Optimal. Leaf size=74 \[ \frac {16 \tanh (x)}{35 a^3 \sqrt {a \text {sech}^2(x)}}+\frac {8 \tanh (x)}{35 a^2 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {6 \tanh (x)}{35 a \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}} \]
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Rubi [A] time = 0.04, 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 = {4122, 192, 191} \[ \frac {16 \tanh (x)}{35 a^3 \sqrt {a \text {sech}^2(x)}}+\frac {8 \tanh (x)}{35 a^2 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {6 \tanh (x)}{35 a \left (a \text {sech}^2(x)\right )^{5/2}}+\frac {\tanh (x)}{7 \left (a \text {sech}^2(x)\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\left (a \text {sech}^2(x)\right )^{7/2}} \, dx &=a \operatorname {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} \operatorname {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 \operatorname {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 \operatorname {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 = 0.05, size = 42, normalized size = 0.57 \[ \frac {(1225 \sinh (x)+245 \sinh (3 x)+49 \sinh (5 x)+5 \sinh (7 x)) \cosh (x) \sqrt {a \text {sech}^2(x)}}{2240 a^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 970, normalized size = 13.11 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 53, normalized size = 0.72 \[ -\frac {{\left (1225 \, e^{\left (6 \, x\right )} + 245 \, e^{\left (4 \, x\right )} + 49 \, e^{\left (2 \, x\right )} + 5\right )} e^{\left (-7 \, x\right )} - 5 \, e^{\left (7 \, x\right )} - 49 \, e^{\left (5 \, x\right )} - 245 \, e^{\left (3 \, x\right )} - 1225 \, e^{x}}{4480 \, a^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 262, normalized size = 3.54 \[ \frac {{\mathrm e}^{8 x}}{896 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}+\frac {7 \,{\mathrm e}^{6 x}}{640 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}+\frac {7 \,{\mathrm e}^{4 x}}{128 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}+\frac {35 \,{\mathrm e}^{2 x}}{128 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}-\frac {35}{128 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right ) a^{3}}-\frac {7 \,{\mathrm e}^{-2 x}}{128 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}-\frac {7 \,{\mathrm e}^{-4 x}}{640 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}-\frac {{\mathrm e}^{-6 x}}{896 a^{3} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 71, normalized size = 0.96 \[ \frac {e^{\left (7 \, x\right )}}{896 \, a^{\frac {7}{2}}} + \frac {7 \, e^{\left (5 \, x\right )}}{640 \, a^{\frac {7}{2}}} + \frac {7 \, e^{\left (3 \, x\right )}}{128 \, a^{\frac {7}{2}}} - \frac {35 \, e^{\left (-x\right )}}{128 \, a^{\frac {7}{2}}} - \frac {7 \, e^{\left (-3 \, x\right )}}{128 \, a^{\frac {7}{2}}} - \frac {7 \, e^{\left (-5 \, x\right )}}{640 \, a^{\frac {7}{2}}} - \frac {e^{\left (-7 \, x\right )}}{896 \, a^{\frac {7}{2}}} + \frac {35 \, e^{x}}{128 \, a^{\frac {7}{2}}} \]
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 \[ \int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\relax (x)}^2}\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 136.54, size = 80, normalized size = 1.08 \[ - \frac {16 \tanh ^{7}{\relax (x )}}{35 a^{\frac {7}{2}} \left (\operatorname {sech}^{2}{\relax (x )}\right )^{\frac {7}{2}}} + \frac {8 \tanh ^{5}{\relax (x )}}{5 a^{\frac {7}{2}} \left (\operatorname {sech}^{2}{\relax (x )}\right )^{\frac {7}{2}}} - \frac {2 \tanh ^{3}{\relax (x )}}{a^{\frac {7}{2}} \left (\operatorname {sech}^{2}{\relax (x )}\right )^{\frac {7}{2}}} + \frac {\tanh {\relax (x )}}{a^{\frac {7}{2}} \left (\operatorname {sech}^{2}{\relax (x )}\right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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