Optimal. Leaf size=151 \[ \frac {x \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{3/2}}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {\left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\sqrt {2} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{a \left (c-\frac {c}{a x}\right )^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 151, 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, 103, 21, 83, 63, 208, 206} \[ \frac {x \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{3/2}}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {\left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\sqrt {2} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{a \left (c-\frac {c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 63
Rule 83
Rule 103
Rule 206
Rule 208
Rule 6179
Rule 6182
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx &=\frac {\left (1-\frac {1}{a x}\right )^{3/2} \int \frac {e^{-\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{3/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{3/2}}\\ &=-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{3/2}}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {\left (1-\frac {1}{a x}\right )^{3/2} \operatorname {Subst}\left (\int \frac {-\frac {1}{2 a}-\frac {x}{2 a^2}}{x \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{3/2}}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{2 a \left (c-\frac {c}{a x}\right )^{3/2}}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \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 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \left (c-\frac {c}{a x}\right )^{3/2}}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \operatorname {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\left (2 \left (1-\frac {1}{a x}\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{3/2}}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {\left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\sqrt {2} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a \left (c-\frac {c}{a x}\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 91, normalized size = 0.60 \[ \frac {\left (1-\frac {1}{a x}\right )^{3/2} \left (a x \sqrt {\frac {1}{a x}+1}+\tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )\right )}{a \left (c-\frac {c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 522, normalized size = 3.46 \[ \left [\frac {{\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} + \frac {\sqrt {2} {\left (a c x - c\right )} \log \left (-\frac {17 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 13 \, a x - \frac {4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{\sqrt {c}} - 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right )}{\sqrt {c}}}{4 \, {\left (a^{2} c^{2} x - a c^{2}\right )}}, \frac {\sqrt {2} {\left (a c x - c\right )} \sqrt {-\frac {1}{c}} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {-\frac {1}{c}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} x^{2} - 2 \, a x - 1}\right ) - {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} c^{2} x - a c^{2}\right )}}\right ] \]
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 [A] time = 0.07, size = 162, normalized size = 1.07 \[ \frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+\ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-\sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{2 a^{\frac {3}{2}} c^{2} \left (a x -1\right ) \sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}} \]
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 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}} \,d x \]
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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