Integrand size = 25, antiderivative size = 311 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{5/2}} \, dx=\frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(a c+b c x)}}+\frac {26 e^{c (a+b x)} \coth (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^3 \sqrt {\coth ^2(a c+b c x)}}-\frac {55 e^{c (a+b x)} \coth (a c+b c x)}{6 b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {25 e^{c (a+b x)} \coth (a c+b c x)}{4 b c \left (1+e^{2 c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)}}-\frac {15 \arctan \left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{4 b c \sqrt {\coth ^2(a c+b c x)}} \]
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Time = 1.22 (sec) , antiderivative size = 311, 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, 209} \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{5/2}} \, dx=-\frac {15 \arctan \left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{4 b c \sqrt {\coth ^2(a c+b c x)}}+\frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}+\frac {25 e^{c (a+b x)} \coth (a c+b c x)}{4 b c \left (e^{2 c (a+b x)}+1\right ) \sqrt {\coth ^2(a c+b c x)}}-\frac {55 e^{c (a+b x)} \coth (a c+b c x)}{6 b c \left (e^{2 c (a+b x)}+1\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {26 e^{c (a+b x)} \coth (a c+b c x)}{3 b c \left (e^{2 c (a+b x)}+1\right )^3 \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right )^4 \sqrt {\coth ^2(a c+b c x)}} \]
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Rule 209
Rule 393
Rule 398
Rule 1171
Rule 1828
Rule 2320
Rule 6852
Rubi steps \begin{align*} \text {integral}& = \frac {\coth (a c+b c x) \int e^{c (a+b x)} \tanh ^5(a c+b c x) \, dx}{\sqrt {\coth ^2(a c+b c x)}} \\ & = \frac {\coth (a c+b c x) \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 \sqrt {\coth ^2(a c+b c x)}} \\ & = \frac {\coth (a c+b c x) \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 \sqrt {\coth ^2(a c+b c x)}} \\ & = \frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {(2 \coth (a c+b c x)) \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 \sqrt {\coth ^2(a c+b c x)}} \\ & = \frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(a c+b c x)}}+\frac {\coth (a c+b c x) \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 \sqrt {\coth ^2(a c+b c x)}} \\ & = \frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(a c+b c x)}}+\frac {26 e^{c (a+b x)} \coth (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^3 \sqrt {\coth ^2(a c+b c x)}}-\frac {\coth (a c+b c x) \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 \sqrt {\coth ^2(a c+b c x)}} \\ & = \frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(a c+b c x)}}+\frac {26 e^{c (a+b x)} \coth (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^3 \sqrt {\coth ^2(a c+b c x)}}-\frac {55 e^{c (a+b x)} \coth (a c+b c x)}{6 b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {\coth (a c+b c x) \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 \sqrt {\coth ^2(a c+b c x)}} \\ & = \frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(a c+b c x)}}+\frac {26 e^{c (a+b x)} \coth (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^3 \sqrt {\coth ^2(a c+b c x)}}-\frac {55 e^{c (a+b x)} \coth (a c+b c x)}{6 b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {25 e^{c (a+b x)} \coth (a c+b c x)}{4 b c \left (1+e^{2 c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)}}-\frac {(15 \coth (a c+b c x)) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^{c (a+b x)}\right )}{4 b c \sqrt {\coth ^2(a c+b c x)}} \\ & = \frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(a c+b c x)}}+\frac {26 e^{c (a+b x)} \coth (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^3 \sqrt {\coth ^2(a c+b c x)}}-\frac {55 e^{c (a+b x)} \coth (a c+b c x)}{6 b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {25 e^{c (a+b x)} \coth (a c+b c x)}{4 b c \left (1+e^{2 c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)}}-\frac {15 \arctan \left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{4 b c \sqrt {\coth ^2(a c+b c x)}} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.43 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{5/2}} \, dx=\frac {\left (e^{c (a+b x)} \left (33+157 e^{2 c (a+b x)}+187 e^{4 c (a+b x)}+123 e^{6 c (a+b x)}+12 e^{8 c (a+b x)}\right )-45 \left (1+e^{2 c (a+b x)}\right )^4 \arctan \left (e^{c (a+b x)}\right )\right ) \coth (c (a+b x))}{12 b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(c (a+b x))}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.59 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.63
method | result | size |
default | \(\frac {\operatorname {csgn}\left (\coth \left (c \left (b x +a \right )\right )\right ) \left (\frac {\sinh \left (b c x +a c \right )^{4}}{\cosh \left (b c x +a c \right )^{3}}+\frac {4 \sinh \left (b c x +a c \right )^{2}}{\cosh \left (b c x +a c \right )^{3}}+\frac {8}{3 \cosh \left (b c x +a c \right )^{3}}+\frac {\sinh \left (b c x +a c \right )^{5}}{\cosh \left (b c x +a c \right )^{4}}+\frac {5 \sinh \left (b c x +a c \right )^{3}}{\cosh \left (b c x +a c \right )^{4}}+\frac {5 \sinh \left (b c x +a c \right )}{\cosh \left (b c x +a c \right )^{4}}-5 \left (\frac {\operatorname {sech}\left (b c x +a c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (b c x +a c \right )}{8}\right ) \tanh \left (b c x +a c \right )-\frac {15 \arctan \left ({\mathrm e}^{b c x +a c}\right )}{4}\right )}{c b}\) | \(195\) |
risch | \(\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) {\mathrm e}^{c \left (b x +a \right )}}{\sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) b c}+\frac {{\mathrm e}^{c \left (b x +a \right )} \left (75 \,{\mathrm e}^{6 c \left (b x +a \right )}+115 \,{\mathrm e}^{4 c \left (b x +a \right )}+109 \,{\mathrm e}^{2 c \left (b x +a \right )}+21\right )}{12 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{3} \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, c b}+\frac {15 i \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \ln \left ({\mathrm e}^{c \left (b x +a \right )}-i\right )}{8 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, c b}-\frac {15 i \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \ln \left ({\mathrm e}^{c \left (b x +a \right )}+i\right )}{8 \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) c b}\) | \(324\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1226 vs. \(2 (281) = 562\).
Time = 0.28 (sec) , antiderivative size = 1226, normalized size of antiderivative = 3.94 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{5/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{5/2}} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.47 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{5/2}} \, dx=-\frac {15 \, \arctan \left (e^{\left (b c x + a c\right )}\right )}{4 \, b c} + \frac {12 \, e^{\left (9 \, b c x + 9 \, a c\right )} + 123 \, e^{\left (7 \, b c x + 7 \, a c\right )} + 187 \, e^{\left (5 \, b c x + 5 \, a c\right )} + 157 \, e^{\left (3 \, b c x + 3 \, a c\right )} + 33 \, e^{\left (b c x + a c\right )}}{12 \, b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} + 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.59 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{5/2}} \, dx=-\frac {{\left (45 \, \arctan \left (e^{\left (b c x + a c\right )}\right ) e^{\left (-a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - 12 \, e^{\left (b c x\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - \frac {75 \, e^{\left (7 \, b c x + 6 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + 115 \, e^{\left (5 \, b c x + 4 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + 109 \, e^{\left (3 \, b c x + 2 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + 21 \, e^{\left (b c x\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}{{\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{4}}\right )} e^{\left (a c\right )}}{12 \, b c} \]
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Timed out. \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{5/2}} \, dx=\int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{{\left ({\mathrm {coth}\left (a\,c+b\,c\,x\right )}^2\right )}^{5/2}} \,d x \]
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