Integrand size = 18, antiderivative size = 18 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx=b \cosh (2 a) \text {Chi}(2 b x)-\frac {\sinh (2 a+2 b x)}{2 x}+b \sinh (2 a) \text {Shi}(2 b x)+\text {Int}\left (\frac {\coth (a+b x)}{x^2},x\right ) \]
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Not integrable
Time = 0.09 (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 = {} \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx=\int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\coth (a+b x)}{x^2} \, dx+\int \frac {\cosh (a+b x) \sinh (a+b x)}{x^2} \, dx \\ & = \int \frac {\coth (a+b x)}{x^2} \, dx+\int \frac {\sinh (2 a+2 b x)}{2 x^2} \, dx \\ & = \frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{x^2} \, dx+\int \frac {\coth (a+b x)}{x^2} \, dx \\ & = -\frac {\sinh (2 a+2 b x)}{2 x}+b \int \frac {\cosh (2 a+2 b x)}{x} \, dx+\int \frac {\coth (a+b x)}{x^2} \, dx \\ & = -\frac {\sinh (2 a+2 b x)}{2 x}+(b \cosh (2 a)) \int \frac {\cosh (2 b x)}{x} \, dx+(b \sinh (2 a)) \int \frac {\sinh (2 b x)}{x} \, dx+\int \frac {\coth (a+b x)}{x^2} \, dx \\ & = b \cosh (2 a) \text {Chi}(2 b x)-\frac {\sinh (2 a+2 b x)}{2 x}+b \sinh (2 a) \text {Shi}(2 b x)+\int \frac {\coth (a+b x)}{x^2} \, dx \\ \end{align*}
Not integrable
Time = 7.40 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx=\int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx \]
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Not integrable
Time = 0.44 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00
\[\int \frac {\cosh \left (b x +a \right )^{3} \operatorname {csch}\left (b x +a \right )}{x^{2}}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )}{x^{2}} \,d x } \]
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Not integrable
Time = 26.59 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx=\int \frac {\cosh ^{3}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}}{x^{2}}\, dx \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 72, normalized size of antiderivative = 4.00 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )}{x^{2}} \,d x } \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )}{x^{2}} \,d x } \]
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Not integrable
Time = 2.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{x^2\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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