Optimal. Leaf size=101 \[ \frac {16 \tanh (a+b x)}{35 b \sqrt {\text {sech}^2(a+b x)}}+\frac {8 \tanh (a+b x)}{35 b \text {sech}^2(a+b x)^{3/2}}+\frac {6 \tanh (a+b x)}{35 b \text {sech}^2(a+b x)^{5/2}}+\frac {\tanh (a+b x)}{7 b \text {sech}^2(a+b x)^{7/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4122, 192, 191} \[ \frac {16 \tanh (a+b x)}{35 b \sqrt {\text {sech}^2(a+b x)}}+\frac {8 \tanh (a+b x)}{35 b \text {sech}^2(a+b x)^{3/2}}+\frac {6 \tanh (a+b x)}{35 b \text {sech}^2(a+b x)^{5/2}}+\frac {\tanh (a+b x)}{7 b \text {sech}^2(a+b x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\text {sech}^2(a+b x)^{7/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{9/2}} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac {\tanh (a+b x)}{7 b \text {sech}^2(a+b x)^{7/2}}+\frac {6 \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{7/2}} \, dx,x,\tanh (a+b x)\right )}{7 b}\\ &=\frac {\tanh (a+b x)}{7 b \text {sech}^2(a+b x)^{7/2}}+\frac {6 \tanh (a+b x)}{35 b \text {sech}^2(a+b x)^{5/2}}+\frac {24 \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{5/2}} \, dx,x,\tanh (a+b x)\right )}{35 b}\\ &=\frac {\tanh (a+b x)}{7 b \text {sech}^2(a+b x)^{7/2}}+\frac {6 \tanh (a+b x)}{35 b \text {sech}^2(a+b x)^{5/2}}+\frac {8 \tanh (a+b x)}{35 b \text {sech}^2(a+b x)^{3/2}}+\frac {16 \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{3/2}} \, dx,x,\tanh (a+b x)\right )}{35 b}\\ &=\frac {\tanh (a+b x)}{7 b \text {sech}^2(a+b x)^{7/2}}+\frac {6 \tanh (a+b x)}{35 b \text {sech}^2(a+b x)^{5/2}}+\frac {8 \tanh (a+b x)}{35 b \text {sech}^2(a+b x)^{3/2}}+\frac {16 \tanh (a+b x)}{35 b \sqrt {\text {sech}^2(a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 57, normalized size = 0.56 \[ \frac {\left (5 \sinh ^6(a+b x)+21 \sinh ^4(a+b x)+35 \sinh ^2(a+b x)+35\right ) \tanh (a+b x)}{35 b \sqrt {\text {sech}^2(a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 108, normalized size = 1.07 \[ \frac {5 \, \sinh \left (b x + a\right )^{7} + 7 \, {\left (15 \, \cosh \left (b x + a\right )^{2} + 7\right )} \sinh \left (b x + a\right )^{5} + 35 \, {\left (5 \, \cosh \left (b x + a\right )^{4} + 14 \, \cosh \left (b x + a\right )^{2} + 7\right )} \sinh \left (b x + a\right )^{3} + 35 \, {\left (\cosh \left (b x + a\right )^{6} + 7 \, \cosh \left (b x + a\right )^{4} + 21 \, \cosh \left (b x + a\right )^{2} + 35\right )} \sinh \left (b x + a\right )}{2240 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 92, normalized size = 0.91 \[ -\frac {{\left (1225 \, e^{\left (6 \, b x + 6 \, a\right )} + 245 \, e^{\left (4 \, b x + 4 \, a\right )} + 49 \, e^{\left (2 \, b x + 2 \, a\right )} + 5\right )} e^{\left (-7 \, b x - 7 \, a\right )} - 5 \, e^{\left (7 \, b x + 7 \, a\right )} - 49 \, e^{\left (5 \, b x + 5 \, a\right )} - 245 \, e^{\left (3 \, b x + 3 \, a\right )} - 1225 \, e^{\left (b x + a\right )}}{4480 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.42, size = 409, normalized size = 4.05 \[ \frac {{\mathrm e}^{8 b x +8 a}}{896 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}+\frac {7 \,{\mathrm e}^{6 b x +6 a}}{640 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}+\frac {7 \,{\mathrm e}^{4 b x +4 a}}{128 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}+\frac {35 \,{\mathrm e}^{2 b x +2 a}}{128 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}-\frac {35}{128 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}-\frac {7 \,{\mathrm e}^{-2 b x -2 a}}{128 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}-\frac {7 \,{\mathrm e}^{-4 b x -4 a}}{640 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}-\frac {{\mathrm e}^{-6 b x -6 a}}{896 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 100, normalized size = 0.99 \[ \frac {{\left (49 \, e^{\left (-2 \, b x - 2 \, a\right )} + 245 \, e^{\left (-4 \, b x - 4 \, a\right )} + 1225 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5\right )} e^{\left (7 \, b x + 7 \, a\right )}}{4480 \, b} - \frac {1225 \, e^{\left (-b x - a\right )} + 245 \, e^{\left (-3 \, b x - 3 \, a\right )} + 49 \, e^{\left (-5 \, b x - 5 \, a\right )} + 5 \, e^{\left (-7 \, b x - 7 \, a\right )}}{4480 \, b} \]
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 \[ \int \frac {1}{{\left (\frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^2}\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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