Optimal. Leaf size=25 \[ \frac{\text{Chi}(b x) \sinh (b x)}{b}-\frac{\text{Shi}(2 b x)}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0486758, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {6541, 12, 5448, 3298} \[ \frac{\text{Chi}(b x) \sinh (b x)}{b}-\frac{\text{Shi}(2 b x)}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6541
Rule 12
Rule 5448
Rule 3298
Rubi steps
\begin{align*} \int \cosh (b x) \text{Chi}(b x) \, dx &=\frac{\text{Chi}(b x) \sinh (b x)}{b}-\int \frac{\cosh (b x) \sinh (b x)}{b x} \, dx\\ &=\frac{\text{Chi}(b x) \sinh (b x)}{b}-\frac{\int \frac{\cosh (b x) \sinh (b x)}{x} \, dx}{b}\\ &=\frac{\text{Chi}(b x) \sinh (b x)}{b}-\frac{\int \frac{\sinh (2 b x)}{2 x} \, dx}{b}\\ &=\frac{\text{Chi}(b x) \sinh (b x)}{b}-\frac{\int \frac{\sinh (2 b x)}{x} \, dx}{2 b}\\ &=\frac{\text{Chi}(b x) \sinh (b x)}{b}-\frac{\text{Shi}(2 b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0105464, size = 25, normalized size = 1. \[ \frac{\text{Chi}(b x) \sinh (b x)}{b}-\frac{\text{Shi}(2 b x)}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 22, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\it Chi} \left ( bx \right ) \sinh \left ( bx \right ) -{\frac{{\it Shi} \left ( 2\,bx \right ) }{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Chi}\left (b x\right ) \cosh \left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cosh \left (b x\right ) \operatorname{Chi}\left (b x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (b x \right )} \operatorname{Chi}\left (b x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Chi}\left (b x\right ) \cosh \left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]