Optimal. Leaf size=33 \[ \frac{\text{Shi}(a+b x) \cosh (a+b x)}{b}-\frac{\text{Shi}(2 a+2 b x)}{2 b} \]
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Rubi [A] time = 0.0619774, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {6540, 5448, 12, 3298} \[ \frac{\text{Shi}(a+b x) \cosh (a+b x)}{b}-\frac{\text{Shi}(2 a+2 b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 6540
Rule 5448
Rule 12
Rule 3298
Rubi steps
\begin{align*} \int \sinh (a+b x) \text{Shi}(a+b x) \, dx &=\frac{\cosh (a+b x) \text{Shi}(a+b x)}{b}-\int \frac{\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx\\ &=\frac{\cosh (a+b x) \text{Shi}(a+b x)}{b}-\int \frac{\sinh (2 a+2 b x)}{2 (a+b x)} \, dx\\ &=\frac{\cosh (a+b x) \text{Shi}(a+b x)}{b}-\frac{1}{2} \int \frac{\sinh (2 a+2 b x)}{a+b x} \, dx\\ &=\frac{\cosh (a+b x) \text{Shi}(a+b x)}{b}-\frac{\text{Shi}(2 a+2 b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0157064, size = 32, normalized size = 0.97 \[ \frac{\text{Shi}(a+b x) \cosh (a+b x)}{b}-\frac{\text{Shi}(2 (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 30, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \cosh \left ( bx+a \right ){\it Shi} \left ( bx+a \right ) -{\frac{{\it Shi} \left ( 2\,bx+2\,a \right ) }{2}} \right ) } \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{\rm Shi}\left (b x + a\right ) \sinh \left (b x + a\right )\,{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 (\sinh \left (b x + a\right ) \operatorname{Shi}\left (b x + a\right ), 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 \sinh{\left (a + b x \right )} \operatorname{Shi}{\left (a + b x \right )}\, 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{\rm Shi}\left (b x + a\right ) \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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