Integrand size = 12, antiderivative size = 172 \[ \int x^3 \cosh ^4(a+b x) \, dx=\frac {45 x^2}{128 b^2}+\frac {3 x^4}{32}-\frac {45 \cosh ^2(a+b x)}{128 b^4}-\frac {9 x^2 \cosh ^2(a+b x)}{16 b^2}-\frac {3 \cosh ^4(a+b x)}{128 b^4}-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {45 x \cosh (a+b x) \sinh (a+b x)}{64 b^3}+\frac {3 x^3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {3 x \cosh ^3(a+b x) \sinh (a+b x)}{32 b^3}+\frac {x^3 \cosh ^3(a+b x) \sinh (a+b x)}{4 b} \]
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Time = 0.11 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3392, 30, 3391} \[ \int x^3 \cosh ^4(a+b x) \, dx=-\frac {3 \cosh ^4(a+b x)}{128 b^4}-\frac {45 \cosh ^2(a+b x)}{128 b^4}+\frac {3 x \sinh (a+b x) \cosh ^3(a+b x)}{32 b^3}+\frac {45 x \sinh (a+b x) \cosh (a+b x)}{64 b^3}-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}-\frac {9 x^2 \cosh ^2(a+b x)}{16 b^2}+\frac {x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3 x^3 \sinh (a+b x) \cosh (a+b x)}{8 b}+\frac {45 x^2}{128 b^2}+\frac {3 x^4}{32} \]
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Rule 30
Rule 3391
Rule 3392
Rubi steps \begin{align*} \text {integral}& = -\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {x^3 \cosh ^3(a+b x) \sinh (a+b x)}{4 b}+\frac {3}{4} \int x^3 \cosh ^2(a+b x) \, dx+\frac {3 \int x \cosh ^4(a+b x) \, dx}{8 b^2} \\ & = -\frac {9 x^2 \cosh ^2(a+b x)}{16 b^2}-\frac {3 \cosh ^4(a+b x)}{128 b^4}-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {3 x^3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {3 x \cosh ^3(a+b x) \sinh (a+b x)}{32 b^3}+\frac {x^3 \cosh ^3(a+b x) \sinh (a+b x)}{4 b}+\frac {3 \int x^3 \, dx}{8}+\frac {9 \int x \cosh ^2(a+b x) \, dx}{32 b^2}+\frac {9 \int x \cosh ^2(a+b x) \, dx}{8 b^2} \\ & = \frac {3 x^4}{32}-\frac {45 \cosh ^2(a+b x)}{128 b^4}-\frac {9 x^2 \cosh ^2(a+b x)}{16 b^2}-\frac {3 \cosh ^4(a+b x)}{128 b^4}-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {45 x \cosh (a+b x) \sinh (a+b x)}{64 b^3}+\frac {3 x^3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {3 x \cosh ^3(a+b x) \sinh (a+b x)}{32 b^3}+\frac {x^3 \cosh ^3(a+b x) \sinh (a+b x)}{4 b}+\frac {9 \int x \, dx}{64 b^2}+\frac {9 \int x \, dx}{16 b^2} \\ & = \frac {45 x^2}{128 b^2}+\frac {3 x^4}{32}-\frac {45 \cosh ^2(a+b x)}{128 b^4}-\frac {9 x^2 \cosh ^2(a+b x)}{16 b^2}-\frac {3 \cosh ^4(a+b x)}{128 b^4}-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {45 x \cosh (a+b x) \sinh (a+b x)}{64 b^3}+\frac {3 x^3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {3 x \cosh ^3(a+b x) \sinh (a+b x)}{32 b^3}+\frac {x^3 \cosh ^3(a+b x) \sinh (a+b x)}{4 b} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.58 \[ \int x^3 \cosh ^4(a+b x) \, dx=\frac {-192 \left (1+2 b^2 x^2\right ) \cosh (2 (a+b x))-3 \left (1+8 b^2 x^2\right ) \cosh (4 (a+b x))+4 b x \left (24 b^3 x^3+32 \left (3+2 b^2 x^2\right ) \sinh (2 (a+b x))+\left (3+8 b^2 x^2\right ) \sinh (4 (a+b x))\right )}{1024 b^4} \]
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Time = 0.37 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.59
method | result | size |
parallelrisch | \(\frac {\left (-384 x^{2} b^{2}-192\right ) \cosh \left (2 b x +2 a \right )+\left (-24 x^{2} b^{2}-3\right ) \cosh \left (4 b x +4 a \right )+\left (256 x^{3} b^{3}+384 b x \right ) \sinh \left (2 b x +2 a \right )+\left (32 x^{3} b^{3}+12 b x \right ) \sinh \left (4 b x +4 a \right )+96 x^{4} b^{4}+195}{1024 b^{4}}\) | \(102\) |
risch | \(\frac {3 x^{4}}{32}+\frac {\left (32 x^{3} b^{3}-24 x^{2} b^{2}+12 b x -3\right ) {\mathrm e}^{4 b x +4 a}}{2048 b^{4}}+\frac {\left (4 x^{3} b^{3}-6 x^{2} b^{2}+6 b x -3\right ) {\mathrm e}^{2 b x +2 a}}{32 b^{4}}-\frac {\left (4 x^{3} b^{3}+6 x^{2} b^{2}+6 b x +3\right ) {\mathrm e}^{-2 b x -2 a}}{32 b^{4}}-\frac {\left (32 x^{3} b^{3}+24 x^{2} b^{2}+12 b x +3\right ) {\mathrm e}^{-4 b x -4 a}}{2048 b^{4}}\) | \(151\) |
derivativedivides | \(\frac {-a^{3} \left (\left (\frac {\cosh \left (b x +a \right )^{3}}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )+\frac {3 b x}{8}+\frac {3 a}{8}\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {3 \left (b x +a \right )^{2}}{16}-\frac {\cosh \left (b x +a \right )^{4}}{16}-\frac {3 \cosh \left (b x +a \right )^{2}}{16}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {\left (b x +a \right )^{3}}{8}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{8}+\frac {\cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{32}+\frac {15 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}+\frac {15 b x}{64}+\frac {15 a}{64}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{8}\right )+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {3 \left (b x +a \right )^{4}}{32}-\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{4}}{16}+\frac {3 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{32}+\frac {45 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}+\frac {45 \left (b x +a \right )^{2}}{128}-\frac {3 \cosh \left (b x +a \right )^{4}}{128}-\frac {45 \cosh \left (b x +a \right )^{2}}{128}-\frac {9 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}{16}}{b^{4}}\) | \(400\) |
default | \(\frac {-a^{3} \left (\left (\frac {\cosh \left (b x +a \right )^{3}}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )+\frac {3 b x}{8}+\frac {3 a}{8}\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {3 \left (b x +a \right )^{2}}{16}-\frac {\cosh \left (b x +a \right )^{4}}{16}-\frac {3 \cosh \left (b x +a \right )^{2}}{16}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {\left (b x +a \right )^{3}}{8}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{8}+\frac {\cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{32}+\frac {15 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}+\frac {15 b x}{64}+\frac {15 a}{64}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{8}\right )+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {3 \left (b x +a \right )^{4}}{32}-\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{4}}{16}+\frac {3 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{32}+\frac {45 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}+\frac {45 \left (b x +a \right )^{2}}{128}-\frac {3 \cosh \left (b x +a \right )^{4}}{128}-\frac {45 \cosh \left (b x +a \right )^{2}}{128}-\frac {9 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}{16}}{b^{4}}\) | \(400\) |
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Time = 0.25 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.13 \[ \int x^3 \cosh ^4(a+b x) \, dx=\frac {96 \, b^{4} x^{4} - 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} + 16 \, {\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{4} - 192 \, {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} - 6 \, {\left (64 \, b^{2} x^{2} + 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} + 32\right )} \sinh \left (b x + a\right )^{2} + 16 \, {\left ({\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{3} + 16 \, {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{1024 \, b^{4}} \]
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Time = 0.56 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.47 \[ \int x^3 \cosh ^4(a+b x) \, dx=\begin {cases} \frac {3 x^{4} \sinh ^{4}{\left (a + b x \right )}}{32} - \frac {3 x^{4} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} + \frac {3 x^{4} \cosh ^{4}{\left (a + b x \right )}}{32} - \frac {3 x^{3} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8 b} + \frac {5 x^{3} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} + \frac {45 x^{2} \sinh ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac {9 x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{64 b^{2}} - \frac {51 x^{2} \cosh ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac {45 x \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{64 b^{3}} + \frac {51 x \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{64 b^{3}} + \frac {45 \sinh ^{4}{\left (a + b x \right )}}{256 b^{4}} - \frac {51 \cosh ^{4}{\left (a + b x \right )}}{256 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \cosh ^{4}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.02 \[ \int x^3 \cosh ^4(a+b x) \, dx=\frac {3}{32} \, x^{4} + \frac {{\left (32 \, b^{3} x^{3} e^{\left (4 \, a\right )} - 24 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 12 \, b x e^{\left (4 \, a\right )} - 3 \, e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{2048 \, b^{4}} + \frac {{\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{32 \, b^{4}} - \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{4}} - \frac {{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.87 \[ \int x^3 \cosh ^4(a+b x) \, dx=\frac {3}{32} \, x^{4} + \frac {{\left (32 \, b^{3} x^{3} - 24 \, b^{2} x^{2} + 12 \, b x - 3\right )} e^{\left (4 \, b x + 4 \, a\right )}}{2048 \, b^{4}} + \frac {{\left (4 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, b x - 3\right )} e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b^{4}} - \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{4}} - \frac {{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \]
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Time = 0.43 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.75 \[ \int x^3 \cosh ^4(a+b x) \, dx=\frac {3\,x^4}{32}-\frac {\frac {3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{16}+\frac {3\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{1024}+b^2\,\left (\frac {3\,x^2\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{8}+\frac {3\,x^2\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{128}\right )-b\,\left (\frac {3\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8}+\frac {3\,x\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{256}\right )-b^3\,\left (\frac {x^3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4}+\frac {x^3\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{32}\right )}{b^4} \]
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