Optimal. Leaf size=31 \[ \frac {\text {sech}^5(a+b x)}{5 b}-\frac {\text {sech}^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2606, 14} \[ \frac {\text {sech}^5(a+b x)}{5 b}-\frac {\text {sech}^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2606
Rubi steps
\begin {align*} \int \text {sech}^3(a+b x) \tanh ^3(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\text {sech}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\text {sech}(a+b x)\right )}{b}\\ &=-\frac {\text {sech}^3(a+b x)}{3 b}+\frac {\text {sech}^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 31, normalized size = 1.00 \[ \frac {\text {sech}^5(a+b x)}{5 b}-\frac {\text {sech}^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 345, normalized size = 11.13 \[ -\frac {8 \, {\left (5 \, \cosh \left (b x + a\right )^{4} + 20 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 5 \, \sinh \left (b x + a\right )^{4} + 2 \, {\left (15 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 5\right )}}{15 \, {\left (b \cosh \left (b x + a\right )^{7} + 7 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{6} + b \sinh \left (b x + a\right )^{7} + 5 \, b \cosh \left (b x + a\right )^{5} + {\left (21 \, b \cosh \left (b x + a\right )^{2} + 5 \, b\right )} \sinh \left (b x + a\right )^{5} + 5 \, {\left (7 \, b \cosh \left (b x + a\right )^{3} + 5 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{4} + 11 \, b \cosh \left (b x + a\right )^{3} + {\left (35 \, b \cosh \left (b x + a\right )^{4} + 50 \, b \cosh \left (b x + a\right )^{2} + 9 \, b\right )} \sinh \left (b x + a\right )^{3} + {\left (21 \, b \cosh \left (b x + a\right )^{5} + 50 \, b \cosh \left (b x + a\right )^{3} + 33 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 15 \, b \cosh \left (b x + a\right ) + {\left (7 \, b \cosh \left (b x + a\right )^{6} + 25 \, b \cosh \left (b x + a\right )^{4} + 27 \, b \cosh \left (b x + a\right )^{2} + 5 \, b\right )} \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 52, normalized size = 1.68 \[ -\frac {8 \, {\left (5 \, e^{\left (7 \, b x + 7 \, a\right )} - 2 \, e^{\left (5 \, b x + 5 \, a\right )} + 5 \, e^{\left (3 \, b x + 3 \, a\right )}\right )}}{15 \, b {\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 34, normalized size = 1.10 \[ \frac {-\frac {\sinh ^{2}\left (b x +a \right )}{3 \cosh \left (b x +a \right )^{5}}-\frac {2}{15 \cosh \left (b x +a \right )^{5}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 214, normalized size = 6.90 \[ -\frac {8 \, e^{\left (-3 \, b x - 3 \, a\right )}}{3 \, b {\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} + 1\right )}} + \frac {16 \, e^{\left (-5 \, b x - 5 \, a\right )}}{15 \, b {\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} + 1\right )}} - \frac {8 \, e^{\left (-7 \, b x - 7 \, a\right )}}{3 \, b {\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.54, size = 251, normalized size = 8.10 \[ \frac {\frac {4\,{\mathrm {e}}^{a+b\,x}}{5\,b}-\frac {12\,{\mathrm {e}}^{3\,a+3\,b\,x}}{5\,b}+\frac {12\,{\mathrm {e}}^{5\,a+5\,b\,x}}{5\,b}-\frac {4\,{\mathrm {e}}^{7\,a+7\,b\,x}}{5\,b}}{5\,{\mathrm {e}}^{2\,a+2\,b\,x}+10\,{\mathrm {e}}^{4\,a+4\,b\,x}+10\,{\mathrm {e}}^{6\,a+6\,b\,x}+5\,{\mathrm {e}}^{8\,a+8\,b\,x}+{\mathrm {e}}^{10\,a+10\,b\,x}+1}-\frac {28\,{\mathrm {e}}^{a+b\,x}}{15\,b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}+\frac {64\,{\mathrm {e}}^{a+b\,x}}{15\,b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}+3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}+1\right )}-\frac {16\,{\mathrm {e}}^{a+b\,x}}{5\,b\,\left (4\,{\mathrm {e}}^{2\,a+2\,b\,x}+6\,{\mathrm {e}}^{4\,a+4\,b\,x}+4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.72, size = 46, normalized size = 1.48 \[ \begin {cases} - \frac {\tanh ^{2}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}}{5 b} - \frac {2 \operatorname {sech}^{3}{\left (a + b x \right )}}{15 b} & \text {for}\: b \neq 0 \\x \tanh ^{3}{\relax (a )} \operatorname {sech}^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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