Integrand size = 25, antiderivative size = 319 \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{5/2} \, dx=\frac {e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c}-\frac {4 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4}+\frac {26 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}-\frac {55 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{6 b c \left (1-e^{2 c (a+b x)}\right )^2}+\frac {25 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{4 b c \left (1-e^{2 c (a+b x)}\right )}-\frac {15 \text {arctanh}\left (e^{c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{4 b c} \]
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Time = 0.64 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6852, 2320, 398, 1828, 1171, 393, 213} \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{5/2} \, dx=-\frac {15 \text {arctanh}\left (e^{c (a+b x)}\right ) \tanh (a c+b c x) \sqrt {\coth ^2(a c+b c x)}}{4 b c}+\frac {e^{c (a+b x)} \tanh (a c+b c x) \sqrt {\coth ^2(a c+b c x)}}{b c}+\frac {25 e^{c (a+b x)} \tanh (a c+b c x) \sqrt {\coth ^2(a c+b c x)}}{4 b c \left (1-e^{2 c (a+b x)}\right )}-\frac {55 e^{c (a+b x)} \tanh (a c+b c x) \sqrt {\coth ^2(a c+b c x)}}{6 b c \left (1-e^{2 c (a+b x)}\right )^2}+\frac {26 e^{c (a+b x)} \tanh (a c+b c x) \sqrt {\coth ^2(a c+b c x)}}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}-\frac {4 e^{c (a+b x)} \tanh (a c+b c x) \sqrt {\coth ^2(a c+b c x)}}{b c \left (1-e^{2 c (a+b x)}\right )^4} \]
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Rule 213
Rule 393
Rule 398
Rule 1171
Rule 1828
Rule 2320
Rule 6852
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)\right ) \int e^{c (a+b x)} \coth ^5(a c+b c x) \, dx \\ & = \frac {\left (\sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)\right ) \text {Subst}\left (\int \frac {\left (1+x^2\right )^5}{\left (-1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c} \\ & = \frac {\left (\sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)\right ) \text {Subst}\left (\int \left (1+\frac {2 \left (1+10 x^4+5 x^8\right )}{\left (-1+x^2\right )^5}\right ) \, dx,x,e^{c (a+b x)}\right )}{b c} \\ & = \frac {e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c}+\frac {\left (2 \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)\right ) \text {Subst}\left (\int \frac {1+10 x^4+5 x^8}{\left (-1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c} \\ & = \frac {e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c}-\frac {4 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4}+\frac {\left (\sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)\right ) \text {Subst}\left (\int \frac {8+120 x^2+40 x^4+40 x^6}{\left (-1+x^2\right )^4} \, dx,x,e^{c (a+b x)}\right )}{4 b c} \\ & = \frac {e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c}-\frac {4 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4}+\frac {26 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}+\frac {\left (\sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)\right ) \text {Subst}\left (\int \frac {160+480 x^2+240 x^4}{\left (-1+x^2\right )^3} \, dx,x,e^{c (a+b x)}\right )}{24 b c} \\ & = \frac {e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c}-\frac {4 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4}+\frac {26 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}-\frac {55 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{6 b c \left (1-e^{2 c (a+b x)}\right )^2}+\frac {\left (\sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)\right ) \text {Subst}\left (\int \frac {240+960 x^2}{\left (-1+x^2\right )^2} \, dx,x,e^{c (a+b x)}\right )}{96 b c} \\ & = \frac {e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c}-\frac {4 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4}+\frac {26 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}-\frac {55 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{6 b c \left (1-e^{2 c (a+b x)}\right )^2}+\frac {25 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{4 b c \left (1-e^{2 c (a+b x)}\right )}+\frac {\left (15 \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,e^{c (a+b x)}\right )}{4 b c} \\ & = \frac {e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c}-\frac {4 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4}+\frac {26 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}-\frac {55 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{6 b c \left (1-e^{2 c (a+b x)}\right )^2}+\frac {25 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{4 b c \left (1-e^{2 c (a+b x)}\right )}-\frac {15 \text {arctanh}\left (e^{c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{4 b c} \\ \end{align*}
Time = 6.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.51 \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{5/2} \, dx=\frac {\sqrt {\coth ^2(c (a+b x))} \left (66 e^{c (a+b x)}-314 e^{3 c (a+b x)}+374 e^{5 c (a+b x)}-246 e^{7 c (a+b x)}+24 e^{9 c (a+b x)}+45 \left (-1+e^{2 c (a+b x)}\right )^4 \log \left (1-e^{c (a+b x)}\right )-45 \left (-1+e^{2 c (a+b x)}\right )^4 \log \left (1+e^{c (a+b x)}\right )\right ) \tanh (c (a+b x))}{24 b c \left (-1+e^{2 c (a+b x)}\right )^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.21 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {\operatorname {csgn}\left (\coth \left (c \left (b x +a \right )\right )\right ) \left (\frac {\cosh \left (b c x +a c \right )^{5}}{\sinh \left (b c x +a c \right )^{4}}-\frac {5 \cosh \left (b c x +a c \right )^{3}}{\sinh \left (b c x +a c \right )^{4}}+\frac {5 \cosh \left (b c x +a c \right )}{\sinh \left (b c x +a c \right )^{4}}+5 \left (-\frac {\operatorname {csch}\left (b c x +a c \right )^{3}}{4}+\frac {3 \,\operatorname {csch}\left (b c x +a c \right )}{8}\right ) \coth \left (b c x +a c \right )-\frac {15 \,\operatorname {arctanh}\left ({\mathrm e}^{b c x +a c}\right )}{4}+\frac {\cosh \left (b c x +a c \right )^{4}}{\sinh \left (b c x +a c \right )^{3}}-\frac {4 \cosh \left (b c x +a c \right )^{2}}{\sinh \left (b c x +a c \right )^{3}}+\frac {8}{3 \sinh \left (b c x +a c \right )^{3}}\right )}{c b}\) | \(195\) |
risch | \(\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, {\mathrm e}^{c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) b c}-\frac {\sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, {\mathrm e}^{c \left (b x +a \right )} \left (75 \,{\mathrm e}^{6 c \left (b x +a \right )}-115 \,{\mathrm e}^{4 c \left (b x +a \right )}+109 \,{\mathrm e}^{2 c \left (b x +a \right )}-21\right )}{12 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{3} c b}-\frac {15 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{c \left (b x +a \right )}+1\right )}{8 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) c b}+\frac {15 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{c \left (b x +a \right )}-1\right )}{8 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) c b}\) | \(320\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1617 vs. \(2 (281) = 562\).
Time = 0.27 (sec) , antiderivative size = 1617, normalized size of antiderivative = 5.07 \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{5/2} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.52 \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{5/2} \, dx=-\frac {15 \, \log \left (e^{\left (b c x + a c\right )} + 1\right )}{8 \, b c} + \frac {15 \, \log \left (e^{\left (b c x + a c\right )} - 1\right )}{8 \, b c} + \frac {12 \, e^{\left (9 \, b c x + 9 \, a c\right )} - 123 \, e^{\left (7 \, b c x + 7 \, a c\right )} + 187 \, e^{\left (5 \, b c x + 5 \, a c\right )} - 157 \, e^{\left (3 \, b c x + 3 \, a c\right )} + 33 \, e^{\left (b c x + a c\right )}}{12 \, b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} - 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
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Time = 0.63 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.57 \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{5/2} \, dx=\frac {\frac {24 \, e^{\left (b c x + a c\right )}}{\mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )} - \frac {45 \, \log \left (e^{\left (b c x + a c\right )} + 1\right )}{\mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )} + \frac {45 \, \log \left ({\left | e^{\left (b c x + a c\right )} - 1 \right |}\right )}{\mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )} - \frac {2 \, {\left (75 \, e^{\left (7 \, b c x + 7 \, a c\right )} - 115 \, e^{\left (5 \, b c x + 5 \, a c\right )} + 109 \, e^{\left (3 \, b c x + 3 \, a c\right )} - 21 \, e^{\left (b c x + a c\right )}\right )}}{{\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}^{4} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}}{24 \, b c} \]
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Timed out. \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{5/2} \, dx=\int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,{\left ({\mathrm {coth}\left (a\,c+b\,c\,x\right )}^2\right )}^{5/2} \,d x \]
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