Integrand size = 25, antiderivative size = 83 \[ \int e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \, dx=\frac {e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c}-\frac {2 \text {arctanh}\left (e^{c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c} \]
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Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6852, 2320, 396, 212} \[ \int e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \, dx=\frac {e^{c (a+b x)} \tanh (a c+b c x) \sqrt {\coth ^2(a c+b c x)}}{b c}-\frac {2 \text {arctanh}\left (e^{c (a+b x)}\right ) \tanh (a c+b c x) \sqrt {\coth ^2(a c+b c x)}}{b c} \]
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Rule 212
Rule 396
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 (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 {-1-x^2}{1-x^2} \, 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}{1-x^2} \, 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 {2 \text {arctanh}\left (e^{c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.61 \[ \int e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \, dx=\frac {\left (e^{c (a+b x)}-2 \text {arctanh}\left (e^{c (a+b x)}\right )\right ) \sqrt {\coth ^2(c (a+b x))} \tanh (c (a+b x))}{b c} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(212\) vs. \(2(77)=154\).
Time = 0.77 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.57
method | result | size |
risch | \(\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, {\mathrm e}^{c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) b c}+\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{c \left (b x +a \right )}-1\right )}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) c b}-\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{c \left (b x +a \right )}+1\right )}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) c b}\) | \(213\) |
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Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.84 \[ \int e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \, dx=\frac {\cosh \left (b c x + a c\right ) - \log \left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right ) + 1\right ) + \log \left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right ) - 1\right ) + \sinh \left (b c x + a c\right )}{b c} \]
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\[ \int e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \, dx=e^{a c} \int \sqrt {\coth ^{2}{\left (a c + b c x \right )}} e^{b c x}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.67 \[ \int e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \, dx=\frac {e^{\left (b c x + a c\right )}}{b c} - \frac {\log \left (e^{\left (b c x + a c\right )} + 1\right )}{b c} + \frac {\log \left (e^{\left (b c x + a c\right )} - 1\right )}{b c} \]
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Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.13 \[ \int e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \, dx=\frac {\frac {e^{\left (b c x + a c\right )}}{\mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )} - \frac {\log \left (e^{\left (b c x + a c\right )} + 1\right )}{\mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )} + \frac {\log \left ({\left | e^{\left (b c x + a c\right )} - 1 \right |}\right )}{\mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}}{b c} \]
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Timed out. \[ \int e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \, dx=\int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,\sqrt {{\mathrm {coth}\left (a\,c+b\,c\,x\right )}^2} \,d x \]
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