\(\int \cosh (a+b x) \text {Shi}(a+b x) \, dx\) [61]

Optimal result

Integrand size = 13, antiderivative size = 46 \[ \int \cosh (a+b x) \text {Shi}(a+b x) \, dx=-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\log (a+b x)}{2 b}+\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b} \]

[Out]

-1/2*Chi(2*b*x+2*a)/b+1/2*ln(b*x+a)/b+Shi(b*x+a)*sinh(b*x+a)/b

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6681, 3393, 3382} \[ \int \cosh (a+b x) \text {Shi}(a+b x) \, dx=-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {\log (a+b x)}{2 b} \]

[In]

Int[Cosh[a + b*x]*SinhIntegral[a + b*x],x]

[Out]

-1/2*CoshIntegral[2*a + 2*b*x]/b + Log[a + b*x]/(2*b) + (Sinh[a + b*x]*SinhIntegral[a + b*x])/b

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 6681

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

Rubi steps \begin{align*} \text {integral}& = \frac {\sinh (a+b x) \text {Shi}(a+b x)}{b}-\int \frac {\sinh ^2(a+b x)}{a+b x} \, dx \\ & = \frac {\sinh (a+b x) \text {Shi}(a+b x)}{b}+\int \left (\frac {1}{2 (a+b x)}-\frac {\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx \\ & = \frac {\log (a+b x)}{2 b}+\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b}-\frac {1}{2} \int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx \\ & = -\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\log (a+b x)}{2 b}+\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \cosh (a+b x) \text {Shi}(a+b x) \, dx=-\frac {\text {Chi}(2 (a+b x))}{2 b}+\frac {\log (a+b x)}{2 b}+\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b} \]

[In]

Integrate[Cosh[a + b*x]*SinhIntegral[a + b*x],x]

[Out]

-1/2*CoshIntegral[2*(a + b*x)]/b + Log[a + b*x]/(2*b) + (Sinh[a + b*x]*SinhIntegral[a + b*x])/b

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (b x +a \right ) \sinh \left (b x +a \right )+\frac {\ln \left (b x +a \right )}{2}-\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}}{b}\)	\(38\)
default	\(\frac {\operatorname {Shi}\left (b x +a \right ) \sinh \left (b x +a \right )+\frac {\ln \left (b x +a \right )}{2}-\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}}{b}\)	\(38\)

[In]

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

[Out]

1/b*(Shi(b*x+a)*sinh(b*x+a)+1/2*ln(b*x+a)-1/2*Chi(2*b*x+2*a))

Fricas [F]

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

[In]

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

[Out]

integral(cosh(b*x + a)*sinh_integral(b*x + a), x)

Sympy [F]

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

[In]

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

[Out]

Integral(cosh(a + b*x)*Shi(a + b*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(sinhint(a + b*x)*cosh(a + b*x),x)

[Out]

int(sinhint(a + b*x)*cosh(a + b*x), x)