\(\int \text {Shi}(b x) \, dx\) [5]

Optimal result

Integrand size = 4, antiderivative size = 16 \[ \int \text {Shi}(b x) \, dx=-\frac {\cosh (b x)}{b}+x \text {Shi}(b x) \]

[Out]

-cosh(b*x)/b+x*Shi(b*x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6663} \[ \int \text {Shi}(b x) \, dx=x \text {Shi}(b x)-\frac {\cosh (b x)}{b} \]

[In]

Int[SinhIntegral[b*x],x]

[Out]

-(Cosh[b*x]/b) + x*SinhIntegral[b*x]

Rule 6663

Int[SinhIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(SinhIntegral[a + b*x]/b), x] - Simp[Cosh[a
+ b*x]/b, x] /; FreeQ[{a, b}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh (b x)}{b}+x \text {Shi}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \text {Shi}(b x) \, dx=-\frac {\cosh (b x)}{b}+x \text {Shi}(b x) \]

[In]

Integrate[SinhIntegral[b*x],x]

[Out]

-(Cosh[b*x]/b) + x*SinhIntegral[b*x]

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
parts	\(-\frac {\cosh \left (b x \right )}{b}+x \,\operatorname {Shi}\left (b x \right )\)	\(17\)
derivativedivides	\(\frac {\operatorname {Shi}\left (b x \right ) b x -\cosh \left (b x \right )}{b}\)	\(19\)
default	\(\frac {\operatorname {Shi}\left (b x \right ) b x -\cosh \left (b x \right )}{b}\)	\(19\)
meijerg	\(-\frac {\sqrt {\pi }\, \left (-\frac {2}{\sqrt {\pi }}+\frac {2 \cosh \left (b x \right )}{\sqrt {\pi }}-\frac {2 b x \,\operatorname {Shi}\left (b x \right )}{\sqrt {\pi }}\right )}{2 b}\)	\(35\)

[In]

int(Shi(b*x),x,method=_RETURNVERBOSE)

[Out]

-cosh(b*x)/b+x*Shi(b*x)

Fricas [F]

\[ \int \text {Shi}(b x) \, dx=\int { {\rm Shi}\left (b x\right ) \,d x } \]

[In]

integrate(Shi(b*x),x, algorithm="fricas")

[Out]

integral(sinh_integral(b*x), x)

Sympy [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \text {Shi}(b x) \, dx=x \operatorname {Shi}{\left (b x \right )} - \frac {\cosh {\left (b x \right )}}{b} \]

[In]

integrate(Shi(b*x),x)

[Out]

x*Shi(b*x) - cosh(b*x)/b

Maxima [F]

\[ \int \text {Shi}(b x) \, dx=\int { {\rm Shi}\left (b x\right ) \,d x } \]

[In]

integrate(Shi(b*x),x, algorithm="maxima")

[Out]

integrate(Shi(b*x), x)

Giac [F]

\[ \int \text {Shi}(b x) \, dx=\int { {\rm Shi}\left (b x\right ) \,d x } \]

[In]

integrate(Shi(b*x),x, algorithm="giac")

[Out]

integrate(Shi(b*x), x)

Mupad [F(-1)]

Timed out. \[ \int \text {Shi}(b x) \, dx=x\,\mathrm {sinhint}\left (b\,x\right )-\frac {\mathrm {cosh}\left (b\,x\right )}{b} \]

[In]

int(sinhint(b*x),x)

[Out]

x*sinhint(b*x) - cosh(b*x)/b