Optimal. Leaf size=163 \[ a^3 \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-2 a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh (x)+3 a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^3(x)-\frac {20}{7} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^5(x)+\frac {5}{3} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^7(x)-\frac {6}{11} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^9(x)+\frac {1}{13} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^{11}(x) \]
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Rubi [A]
time = 0.03, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4208, 3852}
\begin {gather*} a^3 \sinh (x) \cosh (x) \sqrt {a \text {sech}^4(x)}+\frac {1}{13} a^3 \sinh ^2(x) \tanh ^{11}(x) \sqrt {a \text {sech}^4(x)}-\frac {6}{11} a^3 \sinh ^2(x) \tanh ^9(x) \sqrt {a \text {sech}^4(x)}+\frac {5}{3} a^3 \sinh ^2(x) \tanh ^7(x) \sqrt {a \text {sech}^4(x)}-\frac {20}{7} a^3 \sinh ^2(x) \tanh ^5(x) \sqrt {a \text {sech}^4(x)}+3 a^3 \sinh ^2(x) \tanh ^3(x) \sqrt {a \text {sech}^4(x)}-2 a^3 \sinh ^2(x) \tanh (x) \sqrt {a \text {sech}^4(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3852
Rule 4208
Rubi steps
\begin {align*} \int \left (a \text {sech}^4(x)\right )^{7/2} \, dx &=\left (a^3 \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int \text {sech}^{14}(x) \, dx\\ &=\left (i a^3 \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \text {Subst}\left (\int \left (1+6 x^2+15 x^4+20 x^6+15 x^8+6 x^{10}+x^{12}\right ) \, dx,x,-i \tanh (x)\right )\\ &=a^3 \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-2 a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh (x)+3 a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^3(x)-\frac {20}{7} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^5(x)+\frac {5}{3} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^7(x)-\frac {6}{11} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^9(x)+\frac {1}{13} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^{11}(x)\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 54, normalized size = 0.33 \begin {gather*} \frac {\cosh (x) (2048+2380 \cosh (2 x)+1093 \cosh (4 x)+378 \cosh (6 x)+92 \cosh (8 x)+14 \cosh (10 x)+\cosh (12 x)) \left (a \text {sech}^4(x)\right )^{7/2} \sinh (x)}{6006} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.99, size = 72, normalized size = 0.44
method | result | size |
risch | \(-\frac {2048 a^{3} {\mathrm e}^{-2 x} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1716 \,{\mathrm e}^{12 x}+1287 \,{\mathrm e}^{10 x}+715 \,{\mathrm e}^{8 x}+286 \,{\mathrm e}^{6 x}+78 \,{\mathrm e}^{4 x}+13 \,{\mathrm e}^{2 x}+1\right )}{3003 \left (1+{\mathrm e}^{2 x}\right )^{11}}\) | \(72\) |
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. 620 vs.
\(2 (141) = 282\).
time = 0.54, size = 620, normalized size = 3.80 \begin {gather*} \frac {2048 \, a^{\frac {7}{2}} e^{\left (-2 \, x\right )}}{231 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {4096 \, a^{\frac {7}{2}} e^{\left (-4 \, x\right )}}{77 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {4096 \, a^{\frac {7}{2}} e^{\left (-6 \, x\right )}}{21 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {10240 \, a^{\frac {7}{2}} e^{\left (-8 \, x\right )}}{21 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {6144 \, a^{\frac {7}{2}} e^{\left (-10 \, x\right )}}{7 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {8192 \, a^{\frac {7}{2}} e^{\left (-12 \, x\right )}}{7 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, x\right )} + 1\right )}} + \frac {2048 \, a^{\frac {7}{2}}}{3003 \, {\left (13 \, e^{\left (-2 \, x\right )} + 78 \, e^{\left (-4 \, x\right )} + 286 \, e^{\left (-6 \, x\right )} + 715 \, e^{\left (-8 \, x\right )} + 1287 \, e^{\left (-10 \, x\right )} + 1716 \, e^{\left (-12 \, x\right )} + 1716 \, e^{\left (-14 \, x\right )} + 1287 \, e^{\left (-16 \, x\right )} + 715 \, e^{\left (-18 \, x\right )} + 286 \, e^{\left (-20 \, x\right )} + 78 \, e^{\left (-22 \, x\right )} + 13 \, e^{\left (-24 \, x\right )} + e^{\left (-26 \, 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. 2804 vs.
\(2 (141) = 282\).
time = 0.44, size = 2804, normalized size = 17.20 \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 51, normalized size = 0.31 \begin {gather*} -\frac {2048 \, a^{\frac {7}{2}} {\left (1716 \, e^{\left (12 \, x\right )} + 1287 \, e^{\left (10 \, x\right )} + 715 \, e^{\left (8 \, x\right )} + 286 \, e^{\left (6 \, x\right )} + 78 \, e^{\left (4 \, x\right )} + 13 \, e^{\left (2 \, x\right )} + 1\right )}}{3003 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.45, size = 498, normalized size = 3.06 \begin {gather*} \frac {1536\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^8\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {2048\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{7\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^7\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {10240\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^9\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}+\frac {4096\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^{10}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {30720\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{11\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^{11}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}+\frac {1024\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^{12}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {2048\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{13\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^{13}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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