Optimal. Leaf size=108 \[ c x \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{3/2}+\frac {2 c \sqrt {\frac {1}{a x}+1} \sqrt {1-\frac {1}{a x}}}{a}+\frac {c \csc ^{-1}(a x)}{a}-\frac {c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \]
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Rubi [A] time = 0.08, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6194, 97, 154, 21, 105, 41, 216, 92, 208} \[ c x \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{3/2}+\frac {2 c \sqrt {\frac {1}{a x}+1} \sqrt {1-\frac {1}{a x}}}{a}+\frac {c \csc ^{-1}(a x)}{a}-\frac {c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 21
Rule 41
Rule 92
Rule 97
Rule 105
Rule 154
Rule 208
Rule 216
Rule 6194
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx &=-\left (c \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \sqrt {1+\frac {x}{a}}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x-c \operatorname {Subst}\left (\int \frac {\left (-\frac {1}{a}-\frac {2 x}{a^2}\right ) \sqrt {1-\frac {x}{a}}}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{a}+c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x-(a c) \operatorname {Subst}\left (\int \frac {-\frac {1}{a^2}-\frac {x}{a^3}}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{a}+c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x+\frac {c \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {2 c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{a}+c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x+\frac {c \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^2}+\frac {c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {2 c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{a}+c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x+\frac {c \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}-\frac {c \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}\\ &=\frac {2 c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{a}+c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x+\frac {c \csc ^{-1}(a x)}{a}-\frac {c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 55, normalized size = 0.51 \[ \frac {c \left (\sqrt {1-\frac {1}{a^2 x^2}} (a x+1)-\log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+\sin ^{-1}\left (\frac {1}{a x}\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 107, normalized size = 0.99 \[ -\frac {2 \, a c x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (a^{2} c x^{2} + 2 \, a c x + c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 121, normalized size = 1.12 \[ -\frac {2 \, c \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {2 \, c \mathrm {sgn}\left (a x + 1\right )}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 166, normalized size = 1.54 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}\right )}{\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 117, normalized size = 1.08 \[ -a {\left (\frac {4 \, c \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac {2 \, c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 84, normalized size = 0.78 \[ \frac {4\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {2\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {2\,c\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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 \[ \frac {c \left (\int a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{2}}\right )\, dx\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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