Integrand size = 25, antiderivative size = 193 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=\frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {2 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {3 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)}}-\frac {3 \arctan \left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}} \]
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Time = 0.60 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6852, 2320, 398, 1172, 12, 294, 209} \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=-\frac {3 \arctan \left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}+\frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}+\frac {3 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right ) \sqrt {\coth ^2(a c+b c x)}}-\frac {2 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right )^2 \sqrt {\coth ^2(a c+b c x)}} \]
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Rule 12
Rule 209
Rule 294
Rule 398
Rule 1172
Rule 2320
Rule 6852
Rubi steps \begin{align*} \text {integral}& = \frac {\coth (a c+b c x) \int e^{c (a+b x)} \tanh ^3(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 )^3}{\left (1+x^2\right )^3} \, 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+3 x^4\right )}{\left (1+x^2\right )^3}\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+3 x^4}{\left (1+x^2\right )^3} \, 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 e^{c (a+b x)} \coth (a c+b c x)}{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 {12 x^2}{\left (1+x^2\right )^2} \, dx,x,e^{c (a+b x)}\right )}{2 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 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(a c+b c x)}}-\frac {(6 \coth (a c+b c x)) \text {Subst}\left (\int \frac {x^2}{\left (1+x^2\right )^2} \, 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 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {3 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)}}-\frac {(3 \coth (a c+b c x)) \text {Subst}\left (\int \frac {1}{1+x^2} \, 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 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {3 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)}}-\frac {3 \arctan \left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.54 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=\frac {\left (e^{c (a+b x)} \left (2+5 e^{2 c (a+b x)}+e^{4 c (a+b x)}\right )-3 \left (1+e^{2 c (a+b x)}\right )^2 \arctan \left (e^{c (a+b x)}\right )\right ) \coth (c (a+b x))}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(c (a+b x))}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.61 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.68
method | result | size |
default | \(\frac {\operatorname {csgn}\left (\coth \left (c \left (b x +a \right )\right )\right ) \left (\frac {\sinh \left (b c x +a c \right )^{2}}{\cosh \left (b c x +a c \right )}+\frac {2}{\cosh \left (b c x +a c \right )}+\frac {\sinh \left (b c x +a c \right )^{3}}{\cosh \left (b c x +a c \right )^{2}}+\frac {3 \sinh \left (b c x +a c \right )}{\cosh \left (b c x +a c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (b c x +a c \right ) \tanh \left (b c x +a c \right )}{2}-3 \arctan \left ({\mathrm e}^{b c x +a c}\right )\right )}{c b}\) | \(131\) |
risch | \(\frac {3 i {\mathrm e}^{4 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}-i\right )-3 i {\mathrm e}^{4 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}+i\right )+2 \,{\mathrm e}^{5 c \left (b x +a \right )}+6 i {\mathrm e}^{2 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}-i\right )-6 i {\mathrm e}^{2 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}+i\right )+10 \,{\mathrm e}^{3 c \left (b x +a \right )}+3 i \ln \left ({\mathrm e}^{c \left (b x +a \right )}-i\right )-3 i \ln \left ({\mathrm e}^{c \left (b x +a \right )}+i\right )+4 \,{\mathrm e}^{c \left (b x +a \right )}}{2 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, c b}\) | \(223\) |
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Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (179) = 358\).
Time = 0.27 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.37 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=\frac {\cosh \left (b c x + a c\right )^{5} + 5 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{4} + \sinh \left (b c x + a c\right )^{5} + 5 \, {\left (2 \, \cosh \left (b c x + a c\right )^{2} + 1\right )} \sinh \left (b c x + a c\right )^{3} + 5 \, \cosh \left (b c x + a c\right )^{3} + 5 \, {\left (2 \, \cosh \left (b c x + a c\right )^{3} + 3 \, \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{2} - 3 \, {\left (\cosh \left (b c x + a c\right )^{4} + 4 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{3} + \sinh \left (b c x + a c\right )^{4} + 2 \, {\left (3 \, \cosh \left (b c x + a c\right )^{2} + 1\right )} \sinh \left (b c x + a c\right )^{2} + 2 \, \cosh \left (b c x + a c\right )^{2} + 4 \, {\left (\cosh \left (b c x + a c\right )^{3} + \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right ) + 1\right )} \arctan \left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right ) + {\left (5 \, \cosh \left (b c x + a c\right )^{4} + 15 \, \cosh \left (b c x + a c\right )^{2} + 2\right )} \sinh \left (b c x + a c\right ) + 2 \, \cosh \left (b c x + a c\right )}{b c \cosh \left (b c x + a c\right )^{4} + 4 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{3} + b c \sinh \left (b c x + a c\right )^{4} + 2 \, b c \cosh \left (b c x + a c\right )^{2} + 2 \, {\left (3 \, b c \cosh \left (b c x + a c\right )^{2} + b c\right )} \sinh \left (b c x + a c\right )^{2} + b c + 4 \, {\left (b c \cosh \left (b c x + a c\right )^{3} + b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )} \]
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\[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=e^{a c} \int \frac {e^{b c x}}{\left (\coth ^{2}{\left (a c + b c x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.47 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=-\frac {3 \, \arctan \left (e^{\left (b c x + a c\right )}\right )}{b c} + \frac {e^{\left (5 \, b c x + 5 \, a c\right )} + 5 \, e^{\left (3 \, b c x + 3 \, a c\right )} + 2 \, e^{\left (b c x + a c\right )}}{b c {\left (e^{\left (4 \, b c x + 4 \, a c\right )} + 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.67 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=-\frac {{\left (3 \, \arctan \left (e^{\left (b c x + a c\right )}\right ) e^{\left (-a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - e^{\left (b c x\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - \frac {3 \, e^{\left (3 \, b c x + 2 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + e^{\left (b c x\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}{{\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{2}}\right )} e^{\left (a c\right )}}{b c} \]
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Timed out. \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=\int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{{\left ({\mathrm {coth}\left (a\,c+b\,c\,x\right )}^2\right )}^{3/2}} \,d x \]
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