Optimal. Leaf size=23 \[ -\frac {\coth ^{-1}(\tanh (a+b x))}{2 x^2}-\frac {b}{2 x} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2168, 30} \[ -\frac {\coth ^{-1}(\tanh (a+b x))}{2 x^2}-\frac {b}{2 x} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(\tanh (a+b x))}{x^3} \, dx &=-\frac {\coth ^{-1}(\tanh (a+b x))}{2 x^2}+\frac {1}{2} b \int \frac {1}{x^2} \, dx\\ &=-\frac {b}{2 x}-\frac {\coth ^{-1}(\tanh (a+b x))}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.78 \[ -\frac {\coth ^{-1}(\tanh (a+b x))+b x}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 11, normalized size = 0.48 \[ -\frac {2 \, b x + a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 20, normalized size = 0.87 \[ -\frac {b}{2 x}-\frac {\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 19, normalized size = 0.83 \[ -\frac {b}{2 \, x} - \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 16, normalized size = 0.70 \[ -\frac {\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )+b\,x}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 19, normalized size = 0.83 \[ - \frac {b}{2 x} - \frac {\operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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