Optimal. Leaf size=177 \[ \frac {2 a x \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {6 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {3 \sqrt {2} \sqrt {1-\frac {1}{a x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x}} \]
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Rubi [A] time = 0.18, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6176, 6181, 94, 93, 206} \[ \frac {2 a x \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {6 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {3 \sqrt {2} \sqrt {1-\frac {1}{a x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 206
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx &=\frac {\left (\sqrt {1-\frac {1}{a x}} \sqrt {x}\right ) \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a x}} \sqrt {x}} \, dx}{\sqrt {c-a c x}}\\ &=-\frac {\sqrt {1-\frac {1}{a x}} \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^{3/2} \left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{\sqrt {\frac {1}{x}} \sqrt {c-a c x}}\\ &=\frac {2 a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {\left (6 \sqrt {1-\frac {1}{a x}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{\sqrt {x} \left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{a \sqrt {\frac {1}{x}} \sqrt {c-a c x}}\\ &=-\frac {6 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}+\frac {2 a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {\left (3 \sqrt {1-\frac {1}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a \sqrt {\frac {1}{x}} \sqrt {c-a c x}}\\ &=-\frac {6 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}+\frac {2 a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {\left (6 \sqrt {1-\frac {1}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {2 x^2}{a}} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{a \sqrt {\frac {1}{x}} \sqrt {c-a c x}}\\ &=-\frac {6 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}+\frac {2 a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {3 \sqrt {2} \sqrt {1-\frac {1}{a x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 116, normalized size = 0.66 \[ \frac {x \sqrt {1-\frac {1}{a x}} \left (2 \sqrt {a} \sqrt {\frac {1}{a x}+1} (a x-2)-3 \sqrt {2} \sqrt {\frac {1}{x}} (a x-1) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )\right )}{\sqrt {a} (a x-1) \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 288, normalized size = 1.63 \[ \left [\frac {3 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \sqrt {-\frac {1}{c}} \log \left (-\frac {a^{2} x^{2} - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {-\frac {1}{c}} + 2 \, a x - 3}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, {\left (a^{2} x^{2} - a x - 2\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}}, -\frac {2 \, {\left (a^{2} x^{2} - a x - 2\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}} - \frac {3 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x - 1\right )} \sqrt {c}}\right )}{\sqrt {c}}}{a^{3} c x^{2} - 2 \, a^{2} c x + a c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 85, normalized size = 0.48 \[ \frac {3 \, \sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right ) - 2 \, \sqrt {-a c x - c} + \frac {2 \, \sqrt {-a c x - c} c}{a c x - c}}{a c \mathrm {sgn}\left (-a c x - c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 135, normalized size = 0.76 \[ \frac {\sqrt {-c \left (a x -1\right )}\, \left (3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x a c -2 x a \sqrt {-c \left (a x +1\right )}\, \sqrt {c}-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +4 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) c^{\frac {3}{2}} \sqrt {-c \left (a x +1\right )}\, a} \]
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 \[ \int \frac {1}{\sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {c-a\,c\,x}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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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