Optimal. Leaf size=74 \[ \frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}}+\frac {6 \tanh (x)}{35 a \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {8 \tanh (x)}{35 a^2 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {16 \tanh (x)}{35 a^3 \sqrt {a \text {sech}^2(x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {16 \tanh (x)}{35 a^3 \sqrt {a \text {sech}^2(x)}}+\frac {8 \tanh (x)}{35 a^2 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {6 \tanh (x)}{35 a \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}} \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 )^{7/2}} \, dx &=a \text {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{9/2}} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}}+\frac {6}{7} \text {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{7/2}} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}}+\frac {6 \tanh (x)}{35 a \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {24 \text {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )}{35 a}\\ &=\frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}}+\frac {6 \tanh (x)}{35 a \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {8 \tanh (x)}{35 a^2 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {16 \text {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )}{35 a^2}\\ &=\frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}}+\frac {6 \tanh (x)}{35 a \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {8 \tanh (x)}{35 a^2 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {16 \tanh (x)}{35 a^3 \sqrt {a \text {sech}^2(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 42, normalized size = 0.57 \begin {gather*} \frac {\cosh (x) \sqrt {a \text {sech}^2(x)} (1225 \sinh (x)+245 \sinh (3 x)+49 \sinh (5 x)+5 \sinh (7 x))}{2240 a^4} \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. \(261\) vs.
\(2(58)=116\).
time = 1.16, size = 262, normalized size = 3.54
method | result | size |
risch | \(\frac {{\mathrm e}^{8 x}}{896 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}+\frac {7 \,{\mathrm e}^{6 x}}{640 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}+\frac {7 \,{\mathrm e}^{4 x}}{128 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}+\frac {35 \,{\mathrm e}^{2 x}}{128 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}-\frac {35}{128 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right ) a^{3}}-\frac {7 \,{\mathrm e}^{-2 x}}{128 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}-\frac {7 \,{\mathrm e}^{-4 x}}{640 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}-\frac {{\mathrm e}^{-6 x}}{896 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}\) | \(262\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 71, normalized size = 0.96 \begin {gather*} \frac {e^{\left (7 \, x\right )}}{896 \, a^{\frac {7}{2}}} + \frac {7 \, e^{\left (5 \, x\right )}}{640 \, a^{\frac {7}{2}}} + \frac {7 \, e^{\left (3 \, x\right )}}{128 \, a^{\frac {7}{2}}} - \frac {35 \, e^{\left (-x\right )}}{128 \, a^{\frac {7}{2}}} - \frac {7 \, e^{\left (-3 \, x\right )}}{128 \, a^{\frac {7}{2}}} - \frac {7 \, e^{\left (-5 \, x\right )}}{640 \, a^{\frac {7}{2}}} - \frac {e^{\left (-7 \, x\right )}}{896 \, a^{\frac {7}{2}}} + \frac {35 \, e^{x}}{128 \, a^{\frac {7}{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. 970 vs.
\(2 (58) = 116\).
time = 0.40, size = 970, normalized size = 13.11 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 12.86, size = 66, normalized size = 0.89 \begin {gather*} - \frac {16 \tanh ^{7}{\left (x \right )}}{35 \left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {7}{2}}} + \frac {8 \tanh ^{5}{\left (x \right )}}{5 \left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {2 \tanh ^{3}{\left (x \right )}}{\left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {7}{2}}} + \frac {\tanh {\left (x \right )}}{\left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.37, size = 53, normalized size = 0.72 \begin {gather*} -\frac {{\left (1225 \, e^{\left (6 \, x\right )} + 245 \, e^{\left (4 \, x\right )} + 49 \, e^{\left (2 \, x\right )} + 5\right )} e^{\left (-7 \, x\right )} - 5 \, e^{\left (7 \, x\right )} - 49 \, e^{\left (5 \, x\right )} - 245 \, e^{\left (3 \, x\right )} - 1225 \, e^{x}}{4480 \, a^{\frac {7}{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 )}^2}\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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