Optimal. Leaf size=38 \[ \frac {1}{16} \text {ArcTan}(\sinh (x))+\frac {1}{16} \text {sech}(x) \tanh (x)-\frac {1}{8} \text {sech}^3(x) \tanh (x)-\frac {1}{6} \text {sech}^3(x) \tanh ^3(x) \]
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Rubi [A]
time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2691, 3853,
3855} \begin {gather*} \frac {1}{16} \text {ArcTan}(\sinh (x))-\frac {1}{6} \tanh ^3(x) \text {sech}^3(x)-\frac {1}{8} \tanh (x) \text {sech}^3(x)+\frac {1}{16} \tanh (x) \text {sech}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2691
Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int \text {sech}^3(x) \tanh ^4(x) \, dx &=-\frac {1}{6} \text {sech}^3(x) \tanh ^3(x)+\frac {1}{2} \int \text {sech}^3(x) \tanh ^2(x) \, dx\\ &=-\frac {1}{8} \text {sech}^3(x) \tanh (x)-\frac {1}{6} \text {sech}^3(x) \tanh ^3(x)+\frac {1}{8} \int \text {sech}^3(x) \, dx\\ &=\frac {1}{16} \text {sech}(x) \tanh (x)-\frac {1}{8} \text {sech}^3(x) \tanh (x)-\frac {1}{6} \text {sech}^3(x) \tanh ^3(x)+\frac {1}{16} \int \text {sech}(x) \, dx\\ &=\frac {1}{16} \tan ^{-1}(\sinh (x))+\frac {1}{16} \text {sech}(x) \tanh (x)-\frac {1}{8} \text {sech}^3(x) \tanh (x)-\frac {1}{6} \text {sech}^3(x) \tanh ^3(x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 48, normalized size = 1.26 \begin {gather*} \frac {1}{16} \text {ArcTan}(\sinh (x))+\frac {1}{16} \text {sech}(x) \tanh (x)+\frac {1}{24} \text {sech}^3(x) \tanh (x)-\frac {1}{6} \text {sech}^5(x) \tanh (x)-\frac {1}{3} \text {sech}^3(x) \tanh ^3(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.54, size = 64, normalized size = 1.68
method | result | size |
risch | \(\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{10 x}-47 \,{\mathrm e}^{8 x}+78 \,{\mathrm e}^{6 x}-78 \,{\mathrm e}^{4 x}+47 \,{\mathrm e}^{2 x}-3\right )}{24 \left (1+{\mathrm e}^{2 x}\right )^{6}}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{16}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{16}\) | \(64\) |
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. 85 vs.
\(2 (30) = 60\).
time = 0.48, size = 85, normalized size = 2.24 \begin {gather*} \frac {3 \, e^{\left (-x\right )} - 47 \, e^{\left (-3 \, x\right )} + 78 \, e^{\left (-5 \, x\right )} - 78 \, e^{\left (-7 \, x\right )} + 47 \, e^{\left (-9 \, x\right )} - 3 \, e^{\left (-11 \, x\right )}}{24 \, {\left (6 \, e^{\left (-2 \, x\right )} + 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} + e^{\left (-12 \, x\right )} + 1\right )}} - \frac {1}{8} \, \arctan \left (e^{\left (-x\right )}\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. 925 vs.
\(2 (30) = 60\).
time = 0.36, size = 925, normalized size = 24.34 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \tanh ^{4}{\left (x \right )} \operatorname {sech}^{3}{\left (x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs.
\(2 (30) = 60\).
time = 0.40, size = 73, normalized size = 1.92 \begin {gather*} \frac {1}{32} \, \pi - \frac {3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{5} - 32 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - 48 \, e^{\left (-x\right )} + 48 \, e^{x}}{24 \, {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{3}} + \frac {1}{16} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 200, normalized size = 5.26 \begin {gather*} \frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{8}-\frac {10\,{\mathrm {e}}^x}{4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {{\mathrm {e}}^x}{8\,\left ({\mathrm {e}}^{2\,x}+1\right )}+\frac {7\,{\mathrm {e}}^x}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {4\,{\mathrm {e}}^{5\,x}-\frac {8\,{\mathrm {e}}^{3\,x}}{3}-\frac {8\,{\mathrm {e}}^{7\,x}}{3}+\frac {2\,{\mathrm {e}}^{9\,x}}{3}+\frac {2\,{\mathrm {e}}^x}{3}}{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}+\frac {16\,{\mathrm {e}}^x}{3\,\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 {23\,{\mathrm {e}}^x}{12\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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