Optimal. Leaf size=40 \[ \frac {\cosh ^4(a+b x)}{4 b}-\frac {\cosh ^2(a+b x)}{b}+\frac {\log (\cosh (a+b x))}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2590, 266, 43} \[ \frac {\cosh ^4(a+b x)}{4 b}-\frac {\cosh ^2(a+b x)}{b}+\frac {\log (\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2590
Rubi steps
\begin {align*} \int \sinh ^4(a+b x) \tanh (a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(1-x)^2}{x} \, dx,x,\cosh ^2(a+b x)\right )}{2 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-2+\frac {1}{x}+x\right ) \, dx,x,\cosh ^2(a+b x)\right )}{2 b}\\ &=-\frac {\cosh ^2(a+b x)}{b}+\frac {\cosh ^4(a+b x)}{4 b}+\frac {\log (\cosh (a+b x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 34, normalized size = 0.85 \[ \frac {\frac {1}{4} \cosh ^4(a+b x)-\cosh ^2(a+b x)+\log (\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 457, normalized size = 11.42 \[ \frac {\cosh \left (b x + a\right )^{8} + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + \sinh \left (b x + a\right )^{8} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{2} - 3\right )} \sinh \left (b x + a\right )^{6} - 64 \, b x \cosh \left (b x + a\right )^{4} - 12 \, \cosh \left (b x + a\right )^{6} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{3} - 9 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 2 \, {\left (35 \, \cosh \left (b x + a\right )^{4} - 32 \, b x - 90 \, \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{4} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{5} - 32 \, b x \cosh \left (b x + a\right ) - 30 \, \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )^{3} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{6} - 96 \, b x \cosh \left (b x + a\right )^{2} - 45 \, \cosh \left (b x + a\right )^{4} - 3\right )} \sinh \left (b x + a\right )^{2} - 12 \, \cosh \left (b x + a\right )^{2} + 64 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4}\right )} \log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 8 \, {\left (\cosh \left (b x + a\right )^{7} - 32 \, b x \cosh \left (b x + a\right )^{3} - 9 \, \cosh \left (b x + a\right )^{5} - 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{64 \, {\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 86, normalized size = 2.15 \[ -\frac {64 \, b x - {\left (48 \, e^{\left (4 \, b x + 4 \, a\right )} - 12 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )} - {\left (e^{\left (4 \, b x + 16 \, a\right )} - 12 \, e^{\left (2 \, b x + 14 \, a\right )}\right )} e^{\left (-12 \, a\right )} - 64 \, \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{64 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 39, normalized size = 0.98 \[ \frac {\sinh ^{4}\left (b x +a \right )}{4 b}-\frac {\sinh ^{2}\left (b x +a \right )}{2 b}+\frac {\ln \left (\cosh \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 81, normalized size = 2.02 \[ -\frac {{\left (12 \, e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} + \frac {b x + a}{b} - \frac {12 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} + \frac {\log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 77, normalized size = 1.92 \[ \frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1\right )}{b}-x-\frac {3\,{\mathrm {e}}^{-2\,a-2\,b\,x}}{16\,b}-\frac {3\,{\mathrm {e}}^{2\,a+2\,b\,x}}{16\,b}+\frac {{\mathrm {e}}^{-4\,a-4\,b\,x}}{64\,b}+\frac {{\mathrm {e}}^{4\,a+4\,b\,x}}{64\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh ^{4}{\left (a + b x \right )} \tanh {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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