Optimal. Leaf size=96 \[ b^2 \text{Chi}(2 b x)+\frac{1}{4} b^2 \text{Shi}(b x)^2-\frac{\text{Shi}(b x) \sinh (b x)}{2 x^2}-\frac{b \text{Shi}(b x) \cosh (b x)}{2 x}-\frac{\sinh ^2(b x)}{4 x^2}-\frac{b \sinh (2 b x)}{4 x}-\frac{b \sinh (b x) \cosh (b x)}{2 x} \]
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Rubi [A] time = 0.220044, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6544, 6550, 6686, 12, 5448, 3297, 3301, 3314, 29, 3312} \[ b^2 \text{Chi}(2 b x)+\frac{1}{4} b^2 \text{Shi}(b x)^2-\frac{\text{Shi}(b x) \sinh (b x)}{2 x^2}-\frac{b \text{Shi}(b x) \cosh (b x)}{2 x}-\frac{\sinh ^2(b x)}{4 x^2}-\frac{b \sinh (2 b x)}{4 x}-\frac{b \sinh (b x) \cosh (b x)}{2 x} \]
Antiderivative was successfully verified.
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Rule 6544
Rule 6550
Rule 6686
Rule 12
Rule 5448
Rule 3297
Rule 3301
Rule 3314
Rule 29
Rule 3312
Rubi steps
\begin{align*} \int \frac{\sinh (b x) \text{Shi}(b x)}{x^3} \, dx &=-\frac{\sinh (b x) \text{Shi}(b x)}{2 x^2}+\frac{1}{2} b \int \frac{\sinh ^2(b x)}{b x^3} \, dx+\frac{1}{2} b \int \frac{\cosh (b x) \text{Shi}(b x)}{x^2} \, dx\\ &=-\frac{b \cosh (b x) \text{Shi}(b x)}{2 x}-\frac{\sinh (b x) \text{Shi}(b x)}{2 x^2}+\frac{1}{2} \int \frac{\sinh ^2(b x)}{x^3} \, dx+\frac{1}{2} b^2 \int \frac{\cosh (b x) \sinh (b x)}{b x^2} \, dx+\frac{1}{2} b^2 \int \frac{\sinh (b x) \text{Shi}(b x)}{x} \, dx\\ &=-\frac{b \cosh (b x) \sinh (b x)}{2 x}-\frac{\sinh ^2(b x)}{4 x^2}-\frac{b \cosh (b x) \text{Shi}(b x)}{2 x}-\frac{\sinh (b x) \text{Shi}(b x)}{2 x^2}+\frac{1}{4} b^2 \text{Shi}(b x)^2+\frac{1}{2} b \int \frac{\cosh (b x) \sinh (b x)}{x^2} \, dx+\frac{1}{2} b^2 \int \frac{1}{x} \, dx+b^2 \int \frac{\sinh ^2(b x)}{x} \, dx\\ &=\frac{1}{2} b^2 \log (x)-\frac{b \cosh (b x) \sinh (b x)}{2 x}-\frac{\sinh ^2(b x)}{4 x^2}-\frac{b \cosh (b x) \text{Shi}(b x)}{2 x}-\frac{\sinh (b x) \text{Shi}(b x)}{2 x^2}+\frac{1}{4} b^2 \text{Shi}(b x)^2+\frac{1}{2} b \int \frac{\sinh (2 b x)}{2 x^2} \, dx-b^2 \int \left (\frac{1}{2 x}-\frac{\cosh (2 b x)}{2 x}\right ) \, dx\\ &=-\frac{b \cosh (b x) \sinh (b x)}{2 x}-\frac{\sinh ^2(b x)}{4 x^2}-\frac{b \cosh (b x) \text{Shi}(b x)}{2 x}-\frac{\sinh (b x) \text{Shi}(b x)}{2 x^2}+\frac{1}{4} b^2 \text{Shi}(b x)^2+\frac{1}{4} b \int \frac{\sinh (2 b x)}{x^2} \, dx+\frac{1}{2} b^2 \int \frac{\cosh (2 b x)}{x} \, dx\\ &=\frac{1}{2} b^2 \text{Chi}(2 b x)-\frac{b \cosh (b x) \sinh (b x)}{2 x}-\frac{\sinh ^2(b x)}{4 x^2}-\frac{b \sinh (2 b x)}{4 x}-\frac{b \cosh (b x) \text{Shi}(b x)}{2 x}-\frac{\sinh (b x) \text{Shi}(b x)}{2 x^2}+\frac{1}{4} b^2 \text{Shi}(b x)^2+\frac{1}{2} b^2 \int \frac{\cosh (2 b x)}{x} \, dx\\ &=b^2 \text{Chi}(2 b x)-\frac{b \cosh (b x) \sinh (b x)}{2 x}-\frac{\sinh ^2(b x)}{4 x^2}-\frac{b \sinh (2 b x)}{4 x}-\frac{b \cosh (b x) \text{Shi}(b x)}{2 x}-\frac{\sinh (b x) \text{Shi}(b x)}{2 x^2}+\frac{1}{4} b^2 \text{Shi}(b x)^2\\ \end{align*}
Mathematica [A] time = 0.0168658, size = 96, normalized size = 1. \[ b^2 \text{Chi}(2 b x)+\frac{1}{4} b^2 \text{Shi}(b x)^2-\frac{\text{Shi}(b x) \sinh (b x)}{2 x^2}-\frac{b \text{Shi}(b x) \cosh (b x)}{2 x}-\frac{\sinh ^2(b x)}{4 x^2}-\frac{b \sinh (2 b x)}{4 x}-\frac{b \sinh (b x) \cosh (b x)}{2 x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Shi} \left ( bx \right ) \sinh \left ( bx \right ) }{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Shi}\left (b x\right ) \sinh \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sinh \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Shi}\left (b x\right ) \sinh \left (b x\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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