Optimal. Leaf size=35 \[ \frac {\tanh (\pi x)}{\pi }-\frac {2 \tanh ^3(\pi x)}{3 \pi }+\frac {\tanh ^5(\pi x)}{5 \pi } \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3852}
\begin {gather*} \frac {\tanh ^5(\pi x)}{5 \pi }-\frac {2 \tanh ^3(\pi x)}{3 \pi }+\frac {\tanh (\pi x)}{\pi } \end {gather*}
Antiderivative was successfully verified.
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Rule 3852
Rubi steps
\begin {align*} \int \text {sech}^6(\pi x) \, dx &=\frac {i \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \tanh (\pi x)\right )}{\pi }\\ &=\frac {\tanh (\pi x)}{\pi }-\frac {2 \tanh ^3(\pi x)}{3 \pi }+\frac {\tanh ^5(\pi x)}{5 \pi }\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 35, normalized size = 1.00 \begin {gather*} \frac {\tanh (\pi x)}{\pi }-\frac {2 \tanh ^3(\pi x)}{3 \pi }+\frac {\tanh ^5(\pi x)}{5 \pi } \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.19, size = 31, normalized size = 0.89
method | result | size |
risch | \(-\frac {16 \left (10 \,{\mathrm e}^{4 \pi x}+5 \,{\mathrm e}^{2 \pi x}+1\right )}{15 \pi \left ({\mathrm e}^{2 \pi x}+1\right )^{5}}\) | \(31\) |
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. 137 vs.
\(2 (31) = 62\).
time = 0.28, size = 137, normalized size = 3.91 \begin {gather*} \frac {16 \, e^{\left (-2 \, \pi x\right )}}{3 \, \pi {\left (5 \, e^{\left (-2 \, \pi x\right )} + 10 \, e^{\left (-4 \, \pi x\right )} + 10 \, e^{\left (-6 \, \pi x\right )} + 5 \, e^{\left (-8 \, \pi x\right )} + e^{\left (-10 \, \pi x\right )} + 1\right )}} + \frac {32 \, e^{\left (-4 \, \pi x\right )}}{3 \, \pi {\left (5 \, e^{\left (-2 \, \pi x\right )} + 10 \, e^{\left (-4 \, \pi x\right )} + 10 \, e^{\left (-6 \, \pi x\right )} + 5 \, e^{\left (-8 \, \pi x\right )} + e^{\left (-10 \, \pi x\right )} + 1\right )}} + \frac {16}{15 \, \pi {\left (5 \, e^{\left (-2 \, \pi x\right )} + 10 \, e^{\left (-4 \, \pi x\right )} + 10 \, e^{\left (-6 \, \pi x\right )} + 5 \, e^{\left (-8 \, \pi x\right )} + e^{\left (-10 \, \pi 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. 280 vs.
\(2 (31) = 62\).
time = 0.40, size = 280, normalized size = 8.00 \begin {gather*} -\frac {16 \, {\left (11 \, \cosh \left (\pi x\right )^{2} + 18 \, \cosh \left (\pi x\right ) \sinh \left (\pi x\right ) + 11 \, \sinh \left (\pi x\right )^{2} + 5\right )}}{15 \, {\left (5 \, \pi + \pi \cosh \left (\pi x\right )^{8} + 8 \, \pi \cosh \left (\pi x\right ) \sinh \left (\pi x\right )^{7} + \pi \sinh \left (\pi x\right )^{8} + 5 \, \pi \cosh \left (\pi x\right )^{6} + {\left (5 \, \pi + 28 \, \pi \cosh \left (\pi x\right )^{2}\right )} \sinh \left (\pi x\right )^{6} + 2 \, {\left (28 \, \pi \cosh \left (\pi x\right )^{3} + 15 \, \pi \cosh \left (\pi x\right )\right )} \sinh \left (\pi x\right )^{5} + 10 \, \pi \cosh \left (\pi x\right )^{4} + 5 \, {\left (2 \, \pi + 14 \, \pi \cosh \left (\pi x\right )^{4} + 15 \, \pi \cosh \left (\pi x\right )^{2}\right )} \sinh \left (\pi x\right )^{4} + 4 \, {\left (14 \, \pi \cosh \left (\pi x\right )^{5} + 25 \, \pi \cosh \left (\pi x\right )^{3} + 10 \, \pi \cosh \left (\pi x\right )\right )} \sinh \left (\pi x\right )^{3} + 11 \, \pi \cosh \left (\pi x\right )^{2} + {\left (11 \, \pi + 28 \, \pi \cosh \left (\pi x\right )^{6} + 75 \, \pi \cosh \left (\pi x\right )^{4} + 60 \, \pi \cosh \left (\pi x\right )^{2}\right )} \sinh \left (\pi x\right )^{2} + 2 \, {\left (4 \, \pi \cosh \left (\pi x\right )^{7} + 15 \, \pi \cosh \left (\pi x\right )^{5} + 20 \, \pi \cosh \left (\pi x\right )^{3} + 9 \, \pi \cosh \left (\pi x\right )\right )} \sinh \left (\pi x\right )\right )}} \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 \operatorname {sech}^{6}{\left (\pi x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 30, normalized size = 0.86 \begin {gather*} -\frac {16 \, {\left (10 \, e^{\left (4 \, \pi x\right )} + 5 \, e^{\left (2 \, \pi x\right )} + 1\right )}}{15 \, \pi {\left (e^{\left (2 \, \pi x\right )} + 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.52, size = 30, normalized size = 0.86 \begin {gather*} -\frac {16\,\left (5\,{\mathrm {e}}^{2\,\Pi \,x}+10\,{\mathrm {e}}^{4\,\Pi \,x}+1\right )}{15\,\Pi \,{\left ({\mathrm {e}}^{2\,\Pi \,x}+1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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