Optimal. Leaf size=25 \[ -\frac {1}{7} \text {sech}^7(x)+\frac {2 \text {sech}^9(x)}{9}-\frac {\text {sech}^{11}(x)}{11} \]
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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2686, 276}
\begin {gather*} -\frac {1}{11} \text {sech}^{11}(x)+\frac {2 \text {sech}^9(x)}{9}-\frac {\text {sech}^7(x)}{7} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 2686
Rubi steps
\begin {align*} \int \text {sech}^7(x) \tanh ^5(x) \, dx &=-\text {Subst}\left (\int x^6 \left (-1+x^2\right )^2 \, dx,x,\text {sech}(x)\right )\\ &=-\text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\text {sech}(x)\right )\\ &=-\frac {1}{7} \text {sech}^7(x)+\frac {2 \text {sech}^9(x)}{9}-\frac {\text {sech}^{11}(x)}{11}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} -\frac {1}{7} \text {sech}^7(x)+\frac {2 \text {sech}^9(x)}{9}-\frac {\text {sech}^{11}(x)}{11} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 28, normalized size = 1.12
method | result | size |
default | \(-\frac {\sinh ^{4}\left (x \right )}{7 \cosh \left (x \right )^{11}}-\frac {4 \left (\sinh ^{2}\left (x \right )\right )}{63 \cosh \left (x \right )^{11}}-\frac {8}{693 \cosh \left (x \right )^{11}}\) | \(28\) |
risch | \(-\frac {128 \,{\mathrm e}^{7 x} \left (99 \,{\mathrm e}^{8 x}-220 \,{\mathrm e}^{6 x}+370 \,{\mathrm e}^{4 x}-220 \,{\mathrm e}^{2 x}+99\right )}{693 \left (1+{\mathrm e}^{2 x}\right )^{11}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 371 vs.
\(2 (19) = 38\).
time = 0.27, size = 371, normalized size = 14.84 \begin {gather*} -\frac {128 \, e^{\left (-7 \, x\right )}}{7 \, {\left (11 \, e^{\left (-2 \, x\right )} + 55 \, e^{\left (-4 \, x\right )} + 165 \, e^{\left (-6 \, x\right )} + 330 \, e^{\left (-8 \, x\right )} + 462 \, e^{\left (-10 \, x\right )} + 462 \, e^{\left (-12 \, x\right )} + 330 \, e^{\left (-14 \, x\right )} + 165 \, e^{\left (-16 \, x\right )} + 55 \, e^{\left (-18 \, x\right )} + 11 \, e^{\left (-20 \, x\right )} + e^{\left (-22 \, x\right )} + 1\right )}} + \frac {2560 \, e^{\left (-9 \, x\right )}}{63 \, {\left (11 \, e^{\left (-2 \, x\right )} + 55 \, e^{\left (-4 \, x\right )} + 165 \, e^{\left (-6 \, x\right )} + 330 \, e^{\left (-8 \, x\right )} + 462 \, e^{\left (-10 \, x\right )} + 462 \, e^{\left (-12 \, x\right )} + 330 \, e^{\left (-14 \, x\right )} + 165 \, e^{\left (-16 \, x\right )} + 55 \, e^{\left (-18 \, x\right )} + 11 \, e^{\left (-20 \, x\right )} + e^{\left (-22 \, x\right )} + 1\right )}} - \frac {47360 \, e^{\left (-11 \, x\right )}}{693 \, {\left (11 \, e^{\left (-2 \, x\right )} + 55 \, e^{\left (-4 \, x\right )} + 165 \, e^{\left (-6 \, x\right )} + 330 \, e^{\left (-8 \, x\right )} + 462 \, e^{\left (-10 \, x\right )} + 462 \, e^{\left (-12 \, x\right )} + 330 \, e^{\left (-14 \, x\right )} + 165 \, e^{\left (-16 \, x\right )} + 55 \, e^{\left (-18 \, x\right )} + 11 \, e^{\left (-20 \, x\right )} + e^{\left (-22 \, x\right )} + 1\right )}} + \frac {2560 \, e^{\left (-13 \, x\right )}}{63 \, {\left (11 \, e^{\left (-2 \, x\right )} + 55 \, e^{\left (-4 \, x\right )} + 165 \, e^{\left (-6 \, x\right )} + 330 \, e^{\left (-8 \, x\right )} + 462 \, e^{\left (-10 \, x\right )} + 462 \, e^{\left (-12 \, x\right )} + 330 \, e^{\left (-14 \, x\right )} + 165 \, e^{\left (-16 \, x\right )} + 55 \, e^{\left (-18 \, x\right )} + 11 \, e^{\left (-20 \, x\right )} + e^{\left (-22 \, x\right )} + 1\right )}} - \frac {128 \, e^{\left (-15 \, x\right )}}{7 \, {\left (11 \, e^{\left (-2 \, x\right )} + 55 \, e^{\left (-4 \, x\right )} + 165 \, e^{\left (-6 \, x\right )} + 330 \, e^{\left (-8 \, x\right )} + 462 \, e^{\left (-10 \, x\right )} + 462 \, e^{\left (-12 \, x\right )} + 330 \, e^{\left (-14 \, x\right )} + 165 \, e^{\left (-16 \, x\right )} + 55 \, e^{\left (-18 \, x\right )} + 11 \, e^{\left (-20 \, x\right )} + e^{\left (-22 \, x\right )} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 634 vs.
\(2 (19) = 38\).
time = 0.39, size = 634, normalized size = 25.36 \begin {gather*} -\frac {128 \, {\left (99 \, \cosh \left (x\right )^{8} + 792 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + 99 \, \sinh \left (x\right )^{8} + 44 \, {\left (63 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{6} - 220 \, \cosh \left (x\right )^{6} + 264 \, {\left (21 \, \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 10 \, {\left (693 \, \cosh \left (x\right )^{4} - 330 \, \cosh \left (x\right )^{2} + 37\right )} \sinh \left (x\right )^{4} + 370 \, \cosh \left (x\right )^{4} + 8 \, {\left (693 \, \cosh \left (x\right )^{5} - 550 \, \cosh \left (x\right )^{3} + 185 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (693 \, \cosh \left (x\right )^{6} - 825 \, \cosh \left (x\right )^{4} + 555 \, \cosh \left (x\right )^{2} - 55\right )} \sinh \left (x\right )^{2} - 220 \, \cosh \left (x\right )^{2} + 8 \, {\left (99 \, \cosh \left (x\right )^{7} - 165 \, \cosh \left (x\right )^{5} + 185 \, \cosh \left (x\right )^{3} - 55 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 99\right )}}{693 \, {\left (\cosh \left (x\right )^{15} + 15 \, \cosh \left (x\right ) \sinh \left (x\right )^{14} + \sinh \left (x\right )^{15} + {\left (105 \, \cosh \left (x\right )^{2} + 11\right )} \sinh \left (x\right )^{13} + 11 \, \cosh \left (x\right )^{13} + 13 \, {\left (35 \, \cosh \left (x\right )^{3} + 11 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{12} + {\left (1365 \, \cosh \left (x\right )^{4} + 858 \, \cosh \left (x\right )^{2} + 55\right )} \sinh \left (x\right )^{11} + 55 \, \cosh \left (x\right )^{11} + 11 \, {\left (273 \, \cosh \left (x\right )^{5} + 286 \, \cosh \left (x\right )^{3} + 55 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{10} + 55 \, {\left (91 \, \cosh \left (x\right )^{6} + 143 \, \cosh \left (x\right )^{4} + 55 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{9} + 165 \, \cosh \left (x\right )^{9} + 33 \, {\left (195 \, \cosh \left (x\right )^{7} + 429 \, \cosh \left (x\right )^{5} + 275 \, \cosh \left (x\right )^{3} + 45 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{8} + {\left (6435 \, \cosh \left (x\right )^{8} + 18876 \, \cosh \left (x\right )^{6} + 18150 \, \cosh \left (x\right )^{4} + 5940 \, \cosh \left (x\right )^{2} + 329\right )} \sinh \left (x\right )^{7} + 331 \, \cosh \left (x\right )^{7} + {\left (5005 \, \cosh \left (x\right )^{9} + 18876 \, \cosh \left (x\right )^{7} + 25410 \, \cosh \left (x\right )^{5} + 13860 \, \cosh \left (x\right )^{3} + 2317 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{6} + {\left (3003 \, \cosh \left (x\right )^{10} + 14157 \, \cosh \left (x\right )^{8} + 25410 \, \cosh \left (x\right )^{6} + 20790 \, \cosh \left (x\right )^{4} + 6909 \, \cosh \left (x\right )^{2} + 451\right )} \sinh \left (x\right )^{5} + 473 \, \cosh \left (x\right )^{5} + 5 \, {\left (273 \, \cosh \left (x\right )^{11} + 1573 \, \cosh \left (x\right )^{9} + 3630 \, \cosh \left (x\right )^{7} + 4158 \, \cosh \left (x\right )^{5} + 2317 \, \cosh \left (x\right )^{3} + 473 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + {\left (455 \, \cosh \left (x\right )^{12} + 3146 \, \cosh \left (x\right )^{10} + 9075 \, \cosh \left (x\right )^{8} + 13860 \, \cosh \left (x\right )^{6} + 11515 \, \cosh \left (x\right )^{4} + 4510 \, \cosh \left (x\right )^{2} + 407\right )} \sinh \left (x\right )^{3} + 517 \, \cosh \left (x\right )^{3} + {\left (105 \, \cosh \left (x\right )^{13} + 858 \, \cosh \left (x\right )^{11} + 3025 \, \cosh \left (x\right )^{9} + 5940 \, \cosh \left (x\right )^{7} + 6951 \, \cosh \left (x\right )^{5} + 4730 \, \cosh \left (x\right )^{3} + 1551 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (15 \, \cosh \left (x\right )^{14} + 143 \, \cosh \left (x\right )^{12} + 605 \, \cosh \left (x\right )^{10} + 1485 \, \cosh \left (x\right )^{8} + 2303 \, \cosh \left (x\right )^{6} + 2255 \, \cosh \left (x\right )^{4} + 1221 \, \cosh \left (x\right )^{2} + 165\right )} \sinh \left (x\right ) + 495 \, \cosh \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.60, size = 34, normalized size = 1.36 \begin {gather*} - \frac {\tanh ^{4}{\left (x \right )} \operatorname {sech}^{7}{\left (x \right )}}{11} - \frac {4 \tanh ^{2}{\left (x \right )} \operatorname {sech}^{7}{\left (x \right )}}{99} - \frac {8 \operatorname {sech}^{7}{\left (x \right )}}{693} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 35, normalized size = 1.40 \begin {gather*} -\frac {128 \, {\left (99 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 616 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 1008\right )}}{693 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.55, size = 520, normalized size = 20.80 \begin {gather*} \frac {\frac {64\,{\mathrm {e}}^{5\,x}}{11}-\frac {320\,{\mathrm {e}}^{7\,x}}{11}+\frac {640\,{\mathrm {e}}^{9\,x}}{11}-\frac {640\,{\mathrm {e}}^{11\,x}}{11}+\frac {320\,{\mathrm {e}}^{13\,x}}{11}-\frac {64\,{\mathrm {e}}^{15\,x}}{11}}{11\,{\mathrm {e}}^{2\,x}+55\,{\mathrm {e}}^{4\,x}+165\,{\mathrm {e}}^{6\,x}+330\,{\mathrm {e}}^{8\,x}+462\,{\mathrm {e}}^{10\,x}+462\,{\mathrm {e}}^{12\,x}+330\,{\mathrm {e}}^{14\,x}+165\,{\mathrm {e}}^{16\,x}+55\,{\mathrm {e}}^{18\,x}+11\,{\mathrm {e}}^{20\,x}+{\mathrm {e}}^{22\,x}+1}-\frac {38464\,{\mathrm {e}}^x}{693\,\left (6\,{\mathrm {e}}^{2\,x}+15\,{\mathrm {e}}^{4\,x}+20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}+6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1\right )}-\frac {640\,{\mathrm {e}}^x}{33\,\left (8\,{\mathrm {e}}^{2\,x}+28\,{\mathrm {e}}^{4\,x}+56\,{\mathrm {e}}^{6\,x}+70\,{\mathrm {e}}^{8\,x}+56\,{\mathrm {e}}^{10\,x}+28\,{\mathrm {e}}^{12\,x}+8\,{\mathrm {e}}^{14\,x}+{\mathrm {e}}^{16\,x}+1\right )}-\frac {104\,{\mathrm {e}}^x}{21\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {1664\,{\mathrm {e}}^x}{63\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1\right )}+\frac {4096\,{\mathrm {e}}^x}{77\,\left (7\,{\mathrm {e}}^{2\,x}+21\,{\mathrm {e}}^{4\,x}+35\,{\mathrm {e}}^{6\,x}+35\,{\mathrm {e}}^{8\,x}+21\,{\mathrm {e}}^{10\,x}+7\,{\mathrm {e}}^{12\,x}+{\mathrm {e}}^{14\,x}+1\right )}+\frac {\frac {16\,{\mathrm {e}}^{3\,x}}{11}-\frac {112\,{\mathrm {e}}^{5\,x}}{11}+\frac {288\,{\mathrm {e}}^{7\,x}}{11}-32\,{\mathrm {e}}^{9\,x}+\frac {208\,{\mathrm {e}}^{11\,x}}{11}-\frac {48\,{\mathrm {e}}^{13\,x}}{11}}{10\,{\mathrm {e}}^{2\,x}+45\,{\mathrm {e}}^{4\,x}+120\,{\mathrm {e}}^{6\,x}+210\,{\mathrm {e}}^{8\,x}+252\,{\mathrm {e}}^{10\,x}+210\,{\mathrm {e}}^{12\,x}+120\,{\mathrm {e}}^{14\,x}+45\,{\mathrm {e}}^{16\,x}+10\,{\mathrm {e}}^{18\,x}+{\mathrm {e}}^{20\,x}+1}-\frac {\frac {280\,{\mathrm {e}}^{3\,x}}{99}-\frac {112\,{\mathrm {e}}^{5\,x}}{11}+16\,{\mathrm {e}}^{7\,x}-\frac {104\,{\mathrm {e}}^{9\,x}}{9}+\frac {104\,{\mathrm {e}}^{11\,x}}{33}-\frac {8\,{\mathrm {e}}^x}{33}}{9\,{\mathrm {e}}^{2\,x}+36\,{\mathrm {e}}^{4\,x}+84\,{\mathrm {e}}^{6\,x}+126\,{\mathrm {e}}^{8\,x}+126\,{\mathrm {e}}^{10\,x}+84\,{\mathrm {e}}^{12\,x}+36\,{\mathrm {e}}^{14\,x}+9\,{\mathrm {e}}^{16\,x}+{\mathrm {e}}^{18\,x}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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