Integrand size = 12, antiderivative size = 67 \[ \int x \cosh ^7\left (a+b x^2\right ) \, dx=\frac {\sinh \left (a+b x^2\right )}{2 b}+\frac {\sinh ^3\left (a+b x^2\right )}{2 b}+\frac {3 \sinh ^5\left (a+b x^2\right )}{10 b}+\frac {\sinh ^7\left (a+b x^2\right )}{14 b} \]
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Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5429, 2713} \[ \int x \cosh ^7\left (a+b x^2\right ) \, dx=\frac {\sinh ^7\left (a+b x^2\right )}{14 b}+\frac {3 \sinh ^5\left (a+b x^2\right )}{10 b}+\frac {\sinh ^3\left (a+b x^2\right )}{2 b}+\frac {\sinh \left (a+b x^2\right )}{2 b} \]
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Rule 2713
Rule 5429
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \cosh ^7(a+b x) \, dx,x,x^2\right ) \\ & = \frac {i \text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-i \sinh \left (a+b x^2\right )\right )}{2 b} \\ & = \frac {\sinh \left (a+b x^2\right )}{2 b}+\frac {\sinh ^3\left (a+b x^2\right )}{2 b}+\frac {3 \sinh ^5\left (a+b x^2\right )}{10 b}+\frac {\sinh ^7\left (a+b x^2\right )}{14 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int x \cosh ^7\left (a+b x^2\right ) \, dx=\frac {\sinh \left (a+b x^2\right )}{2 b}+\frac {\sinh ^3\left (a+b x^2\right )}{2 b}+\frac {3 \sinh ^5\left (a+b x^2\right )}{10 b}+\frac {\sinh ^7\left (a+b x^2\right )}{14 b} \]
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Time = 0.79 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\left (\frac {16}{35}+\frac {\cosh \left (b \,x^{2}+a \right )^{6}}{7}+\frac {6 \cosh \left (b \,x^{2}+a \right )^{4}}{35}+\frac {8 \cosh \left (b \,x^{2}+a \right )^{2}}{35}\right ) \sinh \left (b \,x^{2}+a \right )}{2 b}\) | \(52\) |
default | \(\frac {\left (\frac {16}{35}+\frac {\cosh \left (b \,x^{2}+a \right )^{6}}{7}+\frac {6 \cosh \left (b \,x^{2}+a \right )^{4}}{35}+\frac {8 \cosh \left (b \,x^{2}+a \right )^{2}}{35}\right ) \sinh \left (b \,x^{2}+a \right )}{2 b}\) | \(52\) |
parallelrisch | \(\frac {5 \sinh \left (7 b \,x^{2}+7 a \right )+49 \sinh \left (5 b \,x^{2}+5 a \right )+245 \sinh \left (3 b \,x^{2}+3 a \right )+1225 \sinh \left (b \,x^{2}+a \right )}{4480 b}\) | \(56\) |
risch | \(\frac {{\mathrm e}^{7 b \,x^{2}+7 a}}{1792 b}+\frac {7 \,{\mathrm e}^{5 b \,x^{2}+5 a}}{1280 b}+\frac {7 \,{\mathrm e}^{3 b \,x^{2}+3 a}}{256 b}+\frac {35 \,{\mathrm e}^{b \,x^{2}+a}}{256 b}-\frac {35 \,{\mathrm e}^{-b \,x^{2}-a}}{256 b}-\frac {7 \,{\mathrm e}^{-3 b \,x^{2}-3 a}}{256 b}-\frac {7 \,{\mathrm e}^{-5 b \,x^{2}-5 a}}{1280 b}-\frac {{\mathrm e}^{-7 b \,x^{2}-7 a}}{1792 b}\) | \(127\) |
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (59) = 118\).
Time = 0.24 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.91 \[ \int x \cosh ^7\left (a+b x^2\right ) \, dx=\frac {5 \, \sinh \left (b x^{2} + a\right )^{7} + 7 \, {\left (15 \, \cosh \left (b x^{2} + a\right )^{2} + 7\right )} \sinh \left (b x^{2} + a\right )^{5} + 35 \, {\left (5 \, \cosh \left (b x^{2} + a\right )^{4} + 14 \, \cosh \left (b x^{2} + a\right )^{2} + 7\right )} \sinh \left (b x^{2} + a\right )^{3} + 35 \, {\left (\cosh \left (b x^{2} + a\right )^{6} + 7 \, \cosh \left (b x^{2} + a\right )^{4} + 21 \, \cosh \left (b x^{2} + a\right )^{2} + 35\right )} \sinh \left (b x^{2} + a\right )}{4480 \, b} \]
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Time = 0.75 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.40 \[ \int x \cosh ^7\left (a+b x^2\right ) \, dx=\begin {cases} - \frac {8 \sinh ^{7}{\left (a + b x^{2} \right )}}{35 b} + \frac {4 \sinh ^{5}{\left (a + b x^{2} \right )} \cosh ^{2}{\left (a + b x^{2} \right )}}{5 b} - \frac {\sinh ^{3}{\left (a + b x^{2} \right )} \cosh ^{4}{\left (a + b x^{2} \right )}}{b} + \frac {\sinh {\left (a + b x^{2} \right )} \cosh ^{6}{\left (a + b x^{2} \right )}}{2 b} & \text {for}\: b \neq 0 \\\frac {x^{2} \cosh ^{7}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (59) = 118\).
Time = 0.18 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.88 \[ \int x \cosh ^7\left (a+b x^2\right ) \, dx=\frac {e^{\left (7 \, b x^{2} + 7 \, a\right )}}{1792 \, b} + \frac {7 \, e^{\left (5 \, b x^{2} + 5 \, a\right )}}{1280 \, b} + \frac {7 \, e^{\left (3 \, b x^{2} + 3 \, a\right )}}{256 \, b} + \frac {35 \, e^{\left (b x^{2} + a\right )}}{256 \, b} - \frac {35 \, e^{\left (-b x^{2} - a\right )}}{256 \, b} - \frac {7 \, e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{256 \, b} - \frac {7 \, e^{\left (-5 \, b x^{2} - 5 \, a\right )}}{1280 \, b} - \frac {e^{\left (-7 \, b x^{2} - 7 \, a\right )}}{1792 \, b} \]
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Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.61 \[ \int x \cosh ^7\left (a+b x^2\right ) \, dx=-\frac {{\left (1225 \, e^{\left (6 \, b x^{2} + 6 \, a\right )} + 245 \, e^{\left (4 \, b x^{2} + 4 \, a\right )} + 49 \, e^{\left (2 \, b x^{2} + 2 \, a\right )} + 5\right )} e^{\left (-7 \, b x^{2} - 7 \, a\right )} - 5 \, e^{\left (7 \, b x^{2} + 7 \, a\right )} - 49 \, e^{\left (5 \, b x^{2} + 5 \, a\right )} - 245 \, e^{\left (3 \, b x^{2} + 3 \, a\right )} - 1225 \, e^{\left (b x^{2} + a\right )}}{8960 \, b} \]
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Time = 1.66 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78 \[ \int x \cosh ^7\left (a+b x^2\right ) \, dx=\frac {5\,{\mathrm {sinh}\left (b\,x^2+a\right )}^7+21\,{\mathrm {sinh}\left (b\,x^2+a\right )}^5+35\,{\mathrm {sinh}\left (b\,x^2+a\right )}^3+35\,\mathrm {sinh}\left (b\,x^2+a\right )}{70\,b} \]
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