Optimal. Leaf size=23 \[ -\frac {\tanh ^{-1}(\tanh (a+b x))}{3 x^3}-\frac {b}{6 x^2} \]
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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} \[ -\frac {\tanh ^{-1}(\tanh (a+b x))}{3 x^3}-\frac {b}{6 x^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x^4} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))}{3 x^3}+\frac {1}{3} b \int \frac {1}{x^3} \, dx\\ &=-\frac {b}{6 x^2}-\frac {\tanh ^{-1}(\tanh (a+b x))}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 0.87 \[ -\frac {2 \tanh ^{-1}(\tanh (a+b x))+b x}{6 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 13, normalized size = 0.57 \[ -\frac {3 \, b x + 2 \, a}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 13, normalized size = 0.57 \[ -\frac {3 \, b x + 2 \, a}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 20, normalized size = 0.87 \[ -\frac {b}{6 x^{2}}-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{3 x^{3}} \]
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 \[ -\frac {b}{6 \, x^{2}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 19, normalized size = 0.83 \[ -\frac {\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{3\,x^3}-\frac {b}{6\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.78, size = 20, normalized size = 0.87 \[ - \frac {b}{6 x^{2}} - \frac {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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