Optimal. Leaf size=61 \[ a \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {2}{3} a \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh (x)+\frac {1}{5} a \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^3(x) \]
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Rubi [A]
time = 0.02, antiderivative size = 61, 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 \sinh (x) \cosh (x) \sqrt {a \text {sech}^4(x)}+\frac {1}{5} a \sinh ^2(x) \tanh ^3(x) \sqrt {a \text {sech}^4(x)}-\frac {2}{3} a \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 )^{3/2} \, dx &=\left (a \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int \text {sech}^6(x) \, dx\\ &=\left (i a \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \tanh (x)\right )\\ &=a \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {2}{3} a \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh (x)+\frac {1}{5} a \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^3(x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 30, normalized size = 0.49 \begin {gather*} \frac {1}{15} \cosh (x) (8+6 \cosh (2 x)+\cosh (4 x)) \left (a \text {sech}^4(x)\right )^{3/2} \sinh (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.86, size = 46, normalized size = 0.75
method | result | size |
risch | \(-\frac {16 a \,{\mathrm e}^{-2 x} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (10 \,{\mathrm e}^{4 x}+5 \,{\mathrm e}^{2 x}+1\right )}{15 \left (1+{\mathrm e}^{2 x}\right )^{3}}\) | \(46\) |
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. 120 vs.
\(2 (51) = 102\).
time = 0.49, size = 120, normalized size = 1.97 \begin {gather*} \frac {16 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} + \frac {32 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} + \frac {16 \, a^{\frac {3}{2}}}{15 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, 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. 516 vs.
\(2 (51) = 102\).
time = 0.37, size = 516, normalized size = 8.46 \begin {gather*} -\frac {16 \, {\left (10 \, a \cosh \left (x\right )^{4} + 10 \, {\left (a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{4} + 40 \, {\left (a \cosh \left (x\right ) e^{\left (4 \, x\right )} + 2 \, a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 5 \, a \cosh \left (x\right )^{2} + 5 \, {\left (12 \, a \cosh \left (x\right )^{2} + {\left (12 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (4 \, x\right )} + 2 \, {\left (12 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} + {\left (10 \, a \cosh \left (x\right )^{4} + 5 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (4 \, x\right )} + 2 \, {\left (10 \, a \cosh \left (x\right )^{4} + 5 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + 10 \, {\left (4 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right ) + {\left (4 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} e^{\left (4 \, x\right )} + 2 \, {\left (4 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right ) + a\right )} \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} e^{\left (2 \, x\right )}}{15 \, {\left (10 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} \sinh \left (x\right )^{9} + e^{\left (2 \, x\right )} \sinh \left (x\right )^{10} + 5 \, {\left (9 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{8} + 40 \, {\left (3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{7} + 10 \, {\left (21 \, \cosh \left (x\right )^{4} + 14 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{6} + 4 \, {\left (63 \, \cosh \left (x\right )^{5} + 70 \, \cosh \left (x\right )^{3} + 15 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{5} + 10 \, {\left (21 \, \cosh \left (x\right )^{6} + 35 \, \cosh \left (x\right )^{4} + 15 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{4} + 40 \, {\left (3 \, \cosh \left (x\right )^{7} + 7 \, \cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{3} + 5 \, {\left (9 \, \cosh \left (x\right )^{8} + 28 \, \cosh \left (x\right )^{6} + 30 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{2} + 10 \, {\left (\cosh \left (x\right )^{9} + 4 \, \cosh \left (x\right )^{7} + 6 \, \cosh \left (x\right )^{5} + 4 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} \sinh \left (x\right ) + {\left (\cosh \left (x\right )^{10} + 5 \, \cosh \left (x\right )^{8} + 10 \, \cosh \left (x\right )^{6} + 10 \, \cosh \left (x\right )^{4} + 5 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, 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 \left (a \operatorname {sech}^{4}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 27, normalized size = 0.44 \begin {gather*} -\frac {16 \, a^{\frac {3}{2}} {\left (10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} + 1\right )}}{15 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.34, size = 46, normalized size = 0.75 \begin {gather*} -\frac {4\,a\,{\mathrm {e}}^{-2\,x}\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+1\right )}{15\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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