Integrand size = 13, antiderivative size = 149 \[ \int \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {6 b^3 n^3 x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-10 b^2 n^2+9 b^4 n^4}+\frac {6 b^2 n^2 x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-10 b^2 n^2+9 b^4 n^4}-\frac {3 b n x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2}+\frac {x \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2} \]
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Time = 0.03 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5630, 5628} \[ \int \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2}-\frac {3 b n x \sinh ^2\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2}+\frac {6 b^2 n^2 x \sinh \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4-10 b^2 n^2+1}-\frac {6 b^3 n^3 x \cosh \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4-10 b^2 n^2+1} \]
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Rule 5628
Rule 5630
Rubi steps \begin{align*} \text {integral}& = -\frac {3 b n x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2}+\frac {x \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2}+\frac {\left (6 b^2 n^2\right ) \int \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx}{1-9 b^2 n^2} \\ & = -\frac {6 b^3 n^3 x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-10 b^2 n^2+9 b^4 n^4}+\frac {6 b^2 n^2 x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-10 b^2 n^2+9 b^4 n^4}-\frac {3 b n x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2}+\frac {x \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.81 \[ \int \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (-3 b n \left (-1+9 b^2 n^2\right ) \cosh \left (a+b \log \left (c x^n\right )\right )+3 b n \left (-1+b^2 n^2\right ) \cosh \left (3 \left (a+b \log \left (c x^n\right )\right )\right )-2 \left (1-13 b^2 n^2+\left (-1+b^2 n^2\right ) \cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )\right )}{4-40 b^2 n^2+36 b^4 n^4} \]
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Time = 5.52 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.11
method | result | size |
parallelrisch | \(\frac {x \left (-27 b^{3} n^{3} \cosh \left (a +b \ln \left (c \,x^{n}\right )\right )+3 b^{3} n^{3} \cosh \left (3 b \ln \left (c \,x^{n}\right )+3 a \right )+27 b^{2} n^{2} \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )-b^{2} n^{2} \sinh \left (3 b \ln \left (c \,x^{n}\right )+3 a \right )+3 b n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right )-3 b n \cosh \left (3 b \ln \left (c \,x^{n}\right )+3 a \right )-3 \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+\sinh \left (3 b \ln \left (c \,x^{n}\right )+3 a \right )\right )}{36 b^{4} n^{4}-40 b^{2} n^{2}+4}\) | \(166\) |
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Time = 0.27 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.34 \[ \int \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {3 \, {\left (b^{3} n^{3} - b n\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 9 \, {\left (b^{3} n^{3} - b n\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - {\left (b^{2} n^{2} - 1\right )} x \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 3 \, {\left (9 \, b^{3} n^{3} - b n\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - 3 \, {\left ({\left (b^{2} n^{2} - 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - {\left (9 \, b^{2} n^{2} - 1\right )} x\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{4 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} \]
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\[ \int \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \int \sinh ^{3}{\left (a - \frac {\log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {1}{n} \\\int \sinh ^{3}{\left (a - \frac {\log {\left (c x^{n} \right )}}{3 n} \right )}\, dx & \text {for}\: b = - \frac {1}{3 n} \\\int \sinh ^{3}{\left (a + \frac {\log {\left (c x^{n} \right )}}{3 n} \right )}\, dx & \text {for}\: b = \frac {1}{3 n} \\\int \sinh ^{3}{\left (a + \frac {\log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {1}{n} \\\frac {9 b^{3} n^{3} x \sinh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} n^{2} + 1} - \frac {6 b^{3} n^{3} x \cosh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} n^{2} + 1} - \frac {7 b^{2} n^{2} x \sinh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} n^{2} + 1} + \frac {6 b^{2} n^{2} x \sinh {\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} n^{2} + 1} - \frac {3 b n x \sinh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} n^{2} + 1} + \frac {x \sinh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} n^{2} + 1} & \text {otherwise} \end {cases} \]
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Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.77 \[ \int \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {c^{3 \, b} x e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )}}{8 \, {\left (3 \, b n + 1\right )}} - \frac {3 \, c^{b} x e^{\left (b \log \left (x^{n}\right ) + a\right )}}{8 \, {\left (b n + 1\right )}} - \frac {3 \, x e^{\left (-b \log \left (x^{n}\right ) - a\right )}}{8 \, {\left (b c^{b} n - c^{b}\right )}} + \frac {x e^{\left (-3 \, b \log \left (x^{n}\right ) - 3 \, a\right )}}{8 \, {\left (3 \, b c^{3 \, b} n - c^{3 \, b}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 665 vs. \(2 (150) = 300\).
Time = 0.30 (sec) , antiderivative size = 665, normalized size of antiderivative = 4.46 \[ \int \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {3 \, b^{3} c^{3 \, b} n^{3} x x^{3 \, b n} e^{\left (3 \, a\right )}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} - \frac {27 \, b^{3} c^{b} n^{3} x x^{b n} e^{a}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} - \frac {b^{2} c^{3 \, b} n^{2} x x^{3 \, b n} e^{\left (3 \, a\right )}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} + \frac {27 \, b^{2} c^{b} n^{2} x x^{b n} e^{a}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} - \frac {3 \, b c^{3 \, b} n x x^{3 \, b n} e^{\left (3 \, a\right )}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} - \frac {27 \, b^{3} n^{3} x e^{\left (-a\right )}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{b} x^{b n}} + \frac {3 \, b^{3} n^{3} x e^{\left (-3 \, a\right )}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{3 \, b} x^{3 \, b n}} + \frac {3 \, b c^{b} n x x^{b n} e^{a}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} + \frac {c^{3 \, b} x x^{3 \, b n} e^{\left (3 \, a\right )}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} - \frac {27 \, b^{2} n^{2} x e^{\left (-a\right )}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{b} x^{b n}} + \frac {b^{2} n^{2} x e^{\left (-3 \, a\right )}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{3 \, b} x^{3 \, b n}} - \frac {3 \, c^{b} x x^{b n} e^{a}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} + \frac {3 \, b n x e^{\left (-a\right )}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{b} x^{b n}} - \frac {3 \, b n x e^{\left (-3 \, a\right )}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{3 \, b} x^{3 \, b n}} + \frac {3 \, x e^{\left (-a\right )}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{b} x^{b n}} - \frac {x e^{\left (-3 \, a\right )}}{8 \, {\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{3 \, b} x^{3 \, b n}} \]
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Time = 1.41 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.62 \[ \int \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x\,{\mathrm {e}}^{-3\,a}}{{\left (c\,x^n\right )}^{3\,b}\,\left (24\,b\,n-8\right )}-\frac {3\,x\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (8\,b\,n-8\right )}+\frac {x\,{\mathrm {e}}^{3\,a}\,{\left (c\,x^n\right )}^{3\,b}}{24\,b\,n+8}-\frac {3\,x\,{\mathrm {e}}^a\,{\left (c\,x^n\right )}^b}{8\,b\,n+8} \]
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