\(\int \cosh (5 x) \text {Shi}(2 x) \, dx\) [54]

Optimal result

Integrand size = 9, antiderivative size = 29 \[ \int \cosh (5 x) \text {Shi}(2 x) \, dx=\frac {\text {Chi}(3 x)}{10}-\frac {\text {Chi}(7 x)}{10}+\frac {1}{5} \sinh (5 x) \text {Shi}(2 x) \]

[Out]

1/10*Chi(3*x)-1/10*Chi(7*x)+1/5*Shi(2*x)*sinh(5*x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6681, 12, 5578, 3382} \[ \int \cosh (5 x) \text {Shi}(2 x) \, dx=\frac {\text {Chi}(3 x)}{10}-\frac {\text {Chi}(7 x)}{10}+\frac {1}{5} \text {Shi}(2 x) \sinh (5 x) \]

[In]

Int[Cosh[5*x]*SinhIntegral[2*x],x]

[Out]

CoshIntegral[3*x]/10 - CoshIntegral[7*x]/10 + (Sinh[5*x]*SinhIntegral[2*x])/5

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 5578

Int[((e_.) + (f_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(p_.)*Sinh[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int
[ExpandTrigReduce[(e + f*x)^m, Sinh[a + b*x]^p*Sinh[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ
[p, 0] && IGtQ[q, 0] && IntegerQ[m]

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 {1}{5} \sinh (5 x) \text {Shi}(2 x)-\frac {2}{5} \int \frac {\sinh (2 x) \sinh (5 x)}{2 x} \, dx \\ & = \frac {1}{5} \sinh (5 x) \text {Shi}(2 x)-\frac {1}{5} \int \frac {\sinh (2 x) \sinh (5 x)}{x} \, dx \\ & = \frac {1}{5} \sinh (5 x) \text {Shi}(2 x)-\frac {1}{5} \int \left (-\frac {\cosh (3 x)}{2 x}+\frac {\cosh (7 x)}{2 x}\right ) \, dx \\ & = \frac {1}{5} \sinh (5 x) \text {Shi}(2 x)+\frac {1}{10} \int \frac {\cosh (3 x)}{x} \, dx-\frac {1}{10} \int \frac {\cosh (7 x)}{x} \, dx \\ & = \frac {\text {Chi}(3 x)}{10}-\frac {\text {Chi}(7 x)}{10}+\frac {1}{5} \sinh (5 x) \text {Shi}(2 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \cosh (5 x) \text {Shi}(2 x) \, dx=\frac {1}{10} (\text {Chi}(3 x)-\text {Chi}(7 x)+2 \sinh (5 x) \text {Shi}(2 x)) \]

[In]

Integrate[Cosh[5*x]*SinhIntegral[2*x],x]

[Out]

(CoshIntegral[3*x] - CoshIntegral[7*x] + 2*Sinh[5*x]*SinhIntegral[2*x])/10

Maple [A] (verified)

Time = 1.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {\operatorname {Chi}\left (3 x \right )}{10}-\frac {\operatorname {Chi}\left (7 x \right )}{10}+\frac {\operatorname {Shi}\left (2 x \right ) \sinh \left (5 x \right )}{5}\)	\(24\)

[In]

int(cosh(5*x)*Shi(2*x),x,method=_RETURNVERBOSE)

[Out]

1/10*Chi(3*x)-1/10*Chi(7*x)+1/5*Shi(2*x)*sinh(5*x)

Fricas [F]

\[ \int \cosh (5 x) \text {Shi}(2 x) \, dx=\int { {\rm Shi}\left (2 \, x\right ) \cosh \left (5 \, x\right ) \,d x } \]

[In]

integrate(cosh(5*x)*Shi(2*x),x, algorithm="fricas")

[Out]

integral(cosh(5*x)*sinh_integral(2*x), x)

Sympy [F]

\[ \int \cosh (5 x) \text {Shi}(2 x) \, dx=\int \cosh {\left (5 x \right )} \operatorname {Shi}{\left (2 x \right )}\, dx \]

[In]

integrate(cosh(5*x)*Shi(2*x),x)

[Out]

Integral(cosh(5*x)*Shi(2*x), x)

Maxima [F]

\[ \int \cosh (5 x) \text {Shi}(2 x) \, dx=\int { {\rm Shi}\left (2 \, x\right ) \cosh \left (5 \, x\right ) \,d x } \]

[In]

integrate(cosh(5*x)*Shi(2*x),x, algorithm="maxima")

[Out]

integrate(Shi(2*x)*cosh(5*x), x)

Giac [F]

\[ \int \cosh (5 x) \text {Shi}(2 x) \, dx=\int { {\rm Shi}\left (2 \, x\right ) \cosh \left (5 \, x\right ) \,d x } \]

[In]

integrate(cosh(5*x)*Shi(2*x),x, algorithm="giac")

[Out]

integrate(Shi(2*x)*cosh(5*x), x)

Mupad [F(-1)]

Timed out. \[ \int \cosh (5 x) \text {Shi}(2 x) \, dx=\int \mathrm {sinhint}\left (2\,x\right )\,\mathrm {cosh}\left (5\,x\right ) \,d x \]

[In]

int(sinhint(2*x)*cosh(5*x),x)

[Out]

int(sinhint(2*x)*cosh(5*x), x)