3.122 \(\int \cosh (5 x) \text {Chi}(2 x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 = {6541, 12, 5472, 3298} \[ \frac {1}{5} \text {Chi}(2 x) \sinh (5 x)-\frac {\text {Shi}(3 x)}{10}-\frac {\text {Shi}(7 x)}{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

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]

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 6541

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 27, normalized size = 0.93 \[ \frac {1}{10} (2 \text {Chi}(2 x) \sinh (5 x)-\text {Shi}(3 x)-\text {Shi}(7 x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred when FriCAS evaluated '(operator(Chi)((x)*(2)))*(cosh((x)*(5)
))':   There are 1 exposed and 1 unexposed library operations named elt having      1 argument(s) but none was
determined to be applicable. Use HyperDoc      Browse, or issue                                )display op el
t      to learn more about the available operations. Perhaps       package-calling the operation or using coer
cions on the arguments       will allow you to apply the operation.    Cannot find application of object of ty
pe BasicOperator to argument(s)       of type(s)                               Polynomial(Integer)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Chi}\left (2 \, x\right ) \cosh \left (5 \, x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Chi(2*x)*cosh(5*x), x)

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maple [A]  time = 0.07, size = 24, normalized size = 0.83 \[ -\frac {\Shi \left (3 x \right )}{10}-\frac {\Shi \left (7 x \right )}{10}+\frac {\Chi \left (2 x \right ) \sinh \left (5 x \right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Chi}\left (2 \, x\right ) \cosh \left (5 \, x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Chi(2*x)*cosh(5*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {coshint}\left (2\,x\right )\,\mathrm {cosh}\left (5\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(coshint(2*x)*cosh(5*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\left (5 x \right )} \operatorname {Chi}\left (2 x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(5*x)*Chi(2*x), x)

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