Optimal. Leaf size=132 \[ \frac {63 x \text {sech}^2(x)}{256 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {21 \cosh (x) \sinh (x)}{128 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {21 \cosh ^3(x) \sinh (x)}{160 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {9 \cosh ^5(x) \sinh (x)}{80 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {\cosh ^7(x) \sinh (x)}{10 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {63 \tanh (x)}{256 a^2 \sqrt {a \text {sech}^4(x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4208, 2715, 8}
\begin {gather*} \frac {63 x \text {sech}^2(x)}{256 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {63 \tanh (x)}{256 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {\sinh (x) \cosh ^7(x)}{10 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {9 \sinh (x) \cosh ^5(x)}{80 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {21 \sinh (x) \cosh ^3(x)}{160 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {21 \sinh (x) \cosh (x)}{128 a^2 \sqrt {a \text {sech}^4(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 4208
Rubi steps
\begin {align*} \int \frac {1}{\left (a \text {sech}^4(x)\right )^{5/2}} \, dx &=\frac {\text {sech}^2(x) \int \cosh ^{10}(x) \, dx}{a^2 \sqrt {a \text {sech}^4(x)}}\\ &=\frac {\cosh ^7(x) \sinh (x)}{10 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {\left (9 \text {sech}^2(x)\right ) \int \cosh ^8(x) \, dx}{10 a^2 \sqrt {a \text {sech}^4(x)}}\\ &=\frac {9 \cosh ^5(x) \sinh (x)}{80 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {\cosh ^7(x) \sinh (x)}{10 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {\left (63 \text {sech}^2(x)\right ) \int \cosh ^6(x) \, dx}{80 a^2 \sqrt {a \text {sech}^4(x)}}\\ &=\frac {21 \cosh ^3(x) \sinh (x)}{160 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {9 \cosh ^5(x) \sinh (x)}{80 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {\cosh ^7(x) \sinh (x)}{10 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {\left (21 \text {sech}^2(x)\right ) \int \cosh ^4(x) \, dx}{32 a^2 \sqrt {a \text {sech}^4(x)}}\\ &=\frac {21 \cosh (x) \sinh (x)}{128 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {21 \cosh ^3(x) \sinh (x)}{160 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {9 \cosh ^5(x) \sinh (x)}{80 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {\cosh ^7(x) \sinh (x)}{10 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {\left (63 \text {sech}^2(x)\right ) \int \cosh ^2(x) \, dx}{128 a^2 \sqrt {a \text {sech}^4(x)}}\\ &=\frac {21 \cosh (x) \sinh (x)}{128 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {21 \cosh ^3(x) \sinh (x)}{160 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {9 \cosh ^5(x) \sinh (x)}{80 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {\cosh ^7(x) \sinh (x)}{10 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {63 \tanh (x)}{256 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {\left (63 \text {sech}^2(x)\right ) \int 1 \, dx}{256 a^2 \sqrt {a \text {sech}^4(x)}}\\ &=\frac {63 x \text {sech}^2(x)}{256 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {21 \cosh (x) \sinh (x)}{128 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {21 \cosh ^3(x) \sinh (x)}{160 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {9 \cosh ^5(x) \sinh (x)}{80 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {\cosh ^7(x) \sinh (x)}{10 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {63 \tanh (x)}{256 a^2 \sqrt {a \text {sech}^4(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 55, normalized size = 0.42 \begin {gather*} \frac {\cosh ^2(x) \sqrt {a \text {sech}^4(x)} (2520 x+2100 \sinh (2 x)+600 \sinh (4 x)+150 \sinh (6 x)+25 \sinh (8 x)+2 \sinh (10 x))}{10240 a^3} \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. \(361\) vs.
\(2(108)=216\).
time = 0.93, size = 362, normalized size = 2.74
method | result | size |
risch | \(\frac {63 \,{\mathrm e}^{2 x} x}{256 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}+\frac {{\mathrm e}^{12 x}}{10240 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}+\frac {5 \,{\mathrm e}^{10 x}}{4096 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}+\frac {15 \,{\mathrm e}^{8 x}}{2048 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}+\frac {15 \,{\mathrm e}^{6 x}}{512 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}+\frac {105 \,{\mathrm e}^{4 x}}{1024 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}-\frac {105}{1024 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2} a^{2}}-\frac {15 \,{\mathrm e}^{-2 x}}{512 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}-\frac {15 \,{\mathrm e}^{-4 x}}{2048 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}-\frac {5 \,{\mathrm e}^{-6 x}}{4096 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}-\frac {{\mathrm e}^{-8 x}}{10240 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}\) | \(362\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 103, normalized size = 0.78 \begin {gather*} \frac {{\left (25 \, \sqrt {a} e^{\left (-2 \, x\right )} + 150 \, \sqrt {a} e^{\left (-4 \, x\right )} + 600 \, \sqrt {a} e^{\left (-6 \, x\right )} + 2100 \, \sqrt {a} e^{\left (-8 \, x\right )} - 2100 \, \sqrt {a} e^{\left (-12 \, x\right )} - 600 \, \sqrt {a} e^{\left (-14 \, x\right )} - 150 \, \sqrt {a} e^{\left (-16 \, x\right )} - 25 \, \sqrt {a} e^{\left (-18 \, x\right )} - 2 \, \sqrt {a} e^{\left (-20 \, x\right )} + 2 \, \sqrt {a}\right )} e^{\left (10 \, x\right )}}{20480 \, a^{3}} + \frac {63 \, x}{256 \, a^{\frac {5}{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. 2600 vs.
\(2 (108) = 216\).
time = 0.38, size = 2600, normalized size = 19.70 \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 \frac {1}{\left (a \operatorname {sech}^{4}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 76, normalized size = 0.58 \begin {gather*} -\frac {{\left (5754 \, e^{\left (10 \, x\right )} + 2100 \, e^{\left (8 \, x\right )} + 600 \, e^{\left (6 \, x\right )} + 150 \, e^{\left (4 \, x\right )} + 25 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-10 \, x\right )} - 5040 \, x - 2 \, e^{\left (10 \, x\right )} - 25 \, e^{\left (8 \, x\right )} - 150 \, e^{\left (6 \, x\right )} - 600 \, e^{\left (4 \, x\right )} - 2100 \, e^{\left (2 \, x\right )}}{20480 \, a^{\frac {5}{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 = -0.01 \begin {gather*} \int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^4}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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