Optimal. Leaf size=64 \[ \frac {b \coth ^{-1}(\tanh (a+b x))^4}{20 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {\coth ^{-1}(\tanh (a+b x))^4}{5 x^5 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A]
time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2202, 2198}
\begin {gather*} \frac {\coth ^{-1}(\tanh (a+b x))^4}{5 x^5 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac {b \coth ^{-1}(\tanh (a+b x))^4}{20 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2198
Rule 2202
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(\tanh (a+b x))^3}{x^6} \, dx &=\frac {\coth ^{-1}(\tanh (a+b x))^4}{5 x^5 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac {b \int \frac {\coth ^{-1}(\tanh (a+b x))^3}{x^5} \, dx}{5 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ &=\frac {b \coth ^{-1}(\tanh (a+b x))^4}{20 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {\coth ^{-1}(\tanh (a+b x))^4}{5 x^5 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 54, normalized size = 0.84 \begin {gather*} -\frac {b^3 x^3+2 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+3 b x \coth ^{-1}(\tanh (a+b x))^2+4 \coth ^{-1}(\tanh (a+b x))^3}{20 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.45, size = 17234, normalized size = 269.28
method | result | size |
risch | \(\text {Expression too large to display}\) | \(17234\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.39, size = 54, normalized size = 0.84 \begin {gather*} -\frac {1}{20} \, b {\left (\frac {b^{2}}{x^{2}} + \frac {2 \, b \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )}{x^{3}}\right )} - \frac {3 \, b \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}{20 \, x^{4}} - \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 49, normalized size = 0.77 \begin {gather*} -\frac {40 \, b^{3} x^{3} + 80 \, a b^{2} x^{2} - 12 \, \pi ^{2} a + 16 \, a^{3} - 15 \, {\left (\pi ^{2} b - 4 \, a^{2} b\right )} x}{80 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.52, size = 60, normalized size = 0.94 \begin {gather*} - \frac {b^{3}}{20 x^{2}} - \frac {b^{2} \operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}{10 x^{3}} - \frac {3 b \operatorname {acoth}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{20 x^{4}} - \frac {\operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{5 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.41, size = 74, normalized size = 1.16 \begin {gather*} -\frac {40 \, b^{3} x^{3} + 40 i \, \pi b^{2} x^{2} + 80 \, a b^{2} x^{2} - 15 \, \pi ^{2} b x + 60 i \, \pi a b x + 60 \, a^{2} b x - 2 i \, \pi ^{3} - 12 \, \pi ^{2} a + 24 i \, \pi a^{2} + 16 \, a^{3}}{80 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 53, normalized size = 0.83 \begin {gather*} -\frac {{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{5\,x^5}-\frac {b^3}{20\,x^2}-\frac {b^2\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{10\,x^3}-\frac {3\,b\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{20\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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