Integrand size = 8, antiderivative size = 46 \[ \int \sinh ^4(a+b x) \, dx=\frac {3 x}{8}-\frac {3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh (a+b x) \sinh ^3(a+b x)}{4 b} \]
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Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 8} \[ \int \sinh ^4(a+b x) \, dx=\frac {\sinh ^3(a+b x) \cosh (a+b x)}{4 b}-\frac {3 \sinh (a+b x) \cosh (a+b x)}{8 b}+\frac {3 x}{8} \]
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Rule 8
Rule 2715
Rubi steps \begin{align*} \text {integral}& = \frac {\cosh (a+b x) \sinh ^3(a+b x)}{4 b}-\frac {3}{4} \int \sinh ^2(a+b x) \, dx \\ & = -\frac {3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh (a+b x) \sinh ^3(a+b x)}{4 b}+\frac {3 \int 1 \, dx}{8} \\ & = \frac {3 x}{8}-\frac {3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh (a+b x) \sinh ^3(a+b x)}{4 b} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.72 \[ \int \sinh ^4(a+b x) \, dx=\frac {12 (a+b x)-8 \sinh (2 (a+b x))+\sinh (4 (a+b x))}{32 b} \]
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Time = 1.43 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67
method | result | size |
parallelrisch | \(\frac {12 b x +\sinh \left (4 b x +4 a \right )-8 \sinh \left (2 b x +2 a \right )}{32 b}\) | \(31\) |
derivativedivides | \(\frac {\left (\frac {\sinh \left (b x +a \right )^{3}}{4}-\frac {3 \sinh \left (b x +a \right )}{8}\right ) \cosh \left (b x +a \right )+\frac {3 b x}{8}+\frac {3 a}{8}}{b}\) | \(39\) |
default | \(\frac {\left (\frac {\sinh \left (b x +a \right )^{3}}{4}-\frac {3 \sinh \left (b x +a \right )}{8}\right ) \cosh \left (b x +a \right )+\frac {3 b x}{8}+\frac {3 a}{8}}{b}\) | \(39\) |
risch | \(\frac {3 x}{8}+\frac {{\mathrm e}^{4 b x +4 a}}{64 b}-\frac {{\mathrm e}^{2 b x +2 a}}{8 b}+\frac {{\mathrm e}^{-2 b x -2 a}}{8 b}-\frac {{\mathrm e}^{-4 b x -4 a}}{64 b}\) | \(61\) |
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Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07 \[ \int \sinh ^4(a+b x) \, dx=\frac {\cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 3 \, b x + {\left (\cosh \left (b x + a\right )^{3} - 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{8 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (41) = 82\).
Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.07 \[ \int \sinh ^4(a+b x) \, dx=\begin {cases} \frac {3 x \sinh ^{4}{\left (a + b x \right )}}{8} - \frac {3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4} + \frac {3 x \cosh ^{4}{\left (a + b x \right )}}{8} + \frac {5 \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8 b} - \frac {3 \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} & \text {for}\: b \neq 0 \\x \sinh ^{4}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int \sinh ^4(a+b x) \, dx=\frac {3}{8} \, x + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} - \frac {e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} \]
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Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int \sinh ^4(a+b x) \, dx=\frac {3}{8} \, x + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} - \frac {e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} \]
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Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.70 \[ \int \sinh ^4(a+b x) \, dx=\frac {3\,x}{8}-\frac {\frac {\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4}-\frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{32}}{b} \]
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