Integrand size = 13, antiderivative size = 191 \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {24 b^4 n^4 x}{1-20 b^2 n^2+64 b^4 n^4}-\frac {24 b^3 n^3 x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{1-20 b^2 n^2+64 b^4 n^4}+\frac {12 b^2 n^2 x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-20 b^2 n^2+64 b^4 n^4}-\frac {4 b n x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac {x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2} \]
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Time = 0.04 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5630, 8} \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}-\frac {4 b n x \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac {12 b^2 n^2 x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4-20 b^2 n^2+1}-\frac {24 b^3 n^3 x \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4-20 b^2 n^2+1}+\frac {24 b^4 n^4 x}{64 b^4 n^4-20 b^2 n^2+1} \]
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Rule 8
Rule 5630
Rubi steps \begin{align*} \text {integral}& = -\frac {4 b n x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac {x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac {\left (12 b^2 n^2\right ) \int \sinh ^2\left (a+b \log \left (c x^n\right )\right ) \, dx}{1-16 b^2 n^2} \\ & = -\frac {24 b^3 n^3 x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{1-20 b^2 n^2+64 b^4 n^4}+\frac {12 b^2 n^2 x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-20 b^2 n^2+64 b^4 n^4}-\frac {4 b n x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac {x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac {\left (24 b^4 n^4\right ) \int 1 \, dx}{1-20 b^2 n^2+64 b^4 n^4} \\ & = \frac {24 b^4 n^4 x}{1-20 b^2 n^2+64 b^4 n^4}-\frac {24 b^3 n^3 x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{1-20 b^2 n^2+64 b^4 n^4}+\frac {12 b^2 n^2 x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-20 b^2 n^2+64 b^4 n^4}-\frac {4 b n x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac {x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.87 \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (3-60 b^2 n^2+192 b^4 n^4+\left (-4+64 b^2 n^2\right ) \cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\left (1-4 b^2 n^2\right ) \cosh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+8 b n \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-128 b^3 n^3 \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-4 b n \sinh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+16 b^3 n^3 \sinh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{8 \left (1-20 b^2 n^2+64 b^4 n^4\right )} \]
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Time = 14.18 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(-\frac {128 x \left (\left (-\frac {1}{8} b^{3} n^{3}+\frac {1}{32} b n \right ) \sinh \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )+\left (\frac {b^{2} n^{2}}{32}-\frac {1}{128}\right ) \cosh \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )+\left (b n -\frac {1}{4}\right ) \left (-\frac {3 b^{2} n^{2}}{2}+b n \sinh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )-\frac {\cosh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )}{2}+\frac {3}{8}\right ) \left (b n +\frac {1}{4}\right )\right )}{512 b^{4} n^{4}-160 b^{2} n^{2}+8}\) | \(132\) |
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Time = 0.26 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.54 \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {{\left (4 \, b^{2} n^{2} - 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} - 16 \, {\left (4 \, b^{3} n^{3} - b n\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + {\left (4 \, b^{2} n^{2} - 1\right )} x \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} - 4 \, {\left (16 \, b^{2} n^{2} - 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, {\left (3 \, {\left (4 \, b^{2} n^{2} - 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 2 \, {\left (16 \, b^{2} n^{2} - 1\right )} x\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 3 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )} x - 16 \, {\left ({\left (4 \, b^{3} n^{3} - b n\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - {\left (16 \, b^{3} n^{3} - b n\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{8 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )}} \]
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\[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \int \sinh ^{4}{\left (a - \frac {\log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = - \frac {1}{2 n} \\\int \sinh ^{4}{\left (a - \frac {\log {\left (c x^{n} \right )}}{4 n} \right )}\, dx & \text {for}\: b = - \frac {1}{4 n} \\\int \sinh ^{4}{\left (a + \frac {\log {\left (c x^{n} \right )}}{4 n} \right )}\, dx & \text {for}\: b = \frac {1}{4 n} \\\int \sinh ^{4}{\left (a + \frac {\log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = \frac {1}{2 n} \\\frac {24 b^{4} n^{4} x \sinh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} - \frac {48 b^{4} n^{4} x \sinh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} + \frac {24 b^{4} n^{4} x \cosh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} + \frac {40 b^{3} n^{3} x \sinh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} - \frac {24 b^{3} n^{3} x \sinh {\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} - \frac {16 b^{2} n^{2} x \sinh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} + \frac {12 b^{2} n^{2} x \sinh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} - \frac {4 b n x \sinh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} + \frac {x \sinh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.68 \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {c^{4 \, b} x e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )}}{16 \, {\left (4 \, b n + 1\right )}} - \frac {c^{2 \, b} x e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}}{4 \, {\left (2 \, b n + 1\right )}} + \frac {3}{8} \, x + \frac {x e^{\left (-2 \, b \log \left (x^{n}\right ) - 2 \, a\right )}}{4 \, {\left (2 \, b c^{2 \, b} n - c^{2 \, b}\right )}} - \frac {x e^{\left (-4 \, a\right )}}{16 \, {\left (4 \, b c^{4 \, b} n - c^{4 \, b}\right )} {\left (x^{n}\right )}^{4 \, b}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (192) = 384\).
Time = 0.32 (sec) , antiderivative size = 777, normalized size of antiderivative = 4.07 \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b^{3} c^{4 \, b} n^{3} x x^{4 \, b n} e^{\left (4 \, a\right )}}{64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1} - \frac {8 \, b^{3} c^{2 \, b} n^{3} x x^{2 \, b n} e^{\left (2 \, a\right )}}{64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1} + \frac {24 \, b^{4} n^{4} x}{64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1} - \frac {b^{2} c^{4 \, b} n^{2} x x^{4 \, b n} e^{\left (4 \, a\right )}}{4 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )}} + \frac {4 \, b^{2} c^{2 \, b} n^{2} x x^{2 \, b n} e^{\left (2 \, a\right )}}{64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1} - \frac {b c^{4 \, b} n x x^{4 \, b n} e^{\left (4 \, a\right )}}{4 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )}} + \frac {b c^{2 \, b} n x x^{2 \, b n} e^{\left (2 \, a\right )}}{2 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )}} + \frac {8 \, b^{3} n^{3} x e^{\left (-2 \, a\right )}}{{\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )} c^{2 \, b} x^{2 \, b n}} - \frac {b^{3} n^{3} x e^{\left (-4 \, a\right )}}{{\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )} c^{4 \, b} x^{4 \, b n}} - \frac {15 \, b^{2} n^{2} x}{2 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )}} + \frac {c^{4 \, b} x x^{4 \, b n} e^{\left (4 \, a\right )}}{16 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )}} - \frac {c^{2 \, b} x x^{2 \, b n} e^{\left (2 \, a\right )}}{4 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )}} + \frac {4 \, b^{2} n^{2} x e^{\left (-2 \, a\right )}}{{\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )} c^{2 \, b} x^{2 \, b n}} - \frac {b^{2} n^{2} x e^{\left (-4 \, a\right )}}{4 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )} c^{4 \, b} x^{4 \, b n}} - \frac {b n x e^{\left (-2 \, a\right )}}{2 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )} c^{2 \, b} x^{2 \, b n}} + \frac {b n x e^{\left (-4 \, a\right )}}{4 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )} c^{4 \, b} x^{4 \, b n}} + \frac {3 \, x}{8 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )}} - \frac {x e^{\left (-2 \, a\right )}}{4 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )} c^{2 \, b} x^{2 \, b n}} + \frac {x e^{\left (-4 \, a\right )}}{16 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )} c^{4 \, b} x^{4 \, b n}} \]
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Time = 1.45 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.53 \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {3\,x}{8}+\frac {x\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}\,\left (8\,b\,n-4\right )}-\frac {x\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}{8\,b\,n+4}-\frac {x\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}\,\left (64\,b\,n-16\right )}+\frac {x\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}{64\,b\,n+16} \]
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