Optimal. Leaf size=16 \[ \frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 b} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2188, 30}
\begin {gather*} \frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2188
Rubi steps
\begin {align*} \int \tanh ^{-1}(\tanh (a+b x))^3 \, dx &=\frac {\text {Subst}\left (\int x^3 \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b}\\ &=\frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 30.43, size = 15, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{4 b}\) | \(15\) |
default | \(\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{4 b}\) | \(15\) |
risch | \(\text {Expression too large to display}\) | \(6400\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (14) = 28\).
time = 0.38, size = 51, normalized size = 3.19 \begin {gather*} -\frac {3}{2} \, b x^{2} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} + x \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac {1}{4} \, {\left (b^{2} x^{4} - 4 \, b x^{3} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (14) = 28\).
time = 0.32, size = 31, normalized size = 1.94 \begin {gather*} \frac {1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac {3}{2} \, a^{2} b x^{2} + a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 20, normalized size = 1.25 \begin {gather*} \begin {cases} \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{4 b} & \text {for}\: b \neq 0 \\x \operatorname {atanh}^{3}{\left (\tanh {\left (a \right )} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (14) = 28\).
time = 0.38, size = 31, normalized size = 1.94 \begin {gather*} \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} a^{2} + \frac {1}{4} \, {\left (b x^{2} + 2 \, a x\right )}^{2} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 47, normalized size = 2.94 \begin {gather*} \frac {x\,\left (2\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )-b\,x\right )\,\left (b^2\,x^2-2\,b\,x\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )+2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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