\(\int \text {Shi}(a+b x) \, dx\) [21]

Optimal result

Integrand size = 6, antiderivative size = 27 \[ \int \text {Shi}(a+b x) \, dx=-\frac {\cosh (a+b x)}{b}+\frac {(a+b x) \text {Shi}(a+b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6663} \[ \int \text {Shi}(a+b x) \, dx=\frac {(a+b x) \text {Shi}(a+b x)}{b}-\frac {\cosh (a+b x)}{b} \]

[In]

Int[SinhIntegral[a + b*x],x]

[Out]

-(Cosh[a + b*x]/b) + ((a + b*x)*SinhIntegral[a + b*x])/b

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 (a+b x)}{b}+\frac {(a+b x) \text {Shi}(a+b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \text {Shi}(a+b x) \, dx=-\frac {\cosh (a) \cosh (b x)}{b}-\frac {\sinh (a) \sinh (b x)}{b}+\frac {a \text {Shi}(a+b x)}{b}+x \text {Shi}(a+b x) \]

[In]

Integrate[SinhIntegral[a + b*x],x]

[Out]

-((Cosh[a]*Cosh[b*x])/b) - (Sinh[a]*Sinh[b*x])/b + (a*SinhIntegral[a + b*x])/b + x*SinhIntegral[a + b*x]

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (b x +a \right ) \left (b x +a \right )-\cosh \left (b x +a \right )}{b}\)	\(26\)
default	\(\frac {\operatorname {Shi}\left (b x +a \right ) \left (b x +a \right )-\cosh \left (b x +a \right )}{b}\)	\(26\)
parts	\(x \,\operatorname {Shi}\left (b x +a \right )-\frac {-a \,\operatorname {Shi}\left (b x +a \right )+\cosh \left (b x +a \right )}{b}\)	\(31\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(sinh_integral(b*x + a), x)

Sympy [F]

\[ \int \text {Shi}(a+b x) \, dx=\int \operatorname {Shi}{\left (a + b x \right )}\, dx \]

[In]

integrate(Shi(b*x+a),x)

[Out]

Integral(Shi(a + b*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(Shi(b*x + a), x)

Giac [F]

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

[In]

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

[Out]

integrate(Shi(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int \text {Shi}(a+b x) \, dx=x\,\mathrm {sinhint}\left (a+b\,x\right )-\frac {{\mathrm {e}}^{a+b\,x}}{2\,b}-\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}+\frac {a\,\mathrm {sinhint}\left (a+b\,x\right )}{b} \]

[In]

int(sinhint(a + b*x),x)

[Out]

x*sinhint(a + b*x) - exp(a + b*x)/(2*b) - exp(- a - b*x)/(2*b) + (a*sinhint(a + b*x))/b