Optimal. Leaf size=65 \[ -\frac {1}{2} a c x^2 \sqrt {1-\frac {1}{a^2 x^2}}+2 c x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {3 c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \]
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Rubi [A] time = 0.17, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6175, 6178, 1807, 807, 266, 63, 208} \[ -\frac {1}{2} a c x^2 \sqrt {1-\frac {1}{a^2 x^2}}+2 c x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {3 c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 1807
Rule 6175
Rule 6178
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} (c-a c x) \, dx &=-\left ((a c) \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right ) x \, dx\right )\\ &=(a c) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {1}{2} (a c) \operatorname {Subst}\left (\int \frac {\frac {4}{a}-\frac {3 x}{a^2}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=2 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {(3 c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=2 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {(3 c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a}\\ &=2 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {1}{2} (3 a c) \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=2 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {3 c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 53, normalized size = 0.82 \[ -\frac {c \left (a x \sqrt {1-\frac {1}{a^2 x^2}} (a x-4)+3 \log \left (a x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )\right )}{2 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.60, size = 81, normalized size = 1.25 \[ -\frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} c x^{2} - 3 \, a c x - 4 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 68, normalized size = 1.05 \[ \frac {3 \, c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{2 \, {\left | a \right |}} - \frac {1}{2} \, \sqrt {a^{2} x^{2} - 1} {\left (c x \mathrm {sgn}\left (a x + 1\right ) - \frac {4 \, c \mathrm {sgn}\left (a x + 1\right )}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 153, normalized size = 2.35 \[ \frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a +4 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -4 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right )}{2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 135, normalized size = 2.08 \[ \frac {1}{2} \, a {\left (\frac {2 \, {\left (5 \, c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 3 \, c \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} - \frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 96, normalized size = 1.48 \[ \frac {3\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}-5\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a-\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {3\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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 \[ - c \left (\int a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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