Integrand size = 19, antiderivative size = 37 \[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=-\frac {\text {csch}^n(a+b x)}{b n}-\frac {\text {csch}^{2+n}(a+b x)}{b (2+n)} \]
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Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2701, 14} \[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=-\frac {\text {csch}^{n+2}(a+b x)}{b (n+2)}-\frac {\text {csch}^n(a+b x)}{b n} \]
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Rule 14
Rule 2701
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^{-1+n} \left (-1-x^2\right ) \, dx,x,\text {csch}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (-x^{-1+n}-x^{1+n}\right ) \, dx,x,\text {csch}(a+b x)\right )}{b} \\ & = -\frac {\text {csch}^n(a+b x)}{b n}-\frac {\text {csch}^{2+n}(a+b x)}{b (2+n)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=-\frac {\text {csch}^n(a+b x) \left (2+n+n \text {csch}^2(a+b x)\right )}{b n (2+n)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.86 (sec) , antiderivative size = 479, normalized size of antiderivative = 12.95
method | result | size |
risch | \(-\frac {\left (n \,{\mathrm e}^{4 b x +4 a}+2 \,{\mathrm e}^{4 b x +4 a}+2 n \,{\mathrm e}^{2 b x +2 a}-4 \,{\mathrm e}^{2 b x +2 a}+n +2\right ) 2^{n} \left ({\mathrm e}^{b x +a}-1\right )^{-n} \left ({\mathrm e}^{b x +a}\right )^{n} \left ({\mathrm e}^{b x +a}+1\right )^{-n} {\mathrm e}^{-\frac {i \pi n \left (\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{b x +a}-1}\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{b x +a}+1}\right ) \operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{b x +a}-1\right ) \left ({\mathrm e}^{b x +a}+1\right )}\right )-\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{b x +a}-1}\right ) {\operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{b x +a}-1\right ) \left ({\mathrm e}^{b x +a}+1\right )}\right )}^{2}+\operatorname {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{\left ({\mathrm e}^{b x +a}+1\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right )^{3}-\operatorname {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{\left ({\mathrm e}^{b x +a}+1\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{b x +a}\right )-\operatorname {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{\left ({\mathrm e}^{b x +a}+1\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{b x +a}-1\right ) \left ({\mathrm e}^{b x +a}+1\right )}\right )+\operatorname {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{\left ({\mathrm e}^{b x +a}+1\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right ) \operatorname {csgn}\left (i {\mathrm e}^{b x +a}\right ) \operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{b x +a}-1\right ) \left ({\mathrm e}^{b x +a}+1\right )}\right )-\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{b x +a}+1}\right ) {\operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{b x +a}-1\right ) \left ({\mathrm e}^{b x +a}+1\right )}\right )}^{2}+{\operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{b x +a}-1\right ) \left ({\mathrm e}^{b x +a}+1\right )}\right )}^{3}\right )}{2}}}{b n \left (n +2\right ) \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}\) | \(479\) |
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Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (37) = 74\).
Time = 0.26 (sec) , antiderivative size = 216, normalized size of antiderivative = 5.84 \[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=\frac {{\left ({\left (n + 2\right )} \cosh \left (b x + a\right )^{2} + {\left (n + 2\right )} \sinh \left (b x + a\right )^{2} + n - 2\right )} \cosh \left (n \log \left (\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )\right ) + {\left ({\left (n + 2\right )} \cosh \left (b x + a\right )^{2} + {\left (n + 2\right )} \sinh \left (b x + a\right )^{2} + n - 2\right )} \sinh \left (n \log \left (\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )\right )}{b n^{2} - {\left (b n^{2} + 2 \, b n\right )} \cosh \left (b x + a\right )^{2} - {\left (b n^{2} + 2 \, b n\right )} \sinh \left (b x + a\right )^{2} + 2 \, b n} \]
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\[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=\int \coth ^{3}{\left (a + b x \right )} \operatorname {csch}^{n}{\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (37) = 74\).
Time = 0.34 (sec) , antiderivative size = 414, normalized size of antiderivative = 11.19 \[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=-\frac {2^{n} n e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right )\right )}}{{\left (n^{2} - 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac {{\left (2^{n + 1} n - 2^{n + 2}\right )} e^{\left (-{\left (b x + a\right )} n - 2 \, b x - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right ) - 2 \, a\right )}}{{\left (n^{2} - 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac {{\left (2^{n} n + 2^{n + 1}\right )} e^{\left (-{\left (b x + a\right )} n - 4 \, b x - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right ) - 4 \, a\right )}}{{\left (n^{2} - 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac {2^{n + 1} e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right )\right )}}{{\left (n^{2} - 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} \]
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\[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=\int { \operatorname {csch}\left (b x + a\right )^{n} \coth \left (b x + a\right )^{3} \,d x } \]
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Time = 2.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.70 \[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=-\frac {{\left (\frac {1}{\frac {{\mathrm {e}}^{a+b\,x}}{2}-\frac {{\mathrm {e}}^{-a-b\,x}}{2}}\right )}^n\,\left (\frac {1}{b\,n}+\frac {{\mathrm {e}}^{4\,a+4\,b\,x}}{b\,n}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}\,\left (2\,n-4\right )}{b\,n\,\left (n+2\right )}\right )}{{\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1} \]
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