Optimal. Leaf size=43 \[ \frac {\cosh ^2(a+b x)}{2 b}-\frac {\text {sech}^2(a+b x)}{2 b}-\frac {2 \log (\cosh (a+b x))}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2590, 266, 43} \[ \frac {\cosh ^2(a+b x)}{2 b}-\frac {\text {sech}^2(a+b x)}{2 b}-\frac {2 \log (\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2590
Rubi steps
\begin {align*} \int \sinh ^2(a+b x) \tanh ^3(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^3} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(1-x)^2}{x^2} \, dx,x,\cosh ^2(a+b x)\right )}{2 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {1}{x^2}-\frac {2}{x}\right ) \, dx,x,\cosh ^2(a+b x)\right )}{2 b}\\ &=\frac {\cosh ^2(a+b x)}{2 b}-\frac {2 \log (\cosh (a+b x))}{b}-\frac {\text {sech}^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 35, normalized size = 0.81 \[ -\frac {-\sinh ^2(a+b x)+\text {sech}^2(a+b x)+4 \log (\cosh (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 742, normalized size = 17.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 96, normalized size = 2.23 \[ \frac {16 \, b x - {\left (8 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + \frac {8 \, {\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} + 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 3\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{2}} + e^{\left (2 \, b x + 2 \, a\right )} - 16 \, \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 47, normalized size = 1.09 \[ \frac {\sinh ^{4}\left (b x +a \right )}{2 b \cosh \left (b x +a \right )^{2}}-\frac {2 \ln \left (\cosh \left (b x +a \right )\right )}{b}+\frac {\tanh ^{2}\left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 103, normalized size = 2.40 \[ -\frac {2 \, {\left (b x + a\right )}}{b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} - \frac {2 \, \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} + \frac {2 \, e^{\left (-2 \, b x - 2 \, a\right )} - 15 \, e^{\left (-4 \, b x - 4 \, a\right )} + 1}{8 \, b {\left (e^{\left (-2 \, b x - 2 \, a\right )} + 2 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 97, normalized size = 2.26 \[ 2\,x-\frac {2\,\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1\right )}{b}-\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}+\frac {2}{b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}+\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{8\,b}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}}{8\,b} \]
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 \[ \int \sinh ^{2}{\left (a + b x \right )} \tanh ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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