Optimal. Leaf size=36 \[ \frac {\tanh (x)}{3 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {2 \tanh (x)}{3 a \sqrt {a \text {sech}^2(x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4207, 198, 197}
\begin {gather*} \frac {2 \tanh (x)}{3 a \sqrt {a \text {sech}^2(x)}}+\frac {\tanh (x)}{3 \left (a \text {sech}^2(x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 4207
Rubi steps
\begin {align*} \int \frac {1}{\left (a \text {sech}^2(x)\right )^{3/2}} \, dx &=a \text {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh (x)}{3 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {2}{3} \text {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh (x)}{3 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {2 \tanh (x)}{3 a \sqrt {a \text {sech}^2(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 27, normalized size = 0.75 \begin {gather*} \frac {\text {sech}^3(x) (9 \sinh (x)+\sinh (3 x))}{12 \left (a \text {sech}^2(x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(129\) vs.
\(2(28)=56\).
time = 0.86, size = 130, normalized size = 3.61
method | result | size |
risch | \(\frac {{\mathrm e}^{4 x}}{24 a \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}+\frac {3 \,{\mathrm e}^{2 x}}{8 a \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}-\frac {3}{8 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right ) a}-\frac {{\mathrm e}^{-2 x}}{24 a \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 35, normalized size = 0.97 \begin {gather*} \frac {e^{\left (3 \, x\right )}}{24 \, a^{\frac {3}{2}}} - \frac {3 \, e^{\left (-x\right )}}{8 \, a^{\frac {3}{2}}} - \frac {e^{\left (-3 \, x\right )}}{24 \, a^{\frac {3}{2}}} + \frac {3 \, e^{x}}{8 \, a^{\frac {3}{2}}} \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. 277 vs.
\(2 (28) = 56\).
time = 0.37, size = 277, normalized size = 7.69 \begin {gather*} \frac {{\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{6} + \cosh \left (x\right )^{6} + 6 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} + {\left (5 \, \cosh \left (x\right )^{2} + 3\right )} e^{\left (2 \, x\right )} + 3\right )} \sinh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + {\left (5 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} + 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} + 18 \, \cosh \left (x\right )^{2} + {\left (5 \, \cosh \left (x\right )^{4} + 18 \, \cosh \left (x\right )^{2} - 3\right )} e^{\left (2 \, x\right )} - 3\right )} \sinh \left (x\right )^{2} - 9 \, \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{6} + 9 \, \cosh \left (x\right )^{4} - 9 \, \cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )} + 6 \, {\left (\cosh \left (x\right )^{5} + 6 \, \cosh \left (x\right )^{3} + {\left (\cosh \left (x\right )^{5} + 6 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{24 \, {\left (a^{2} \cosh \left (x\right )^{3} e^{x} + 3 \, a^{2} \cosh \left (x\right )^{2} e^{x} \sinh \left (x\right ) + 3 \, a^{2} \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{2} + a^{2} e^{x} \sinh \left (x\right )^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.38, size = 31, normalized size = 0.86 \begin {gather*} - \frac {2 \tanh ^{3}{\left (x \right )}}{3 \left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {3}{2}}} + \frac {\tanh {\left (x \right )}}{\left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 29, normalized size = 0.81 \begin {gather*} -\frac {{\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} - e^{\left (3 \, x\right )} - 9 \, e^{x}}{24 \, a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^2}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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