Integrand size = 12, antiderivative size = 134 \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {15 x}{64 b^2}+\frac {x^3}{8}-\frac {3 x \cosh ^2(a+b x)}{8 b^2}-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {15 \cosh (a+b x) \sinh (a+b x)}{64 b^3}+\frac {3 x^2 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{32 b^3}+\frac {x^2 \cosh ^3(a+b x) \sinh (a+b x)}{4 b} \]
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Time = 0.09 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3392, 30, 2715, 8} \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {\sinh (a+b x) \cosh ^3(a+b x)}{32 b^3}+\frac {15 \sinh (a+b x) \cosh (a+b x)}{64 b^3}-\frac {x \cosh ^4(a+b x)}{8 b^2}-\frac {3 x \cosh ^2(a+b x)}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3 x^2 \sinh (a+b x) \cosh (a+b x)}{8 b}+\frac {15 x}{64 b^2}+\frac {x^3}{8} \]
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Rule 8
Rule 30
Rule 2715
Rule 3392
Rubi steps \begin{align*} \text {integral}& = -\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {x^2 \cosh ^3(a+b x) \sinh (a+b x)}{4 b}+\frac {3}{4} \int x^2 \cosh ^2(a+b x) \, dx+\frac {\int \cosh ^4(a+b x) \, dx}{8 b^2} \\ & = -\frac {3 x \cosh ^2(a+b x)}{8 b^2}-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {3 x^2 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{32 b^3}+\frac {x^2 \cosh ^3(a+b x) \sinh (a+b x)}{4 b}+\frac {3 \int x^2 \, dx}{8}+\frac {3 \int \cosh ^2(a+b x) \, dx}{32 b^2}+\frac {3 \int \cosh ^2(a+b x) \, dx}{8 b^2} \\ & = \frac {x^3}{8}-\frac {3 x \cosh ^2(a+b x)}{8 b^2}-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {15 \cosh (a+b x) \sinh (a+b x)}{64 b^3}+\frac {3 x^2 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{32 b^3}+\frac {x^2 \cosh ^3(a+b x) \sinh (a+b x)}{4 b}+\frac {3 \int 1 \, dx}{64 b^2}+\frac {3 \int 1 \, dx}{16 b^2} \\ & = \frac {15 x}{64 b^2}+\frac {x^3}{8}-\frac {3 x \cosh ^2(a+b x)}{8 b^2}-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {15 \cosh (a+b x) \sinh (a+b x)}{64 b^3}+\frac {3 x^2 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{32 b^3}+\frac {x^2 \cosh ^3(a+b x) \sinh (a+b x)}{4 b} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.67 \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {32 b^3 x^3-64 b x \cosh (2 (a+b x))-4 b x \cosh (4 (a+b x))+32 \sinh (2 (a+b x))+64 b^2 x^2 \sinh (2 (a+b x))+\sinh (4 (a+b x))+8 b^2 x^2 \sinh (4 (a+b x))}{256 b^3} \]
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Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.60
method | result | size |
parallelrisch | \(\frac {\left (64 x^{2} b^{2}+32\right ) \sinh \left (2 b x +2 a \right )+\left (8 x^{2} b^{2}+1\right ) \sinh \left (4 b x +4 a \right )+32 x \left (x^{2} b^{2}-2 \cosh \left (2 b x +2 a \right )-\frac {\cosh \left (4 b x +4 a \right )}{8}\right ) b}{256 b^{3}}\) | \(81\) |
risch | \(\frac {x^{3}}{8}+\frac {\left (8 x^{2} b^{2}-4 b x +1\right ) {\mathrm e}^{4 b x +4 a}}{512 b^{3}}+\frac {\left (2 x^{2} b^{2}-2 b x +1\right ) {\mathrm e}^{2 b x +2 a}}{16 b^{3}}-\frac {\left (2 x^{2} b^{2}+2 b x +1\right ) {\mathrm e}^{-2 b x -2 a}}{16 b^{3}}-\frac {\left (8 x^{2} b^{2}+4 b x +1\right ) {\mathrm e}^{-4 b x -4 a}}{512 b^{3}}\) | \(119\) |
derivativedivides | \(\frac {a^{2} \left (\left (\frac {\cosh \left (b x +a \right )^{3}}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )+\frac {3 b x}{8}+\frac {3 a}{8}\right )-2 a \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {3 \left (b x +a \right )^{2}}{16}-\frac {\cosh \left (b x +a \right )^{4}}{16}-\frac {3 \cosh \left (b x +a \right )^{2}}{16}\right )+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {\left (b x +a \right )^{3}}{8}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{8}+\frac {\cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{32}+\frac {15 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}+\frac {15 b x}{64}+\frac {15 a}{64}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{8}}{b^{3}}\) | \(237\) |
default | \(\frac {a^{2} \left (\left (\frac {\cosh \left (b x +a \right )^{3}}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )+\frac {3 b x}{8}+\frac {3 a}{8}\right )-2 a \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {3 \left (b x +a \right )^{2}}{16}-\frac {\cosh \left (b x +a \right )^{4}}{16}-\frac {3 \cosh \left (b x +a \right )^{2}}{16}\right )+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {\left (b x +a \right )^{3}}{8}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{8}+\frac {\cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{32}+\frac {15 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}+\frac {15 b x}{64}+\frac {15 a}{64}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{8}}{b^{3}}\) | \(237\) |
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Time = 0.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.10 \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {8 \, b^{3} x^{3} - b x \cosh \left (b x + a\right )^{4} - b x \sinh \left (b x + a\right )^{4} + {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - 16 \, b x \cosh \left (b x + a\right )^{2} - 2 \, {\left (3 \, b x \cosh \left (b x + a\right )^{2} + 8 \, b x\right )} \sinh \left (b x + a\right )^{2} + {\left ({\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{3} + 16 \, {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{64 \, b^{3}} \]
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Time = 0.41 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.56 \[ \int x^2 \cosh ^4(a+b x) \, dx=\begin {cases} \frac {x^{3} \sinh ^{4}{\left (a + b x \right )}}{8} - \frac {x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4} + \frac {x^{3} \cosh ^{4}{\left (a + b x \right )}}{8} - \frac {3 x^{2} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8 b} + \frac {5 x^{2} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} + \frac {15 x \sinh ^{4}{\left (a + b x \right )}}{64 b^{2}} - \frac {3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{32 b^{2}} - \frac {17 x \cosh ^{4}{\left (a + b x \right )}}{64 b^{2}} - \frac {15 \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{64 b^{3}} + \frac {17 \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{64 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \cosh ^{4}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.99 \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {1}{8} \, x^{3} + \frac {{\left (8 \, b^{2} x^{2} e^{\left (4 \, a\right )} - 4 \, b x e^{\left (4 \, a\right )} + e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{512 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{16 \, b^{3}} - \frac {{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{3}} - \frac {{\left (8 \, b^{2} x^{2} + 4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{512 \, b^{3}} \]
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Time = 0.26 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.88 \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {1}{8} \, x^{3} + \frac {{\left (8 \, b^{2} x^{2} - 4 \, b x + 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{512 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{3}} - \frac {{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{3}} - \frac {{\left (8 \, b^{2} x^{2} + 4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{512 \, b^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.70 \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {\frac {\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8}+\frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{256}-b\,\left (\frac {x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4}+\frac {x\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{64}\right )+b^2\,\left (\frac {x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4}+\frac {x^2\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{32}\right )}{b^3}+\frac {x^3}{8} \]
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