Optimal. Leaf size=104 \[ \frac {x \sqrt {\frac {1}{a x}+1}}{c \sqrt {1-\frac {1}{a x}}}-\frac {2 \sqrt {\frac {1}{a x}+1}}{a c \sqrt {1-\frac {1}{a x}}}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c} \]
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Rubi [A] time = 0.08, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6194, 103, 21, 94, 92, 208} \[ \frac {x \sqrt {\frac {1}{a x}+1}}{c \sqrt {1-\frac {1}{a x}}}-\frac {2 \sqrt {\frac {1}{a x}+1}}{a c \sqrt {1-\frac {1}{a x}}}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c} \]
Antiderivative was successfully verified.
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Rule 21
Rule 92
Rule 94
Rule 103
Rule 208
Rule 6194
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\sqrt {1+\frac {1}{a x}} x}{c \sqrt {1-\frac {1}{a x}}}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{a}-\frac {x}{a^2}}{x \left (1-\frac {x}{a}\right )^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\sqrt {1+\frac {1}{a x}} x}{c \sqrt {1-\frac {1}{a x}}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x \left (1-\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{a c}\\ &=-\frac {2 \sqrt {1+\frac {1}{a x}}}{a c \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \sqrt {1-\frac {1}{a x}}}-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a c}\\ &=-\frac {2 \sqrt {1+\frac {1}{a x}}}{a c \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \sqrt {1-\frac {1}{a x}}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a^2 c}\\ &=-\frac {2 \sqrt {1+\frac {1}{a x}}}{a c \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \sqrt {1-\frac {1}{a x}}}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 56, normalized size = 0.54 \[ \frac {\frac {x \sqrt {1-\frac {1}{a^2 x^2}} (a x-2)}{a x-1}+\frac {\log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{a}}{c} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.50, size = 93, normalized size = 0.89 \[ \frac {{\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} x^{2} - a x - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c x - a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 127, normalized size = 1.22 \[ a {\left (\frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {\log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c} - \frac {\frac {3 \, {\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c {\left (\frac {{\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} - \sqrt {\frac {a x - 1}{a x + 1}}\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 251, normalized size = 2.41 \[ \frac {2 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+3 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}-4 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-\left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a +2 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )+3 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{2 a \sqrt {a^{2}}\, \left (a x -1\right ) c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\frac {a x -1}{a x +1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 116, normalized size = 1.12 \[ -a {\left (\frac {\frac {3 \, {\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{2} c \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 62, normalized size = 0.60 \[ \frac {2\,a\,x+4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}-4}{2\,a\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]
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 \[ \frac {a^{2} \int \frac {x^{2}}{a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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