3.53 \(\int \sinh (5 x) \text{Shi}(2 x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0560017, 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 = {6540, 12, 5472, 3298} \[ \frac{\text{Shi}(3 x)}{10}-\frac{\text{Shi}(7 x)}{10}+\frac{1}{5} \text{Shi}(2 x) \cosh (5 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6540

Int[Sinh[(a_.) + (b_.)*(x_)]*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(Cosh[a + b*x]*SinhIntegral[c
+ d*x])/b, x] - Dist[d/b, Int[(Cosh[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 5472

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

Rule 3298

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

Rubi steps

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

Mathematica [A]  time = 0.0276211, size = 25, normalized size = 0.86 \[ \frac{1}{10} (\text{Shi}(3 x)-\text{Shi}(7 x)+2 \text{Shi}(2 x) \cosh (5 x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.203, size = 24, normalized size = 0.8 \begin{align*}{\frac{\cosh \left ( 5\,x \right ){\it Shi} \left ( 2\,x \right ) }{5}}+{\frac{{\it Shi} \left ( 3\,x \right ) }{10}}-{\frac{{\it Shi} \left ( 7\,x \right ) }{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\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(Shi(2*x)*sinh(5*x),x)

[Out]

Integral(sinh(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 ) \sinh \left (5 \, x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi(2*x)*sinh(5*x), x)