Integrand size = 12, antiderivative size = 101 \[ \int \frac {1}{\text {sech}^2(a+b x)^{7/2}} \, dx=\frac {\tanh (a+b x)}{7 b \text {sech}^2(a+b x)^{7/2}}+\frac {6 \tanh (a+b x)}{35 b \text {sech}^2(a+b x)^{5/2}}+\frac {8 \tanh (a+b x)}{35 b \text {sech}^2(a+b x)^{3/2}}+\frac {16 \tanh (a+b x)}{35 b \sqrt {\text {sech}^2(a+b x)}} \]
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Time = 0.03 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4207, 198, 197} \[ \int \frac {1}{\text {sech}^2(a+b x)^{7/2}} \, dx=\frac {16 \tanh (a+b x)}{35 b \sqrt {\text {sech}^2(a+b x)}}+\frac {8 \tanh (a+b x)}{35 b \text {sech}^2(a+b x)^{3/2}}+\frac {6 \tanh (a+b x)}{35 b \text {sech}^2(a+b x)^{5/2}}+\frac {\tanh (a+b x)}{7 b \text {sech}^2(a+b x)^{7/2}} \]
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Rule 197
Rule 198
Rule 4207
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 \text {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 \text {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 \text {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*}
Time = 0.15 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\text {sech}^2(a+b x)^{7/2}} \, dx=\frac {\left (35+35 \sinh ^2(a+b x)+21 \sinh ^4(a+b x)+5 \sinh ^6(a+b x)\right ) \tanh (a+b x)}{35 b \sqrt {\text {sech}^2(a+b x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(408\) vs. \(2(85)=170\).
Time = 0.48 (sec) , antiderivative size = 409, normalized size of antiderivative = 4.05
method | result | size |
risch | \(\frac {{\mathrm e}^{8 b x +8 a}}{896 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}+\frac {7 \,{\mathrm e}^{6 b x +6 a}}{640 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}+\frac {7 \,{\mathrm e}^{4 b x +4 a}}{128 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}+\frac {35 \,{\mathrm e}^{2 b x +2 a}}{128 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}-\frac {35}{128 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}-\frac {7 \,{\mathrm e}^{-2 b x -2 a}}{128 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}-\frac {7 \,{\mathrm e}^{-4 b x -4 a}}{640 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}-\frac {{\mathrm e}^{-6 b x -6 a}}{896 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}\) | \(409\) |
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Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\text {sech}^2(a+b x)^{7/2}} \, dx=\frac {5 \, \sinh \left (b x + a\right )^{7} + 7 \, {\left (15 \, \cosh \left (b x + a\right )^{2} + 7\right )} \sinh \left (b x + a\right )^{5} + 35 \, {\left (5 \, \cosh \left (b x + a\right )^{4} + 14 \, \cosh \left (b x + a\right )^{2} + 7\right )} \sinh \left (b x + a\right )^{3} + 35 \, {\left (\cosh \left (b x + a\right )^{6} + 7 \, \cosh \left (b x + a\right )^{4} + 21 \, \cosh \left (b x + a\right )^{2} + 35\right )} \sinh \left (b x + a\right )}{2240 \, b} \]
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Time = 17.86 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\text {sech}^2(a+b x)^{7/2}} \, dx=\begin {cases} - \frac {16 \tanh ^{7}{\left (a + b x \right )}}{35 b \left (\operatorname {sech}^{2}{\left (a + b x \right )}\right )^{\frac {7}{2}}} + \frac {8 \tanh ^{5}{\left (a + b x \right )}}{5 b \left (\operatorname {sech}^{2}{\left (a + b x \right )}\right )^{\frac {7}{2}}} - \frac {2 \tanh ^{3}{\left (a + b x \right )}}{b \left (\operatorname {sech}^{2}{\left (a + b x \right )}\right )^{\frac {7}{2}}} + \frac {\tanh {\left (a + b x \right )}}{b \left (\operatorname {sech}^{2}{\left (a + b x \right )}\right )^{\frac {7}{2}}} & \text {for}\: b \neq 0 \\\frac {x}{\left (\operatorname {sech}^{2}{\left (a \right )}\right )^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\text {sech}^2(a+b x)^{7/2}} \, dx=\frac {{\left (49 \, e^{\left (-2 \, b x - 2 \, a\right )} + 245 \, e^{\left (-4 \, b x - 4 \, a\right )} + 1225 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5\right )} e^{\left (7 \, b x + 7 \, a\right )}}{4480 \, b} - \frac {1225 \, e^{\left (-b x - a\right )} + 245 \, e^{\left (-3 \, b x - 3 \, a\right )} + 49 \, e^{\left (-5 \, b x - 5 \, a\right )} + 5 \, e^{\left (-7 \, b x - 7 \, a\right )}}{4480 \, b} \]
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Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\text {sech}^2(a+b x)^{7/2}} \, dx=-\frac {{\left (1225 \, e^{\left (6 \, b x + 6 \, a\right )} + 245 \, e^{\left (4 \, b x + 4 \, a\right )} + 49 \, e^{\left (2 \, b x + 2 \, a\right )} + 5\right )} e^{\left (-7 \, b x - 7 \, a\right )} - 5 \, e^{\left (7 \, b x + 7 \, a\right )} - 49 \, e^{\left (5 \, b x + 5 \, a\right )} - 245 \, e^{\left (3 \, b x + 3 \, a\right )} - 1225 \, e^{\left (b x + a\right )}}{4480 \, b} \]
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Timed out. \[ \int \frac {1}{\text {sech}^2(a+b x)^{7/2}} \, dx=\int \frac {1}{{\left (\frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^2}\right )}^{7/2}} \,d x \]
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