Integrand size = 12, antiderivative size = 12 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=-\frac {b \cosh (2 b x)}{4 x}-\frac {b \sinh ^2(b x)}{2 x}-\frac {\sinh (2 b x)}{8 x^2}-\frac {\cosh (b x) \text {Shi}(b x)}{2 x^2}-\frac {b \sinh (b x) \text {Shi}(b x)}{2 x}+b^2 \text {Shi}(2 b x)+\frac {1}{2} b^2 \text {Int}\left (\frac {\cosh (b x) \text {Shi}(b x)}{x},x\right ) \]
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Not integrable
Time = 0.14 (sec) , antiderivative size = 12, 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 = {} \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh (b x) \text {Shi}(b x)}{2 x^2}+\frac {1}{2} b \int \frac {\cosh (b x) \sinh (b x)}{b x^3} \, dx+\frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2} \, dx \\ & = -\frac {\cosh (b x) \text {Shi}(b x)}{2 x^2}-\frac {b \sinh (b x) \text {Shi}(b x)}{2 x}+\frac {1}{2} \int \frac {\cosh (b x) \sinh (b x)}{x^3} \, dx+\frac {1}{2} b^2 \int \frac {\sinh ^2(b x)}{b x^2} \, dx+\frac {1}{2} b^2 \int \frac {\cosh (b x) \text {Shi}(b x)}{x} \, dx \\ & = -\frac {\cosh (b x) \text {Shi}(b x)}{2 x^2}-\frac {b \sinh (b x) \text {Shi}(b x)}{2 x}+\frac {1}{2} \int \frac {\sinh (2 b x)}{2 x^3} \, dx+\frac {1}{2} b \int \frac {\sinh ^2(b x)}{x^2} \, dx+\frac {1}{2} b^2 \int \frac {\cosh (b x) \text {Shi}(b x)}{x} \, dx \\ & = -\frac {b \sinh ^2(b x)}{2 x}-\frac {\cosh (b x) \text {Shi}(b x)}{2 x^2}-\frac {b \sinh (b x) \text {Shi}(b x)}{2 x}+\frac {1}{4} \int \frac {\sinh (2 b x)}{x^3} \, dx-\left (i b^2\right ) \int \frac {i \sinh (2 b x)}{2 x} \, dx+\frac {1}{2} b^2 \int \frac {\cosh (b x) \text {Shi}(b x)}{x} \, dx \\ & = -\frac {b \sinh ^2(b x)}{2 x}-\frac {\sinh (2 b x)}{8 x^2}-\frac {\cosh (b x) \text {Shi}(b x)}{2 x^2}-\frac {b \sinh (b x) \text {Shi}(b x)}{2 x}+\frac {1}{4} b \int \frac {\cosh (2 b x)}{x^2} \, dx+\frac {1}{2} b^2 \int \frac {\sinh (2 b x)}{x} \, dx+\frac {1}{2} b^2 \int \frac {\cosh (b x) \text {Shi}(b x)}{x} \, dx \\ & = -\frac {b \cosh (2 b x)}{4 x}-\frac {b \sinh ^2(b x)}{2 x}-\frac {\sinh (2 b x)}{8 x^2}-\frac {\cosh (b x) \text {Shi}(b x)}{2 x^2}-\frac {b \sinh (b x) \text {Shi}(b x)}{2 x}+\frac {1}{2} b^2 \text {Shi}(2 b x)+\frac {1}{2} b^2 \int \frac {\sinh (2 b x)}{x} \, dx+\frac {1}{2} b^2 \int \frac {\cosh (b x) \text {Shi}(b x)}{x} \, dx \\ & = -\frac {b \cosh (2 b x)}{4 x}-\frac {b \sinh ^2(b x)}{2 x}-\frac {\sinh (2 b x)}{8 x^2}-\frac {\cosh (b x) \text {Shi}(b x)}{2 x^2}-\frac {b \sinh (b x) \text {Shi}(b x)}{2 x}+b^2 \text {Shi}(2 b x)+\frac {1}{2} b^2 \int \frac {\cosh (b x) \text {Shi}(b x)}{x} \, dx \\ \end{align*}
Not integrable
Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx \]
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Not integrable
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00
\[\int \frac {\cosh \left (b x \right ) \operatorname {Shi}\left (b x \right )}{x^{3}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}} \,d x } \]
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Not integrable
Time = 2.73 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\cosh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}}{x^{3}}\, dx \]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}} \,d x } \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}} \,d x } \]
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Not integrable
Time = 5.16 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right )}{x^3} \,d x \]
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