Optimal. Leaf size=46 \[ \frac{b \sinh (a) \text{Chi}(b x)}{a}-\frac{b \text{Shi}(a+b x)}{a}-\frac{\text{Shi}(a+b x)}{x}+\frac{b \cosh (a) \text{Shi}(b x)}{a} \]
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Rubi [A] time = 0.228159, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6532, 6742, 3303, 3298, 3301} \[ \frac{b \sinh (a) \text{Chi}(b x)}{a}-\frac{b \text{Shi}(a+b x)}{a}-\frac{\text{Shi}(a+b x)}{x}+\frac{b \cosh (a) \text{Shi}(b x)}{a} \]
Antiderivative was successfully verified.
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Rule 6532
Rule 6742
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\text{Shi}(a+b x)}{x^2} \, dx &=-\frac{\text{Shi}(a+b x)}{x}+b \int \frac{\sinh (a+b x)}{x (a+b x)} \, dx\\ &=-\frac{\text{Shi}(a+b x)}{x}+b \int \left (\frac{\sinh (a+b x)}{a x}-\frac{b \sinh (a+b x)}{a (a+b x)}\right ) \, dx\\ &=-\frac{\text{Shi}(a+b x)}{x}+\frac{b \int \frac{\sinh (a+b x)}{x} \, dx}{a}-\frac{b^2 \int \frac{\sinh (a+b x)}{a+b x} \, dx}{a}\\ &=-\frac{b \text{Shi}(a+b x)}{a}-\frac{\text{Shi}(a+b x)}{x}+\frac{(b \cosh (a)) \int \frac{\sinh (b x)}{x} \, dx}{a}+\frac{(b \sinh (a)) \int \frac{\cosh (b x)}{x} \, dx}{a}\\ &=\frac{b \text{Chi}(b x) \sinh (a)}{a}+\frac{b \cosh (a) \text{Shi}(b x)}{a}-\frac{b \text{Shi}(a+b x)}{a}-\frac{\text{Shi}(a+b x)}{x}\\ \end{align*}
Mathematica [A] time = 0.106369, size = 39, normalized size = 0.85 \[ \frac{b x \sinh (a) \text{Chi}(b x)-(a+b x) \text{Shi}(a+b x)+b x \cosh (a) \text{Shi}(b x)}{a x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Shi} \left ( bx+a \right ) }{{x}^{2}}}\, 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 + a\right )}{x^{2}}\,{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{\operatorname{Shi}\left (b x + a\right )}{x^{2}}, 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{\operatorname{Shi}{\left (a + b x \right )}}{x^{2}}\, 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 + a\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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