Optimal. Leaf size=42 \[ -\frac {\tanh ^4(a+b x)}{4 b}-\frac {\tanh ^2(a+b x)}{2 b}+\frac {\log (\cosh (a+b x))}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3473, 3475} \[ -\frac {\tanh ^4(a+b x)}{4 b}-\frac {\tanh ^2(a+b x)}{2 b}+\frac {\log (\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin {align*} \int \tanh ^5(a+b x) \, dx &=-\frac {\tanh ^4(a+b x)}{4 b}+\int \tanh ^3(a+b x) \, dx\\ &=-\frac {\tanh ^2(a+b x)}{2 b}-\frac {\tanh ^4(a+b x)}{4 b}+\int \tanh (a+b x) \, dx\\ &=\frac {\log (\cosh (a+b x))}{b}-\frac {\tanh ^2(a+b x)}{2 b}-\frac {\tanh ^4(a+b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 37, normalized size = 0.88 \[ \frac {-\tanh ^4(a+b x)-2 \tanh ^2(a+b x)+4 \log (\cosh (a+b x))}{4 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 968, normalized size = 23.05 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 67, normalized size = 1.60 \[ -\frac {b x + a - \frac {4 \, {\left (e^{\left (6 \, b x + 6 \, a\right )} + e^{\left (4 \, b x + 4 \, a\right )} + e^{\left (2 \, b x + 2 \, a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{4}} - \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 1.33 \[ -\frac {\tanh ^{4}\left (b x +a \right )}{4 b}-\frac {\tanh ^{2}\left (b x +a \right )}{2 b}-\frac {\ln \left (-1+\tanh \left (b x +a \right )\right )}{2 b}-\frac {\ln \left (1+\tanh \left (b x +a \right )\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.17, size = 102, normalized size = 2.43 \[ x + \frac {a}{b} + \frac {\log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} + \frac {4 \, {\left (e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}}{b {\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} + 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 37, normalized size = 0.88 \[ x-\frac {\ln \left (\mathrm {tanh}\left (a+b\,x\right )+1\right )+\frac {{\mathrm {tanh}\left (a+b\,x\right )}^2}{2}+\frac {{\mathrm {tanh}\left (a+b\,x\right )}^4}{4}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 42, normalized size = 1.00 \[ \begin {cases} x - \frac {\log {\left (\tanh {\left (a + b x \right )} + 1 \right )}}{b} - \frac {\tanh ^{4}{\left (a + b x \right )}}{4 b} - \frac {\tanh ^{2}{\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\x \tanh ^{5}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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