Integrand size = 11, antiderivative size = 61 \[ \int (a \cosh (x)+b \sinh (x))^5 \, dx=\left (a^2-b^2\right )^2 (b \cosh (x)+a \sinh (x))+\frac {2}{3} \left (a^2-b^2\right ) (b \cosh (x)+a \sinh (x))^3+\frac {1}{5} (b \cosh (x)+a \sinh (x))^5 \]
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Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3151, 200} \[ \int (a \cosh (x)+b \sinh (x))^5 \, dx=\frac {2}{3} \left (a^2-b^2\right ) (a \sinh (x)+b \cosh (x))^3+\left (a^2-b^2\right )^2 (a \sinh (x)+b \cosh (x))+\frac {1}{5} (a \sinh (x)+b \cosh (x))^5 \]
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Rule 200
Rule 3151
Rubi steps \begin{align*} \text {integral}& = i \text {Subst}\left (\int \left (a^2-b^2-x^2\right )^2 \, dx,x,-i b \cosh (x)-i a \sinh (x)\right ) \\ & = i \text {Subst}\left (\int \left (a^4 \left (1+\frac {-2 a^2 b^2+b^4}{a^4}\right )-2 a^2 \left (1-\frac {b^2}{a^2}\right ) x^2+x^4\right ) \, dx,x,-i b \cosh (x)-i a \sinh (x)\right ) \\ & = \left (a^2-b^2\right )^2 (b \cosh (x)+a \sinh (x))+\frac {2}{3} \left (a^2-b^2\right ) (b \cosh (x)+a \sinh (x))^3+\frac {1}{5} (b \cosh (x)+a \sinh (x))^5 \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(133\) vs. \(2(61)=122\).
Time = 0.20 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.18 \[ \int (a \cosh (x)+b \sinh (x))^5 \, dx=\frac {1}{240} \left (150 b \left (a^2-b^2\right )^2 \cosh (x)-25 b \left (-3 a^4+2 a^2 b^2+b^4\right ) \cosh (3 x)+3 b \left (5 a^4+10 a^2 b^2+b^4\right ) \cosh (5 x)+150 a \left (a^2-b^2\right )^2 \sinh (x)+25 a \left (a^4+2 a^2 b^2-3 b^4\right ) \sinh (3 x)+3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \sinh (5 x)\right ) \]
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Time = 180.66 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.67
method | result | size |
parts | \(a^{5} \left (\frac {8}{15}+\frac {\cosh \left (x \right )^{4}}{5}+\frac {4 \cosh \left (x \right )^{2}}{15}\right ) \sinh \left (x \right )+b^{5} \left (\frac {8}{15}+\frac {\sinh \left (x \right )^{4}}{5}-\frac {4 \sinh \left (x \right )^{2}}{15}\right ) \cosh \left (x \right )+a^{4} b \cosh \left (x \right )^{5}+10 a^{3} b^{2} \left (\frac {\sinh \left (x \right )^{5}}{5}+\frac {\sinh \left (x \right )^{3}}{3}\right )+10 a^{2} b^{3} \left (\frac {\cosh \left (x \right )^{5}}{5}-\frac {\cosh \left (x \right )^{3}}{3}\right )+a \,b^{4} \sinh \left (x \right )^{5}\) | \(102\) |
default | \(a^{5} \left (\frac {8}{15}+\frac {\cosh \left (x \right )^{4}}{5}+\frac {4 \cosh \left (x \right )^{2}}{15}\right ) \sinh \left (x \right )+a^{4} b \cosh \left (x \right )^{5}+10 a^{3} b^{2} \left (\frac {\cosh \left (x \right )^{4} \sinh \left (x \right )}{5}-\frac {\left (\frac {2}{3}+\frac {\cosh \left (x \right )^{2}}{3}\right ) \sinh \left (x \right )}{5}\right )+10 a^{2} b^{3} \left (\frac {\sinh \left (x \right )^{2} \cosh \left (x \right )^{3}}{5}-\frac {2 \cosh \left (x \right )^{3}}{15}\right )+a \,b^{4} \sinh \left (x \right )^{5}+b^{5} \left (\frac {8}{15}+\frac {\sinh \left (x \right )^{4}}{5}-\frac {4 \sinh \left (x \right )^{2}}{15}\right ) \cosh \left (x \right )\) | \(114\) |
risch | \(\frac {5 b^{5} {\mathrm e}^{x}}{16}+\frac {5 \,{\mathrm e}^{x} a \,b^{4}}{16}+\frac {5 \,{\mathrm e}^{x} a^{4} b}{16}-\frac {5 \,{\mathrm e}^{x} a^{3} b^{2}}{8}-\frac {5 \,{\mathrm e}^{x} a^{2} b^{3}}{8}+\frac {5 \,{\mathrm e}^{x} a^{5}}{16}+\frac {{\mathrm e}^{-5 x} b^{5}}{160}+\frac {{\mathrm e}^{5 x} a^{5}}{160}+\frac {5 \,{\mathrm e}^{3 x} a^{5}}{96}+\frac {5 \,{\mathrm e}^{-x} a^{4} b}{16}+\frac {5 \,{\mathrm e}^{-x} a^{3} b^{2}}{8}-\frac {5 \,{\mathrm e}^{-x} a^{2} b^{3}}{8}-\frac {5 \,{\mathrm e}^{-x} a \,b^{4}}{16}+\frac {5 \,{\mathrm e}^{-3 x} a^{4} b}{32}-\frac {5 \,{\mathrm e}^{-3 x} a^{3} b^{2}}{48}-\frac {5 \,{\mathrm e}^{-3 x} a^{2} b^{3}}{48}+\frac {5 \,{\mathrm e}^{-3 x} a \,b^{4}}{32}+\frac {{\mathrm e}^{-5 x} a^{4} b}{32}-\frac {{\mathrm e}^{-5 x} a^{3} b^{2}}{16}+\frac {{\mathrm e}^{-5 x} a^{2} b^{3}}{16}-\frac {{\mathrm e}^{-5 x} a \,b^{4}}{32}+\frac {a^{4} b \,{\mathrm e}^{5 x}}{32}+\frac {a^{2} b^{3} {\mathrm e}^{5 x}}{16}-\frac {5 a^{2} b^{3} {\mathrm e}^{3 x}}{48}+\frac {{\mathrm e}^{5 x} a^{3} b^{2}}{16}+\frac {{\mathrm e}^{5 x} a \,b^{4}}{32}+\frac {5 \,{\mathrm e}^{3 x} a^{4} b}{32}+\frac {5 \,{\mathrm e}^{3 x} a^{3} b^{2}}{48}-\frac {5 \,{\mathrm e}^{3 x} a \,b^{4}}{32}+\frac {b^{5} {\mathrm e}^{5 x}}{160}-\frac {5 b^{5} {\mathrm e}^{3 x}}{96}-\frac {5 \,{\mathrm e}^{-x} a^{5}}{16}+\frac {5 \,{\mathrm e}^{-x} b^{5}}{16}-\frac {5 \,{\mathrm e}^{-3 x} a^{5}}{96}-\frac {5 \,{\mathrm e}^{-3 x} b^{5}}{96}-\frac {{\mathrm e}^{-5 x} a^{5}}{160}\) | \(362\) |
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Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (57) = 114\).
Time = 0.25 (sec) , antiderivative size = 298, normalized size of antiderivative = 4.89 \[ \int (a \cosh (x)+b \sinh (x))^5 \, dx=\frac {1}{80} \, {\left (5 \, a^{4} b + 10 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )^{5} + \frac {1}{16} \, {\left (5 \, a^{4} b + 10 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{4} + \frac {1}{80} \, {\left (a^{5} + 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sinh \left (x\right )^{5} + \frac {5}{48} \, {\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{3} + \frac {1}{48} \, {\left (5 \, a^{5} + 10 \, a^{3} b^{2} - 15 \, a b^{4} + 6 \, {\left (a^{5} + 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{3} + \frac {1}{16} \, {\left (2 \, {\left (5 \, a^{4} b + 10 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )^{3} + 5 \, {\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + \frac {5}{8} \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right ) + \frac {1}{16} \, {\left (10 \, a^{5} - 20 \, a^{3} b^{2} + 10 \, a b^{4} + {\left (a^{5} + 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cosh \left (x\right )^{4} + 5 \, {\left (a^{5} + 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (56) = 112\).
Time = 0.25 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.82 \[ \int (a \cosh (x)+b \sinh (x))^5 \, dx=\frac {8 a^{5} \sinh ^{5}{\left (x \right )}}{15} - \frac {4 a^{5} \sinh ^{3}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{3} + a^{5} \sinh {\left (x \right )} \cosh ^{4}{\left (x \right )} + a^{4} b \cosh ^{5}{\left (x \right )} - \frac {4 a^{3} b^{2} \sinh ^{5}{\left (x \right )}}{3} + \frac {10 a^{3} b^{2} \sinh ^{3}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{3} + \frac {10 a^{2} b^{3} \sinh ^{2}{\left (x \right )} \cosh ^{3}{\left (x \right )}}{3} - \frac {4 a^{2} b^{3} \cosh ^{5}{\left (x \right )}}{3} + a b^{4} \sinh ^{5}{\left (x \right )} + b^{5} \sinh ^{4}{\left (x \right )} \cosh {\left (x \right )} - \frac {4 b^{5} \sinh ^{2}{\left (x \right )} \cosh ^{3}{\left (x \right )}}{3} + \frac {8 b^{5} \cosh ^{5}{\left (x \right )}}{15} \]
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Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (57) = 114\).
Time = 0.21 (sec) , antiderivative size = 191, normalized size of antiderivative = 3.13 \[ \int (a \cosh (x)+b \sinh (x))^5 \, dx=a^{4} b \cosh \left (x\right )^{5} + a b^{4} \sinh \left (x\right )^{5} + \frac {1}{48} \, {\left ({\left (5 \, e^{\left (-2 \, x\right )} - 30 \, e^{\left (-4 \, x\right )} + 3\right )} e^{\left (5 \, x\right )} + 30 \, e^{\left (-x\right )} - 5 \, e^{\left (-3 \, x\right )} - 3 \, e^{\left (-5 \, x\right )}\right )} a^{3} b^{2} - \frac {1}{48} \, {\left ({\left (5 \, e^{\left (-2 \, x\right )} + 30 \, e^{\left (-4 \, x\right )} - 3\right )} e^{\left (5 \, x\right )} + 30 \, e^{\left (-x\right )} + 5 \, e^{\left (-3 \, x\right )} - 3 \, e^{\left (-5 \, x\right )}\right )} a^{2} b^{3} + \frac {1}{480} \, a^{5} {\left (3 \, e^{\left (5 \, x\right )} + 25 \, e^{\left (3 \, x\right )} - 150 \, e^{\left (-x\right )} - 25 \, e^{\left (-3 \, x\right )} - 3 \, e^{\left (-5 \, x\right )} + 150 \, e^{x}\right )} + \frac {1}{480} \, b^{5} {\left (3 \, e^{\left (5 \, x\right )} - 25 \, e^{\left (3 \, x\right )} + 150 \, e^{\left (-x\right )} - 25 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-5 \, x\right )} + 150 \, e^{x}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (57) = 114\).
Time = 0.27 (sec) , antiderivative size = 344, normalized size of antiderivative = 5.64 \[ \int (a \cosh (x)+b \sinh (x))^5 \, dx=\frac {1}{160} \, a^{5} e^{\left (5 \, x\right )} + \frac {1}{32} \, a^{4} b e^{\left (5 \, x\right )} + \frac {1}{16} \, a^{3} b^{2} e^{\left (5 \, x\right )} + \frac {1}{16} \, a^{2} b^{3} e^{\left (5 \, x\right )} + \frac {1}{32} \, a b^{4} e^{\left (5 \, x\right )} + \frac {1}{160} \, b^{5} e^{\left (5 \, x\right )} + \frac {5}{96} \, a^{5} e^{\left (3 \, x\right )} + \frac {5}{32} \, a^{4} b e^{\left (3 \, x\right )} + \frac {5}{48} \, a^{3} b^{2} e^{\left (3 \, x\right )} - \frac {5}{48} \, a^{2} b^{3} e^{\left (3 \, x\right )} - \frac {5}{32} \, a b^{4} e^{\left (3 \, x\right )} - \frac {5}{96} \, b^{5} e^{\left (3 \, x\right )} + \frac {5}{16} \, a^{5} e^{x} + \frac {5}{16} \, a^{4} b e^{x} - \frac {5}{8} \, a^{3} b^{2} e^{x} - \frac {5}{8} \, a^{2} b^{3} e^{x} + \frac {5}{16} \, a b^{4} e^{x} + \frac {5}{16} \, b^{5} e^{x} - \frac {1}{480} \, {\left (150 \, a^{5} e^{\left (4 \, x\right )} - 150 \, a^{4} b e^{\left (4 \, x\right )} - 300 \, a^{3} b^{2} e^{\left (4 \, x\right )} + 300 \, a^{2} b^{3} e^{\left (4 \, x\right )} + 150 \, a b^{4} e^{\left (4 \, x\right )} - 150 \, b^{5} e^{\left (4 \, x\right )} + 25 \, a^{5} e^{\left (2 \, x\right )} - 75 \, a^{4} b e^{\left (2 \, x\right )} + 50 \, a^{3} b^{2} e^{\left (2 \, x\right )} + 50 \, a^{2} b^{3} e^{\left (2 \, x\right )} - 75 \, a b^{4} e^{\left (2 \, x\right )} + 25 \, b^{5} e^{\left (2 \, x\right )} + 3 \, a^{5} - 15 \, a^{4} b + 30 \, a^{3} b^{2} - 30 \, a^{2} b^{3} + 15 \, a b^{4} - 3 \, b^{5}\right )} e^{\left (-5 \, x\right )} \]
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Time = 0.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.92 \[ \int (a \cosh (x)+b \sinh (x))^5 \, dx={\mathrm {cosh}\left (x\right )}^5\,\left (a^4\,b-\frac {4\,a^2\,b^3}{3}+\frac {8\,b^5}{15}\right )+{\mathrm {sinh}\left (x\right )}^5\,\left (\frac {8\,a^5}{15}-\frac {4\,a^3\,b^2}{3}+a\,b^4\right )-{\mathrm {cosh}\left (x\right )}^2\,{\mathrm {sinh}\left (x\right )}^3\,\left (\frac {4\,a^5}{3}-\frac {10\,a^3\,b^2}{3}\right )+a^5\,{\mathrm {cosh}\left (x\right )}^4\,\mathrm {sinh}\left (x\right )-{\mathrm {cosh}\left (x\right )}^3\,{\mathrm {sinh}\left (x\right )}^2\,\left (\frac {4\,b^5}{3}-\frac {10\,a^2\,b^3}{3}\right )+b^5\,\mathrm {cosh}\left (x\right )\,{\mathrm {sinh}\left (x\right )}^4 \]
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