Optimal. Leaf size=55 \[ \frac {\tan ^{-1}(\sinh (a+b x))}{8 b}-\frac {\tanh (a+b x) \text {sech}^3(a+b x)}{4 b}+\frac {\tanh (a+b x) \text {sech}(a+b x)}{8 b} \]
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Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2611, 3768, 3770} \[ \frac {\tan ^{-1}(\sinh (a+b x))}{8 b}-\frac {\tanh (a+b x) \text {sech}^3(a+b x)}{4 b}+\frac {\tanh (a+b x) \text {sech}(a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \text {sech}^3(a+b x) \tanh ^2(a+b x) \, dx &=-\frac {\text {sech}^3(a+b x) \tanh (a+b x)}{4 b}+\frac {1}{4} \int \text {sech}^3(a+b x) \, dx\\ &=\frac {\text {sech}(a+b x) \tanh (a+b x)}{8 b}-\frac {\text {sech}^3(a+b x) \tanh (a+b x)}{4 b}+\frac {1}{8} \int \text {sech}(a+b x) \, dx\\ &=\frac {\tan ^{-1}(\sinh (a+b x))}{8 b}+\frac {\text {sech}(a+b x) \tanh (a+b x)}{8 b}-\frac {\text {sech}^3(a+b x) \tanh (a+b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 55, normalized size = 1.00 \[ \frac {\tan ^{-1}(\sinh (a+b x))}{8 b}-\frac {\tanh (a+b x) \text {sech}^3(a+b x)}{4 b}+\frac {\tanh (a+b x) \text {sech}(a+b x)}{8 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 808, normalized size = 14.69 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 67, normalized size = 1.22 \[ \frac {\frac {e^{\left (7 \, b x + 7 \, a\right )} - 7 \, e^{\left (5 \, b x + 5 \, a\right )} + 7 \, e^{\left (3 \, b x + 3 \, a\right )} - e^{\left (b x + a\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{4}} + \arctan \left (e^{\left (b x + a\right )}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 69, normalized size = 1.25 \[ -\frac {\sinh \left (b x +a \right )}{3 b \cosh \left (b x +a \right )^{4}}+\frac {\mathrm {sech}\left (b x +a \right )^{3} \tanh \left (b x +a \right )}{12 b}+\frac {\mathrm {sech}\left (b x +a \right ) \tanh \left (b x +a \right )}{8 b}+\frac {\arctan \left ({\mathrm e}^{b x +a}\right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 110, normalized size = 2.00 \[ -\frac {\arctan \left (e^{\left (-b x - a\right )}\right )}{4 \, b} + \frac {e^{\left (-b x - a\right )} - 7 \, e^{\left (-3 \, b x - 3 \, a\right )} + 7 \, e^{\left (-5 \, b x - 5 \, a\right )} - e^{\left (-7 \, b x - 7 \, a\right )}}{4 \, b {\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} + 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 215, normalized size = 3.91 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{4\,\sqrt {b^2}}-\frac {\frac {{\mathrm {e}}^{a+b\,x}}{b}-\frac {2\,{\mathrm {e}}^{3\,a+3\,b\,x}}{b}+\frac {{\mathrm {e}}^{5\,a+5\,b\,x}}{b}}{4\,{\mathrm {e}}^{2\,a+2\,b\,x}+6\,{\mathrm {e}}^{4\,a+4\,b\,x}+4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1}-\frac {3\,{\mathrm {e}}^{a+b\,x}}{2\,b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}+\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}+3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}+1\right )}+\frac {{\mathrm {e}}^{a+b\,x}}{4\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
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 \[ \int \tanh ^{2}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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