Optimal. Leaf size=101 \[ \frac{1}{2} b^2 \text{CannotIntegrate}\left (\frac{\text{Shi}(b x) \cosh (b x)}{x},x\right )+b^2 \text{Shi}(2 b x)-\frac{\text{Shi}(b x) \cosh (b x)}{2 x^2}-\frac{b \text{Shi}(b x) \sinh (b x)}{2 x}-\frac{\sinh (2 b x)}{8 x^2}-\frac{b \sinh ^2(b x)}{2 x}-\frac{b \cosh (2 b x)}{4 x} \]
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Rubi [A] time = 0.196629, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\cosh (b x) \text{Shi}(b x)}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\cosh (b x) \text{Shi}(b x)}{x^3} \, dx &=-\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*}
Mathematica [A] time = 0.445615, size = 0, normalized size = 0. \[ \int \frac{\cosh (b x) \text{Shi}(b x)}{x^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cosh \left ( bx \right ){\it Shi} \left ( bx \right ) }{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cosh \left (b x\right ) \operatorname{Shi}\left (b x\right )}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (b x \right )} \operatorname{Shi}{\left (b x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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