Integrand size = 18, antiderivative size = 18 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\frac {1}{2} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x)+\text {Int}\left (\frac {\coth (a+b x)}{x},x\right ) \]
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Not integrable
Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\coth (a+b x)}{x} \, dx+\int \frac {\cosh (a+b x) \sinh (a+b x)}{x} \, dx \\ & = \int \frac {\coth (a+b x)}{x} \, dx+\int \frac {\sinh (2 a+2 b x)}{2 x} \, dx \\ & = \frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{x} \, dx+\int \frac {\coth (a+b x)}{x} \, dx \\ & = \frac {1}{2} \cosh (2 a) \int \frac {\sinh (2 b x)}{x} \, dx+\frac {1}{2} \sinh (2 a) \int \frac {\cosh (2 b x)}{x} \, dx+\int \frac {\coth (a+b x)}{x} \, dx \\ & = \frac {1}{2} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x)+\int \frac {\coth (a+b x)}{x} \, dx \\ \end{align*}
Not integrable
Time = 7.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx \]
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Not integrable
Time = 0.40 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00
\[\int \frac {\cosh \left (b x +a \right )^{3} \operatorname {csch}\left (b x +a \right )}{x}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )}{x} \,d x } \]
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Not integrable
Time = 21.80 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int \frac {\cosh ^{3}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}}{x}\, dx \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.17 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )}{x} \,d x } \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )}{x} \,d x } \]
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Not integrable
Time = 2.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{x\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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