {\frac {1}{a x}+1} (a x-3)+7 (a x-1) \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )-5 \sqrt {2} (a x-1) \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )\right )}{a (a x-1) \sqrt {c-\frac {c}{a x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 581, normalized size = 2.70 \[ \left [\frac {7 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} + \frac {5 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \log \left (-\frac {17 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 13 \, a x - \frac {4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{\sqrt {c}} - 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right )}{\sqrt {c}}}{4 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}}, \frac {5 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \sqrt {-\frac {1}{c}} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {-\frac {1}{c}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} x^{2} - 2 \, a x - 1}\right ) - 7 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 259, normalized size = 1.20 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, x -5 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x +7 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {1}{a}}\, x -6 \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-7 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+5 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) a^{\frac {3}{2}} c \sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}} \]
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 \[ \int \frac {1}{\sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {c-\frac {c}{a\,x}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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