3.54 \(\int \cosh (5 x) \text{Shi}(2 x) \, dx\)

Optimal. Leaf size=29 \[ \frac{\text{Chi}(3 x)}{10}-\frac{\text{Chi}(7 x)}{10}+\frac{1}{5} \text{Shi}(2 x) \sinh (5 x) \]

[Out]

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

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Rubi [A]  time = 0.0574705, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {6546, 12, 5470, 3301} \[ \frac{\text{Chi}(3 x)}{10}-\frac{\text{Chi}(7 x)}{10}+\frac{1}{5} \text{Shi}(2 x) \sinh (5 x) \]

Antiderivative was successfully verified.

[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 6546

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]

Rule 12

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

Rule 5470

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 3301

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]

Rubi steps

\begin{align*} \int \cosh (5 x) \text{Shi}(2 x) \, dx &=\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]  time = 0.0278216, size = 25, normalized size = 0.86 \[ \frac{1}{10} (\text{Chi}(3 x)-\text{Chi}(7 x)+2 \text{Shi}(2 x) \sinh (5 x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.207, size = 24, normalized size = 0.8 \begin{align*}{\frac{{\it Chi} \left ( 3\,x \right ) }{10}}-{\frac{{\it Chi} \left ( 7\,x \right ) }{10}}+{\frac{{\it Shi} \left ( 2\,x \right ) \sinh \left ( 5\,x \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Shi}\left (2 \, x\right ) \cosh \left (5 \, x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi(2*x)*cosh(5*x), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (5 x \right )} \operatorname{Shi}{\left (2 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(5*x)*Shi(2*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Shi}\left (2 \, x\right ) \cosh \left (5 \, x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi(2*x)*cosh(5*x), x)