\(\int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx\) [115]

Optimal result

Integrand size = 12, antiderivative size = 44 \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\frac {1}{2} b \text {Chi}(b x)^2+b \text {Chi}(2 b x)-\frac {\text {Chi}(b x) \sinh (b x)}{x}-\frac {\sinh (2 b x)}{2 x} \] Output:

1/2*b*Chi(b*x)^2+b*Chi(2*b*x)-Chi(b*x)*sinh(b*x)/x-1/2*sinh(2*b*x)/x
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\frac {1}{2} b \text {Chi}(b x)^2+b \text {Chi}(2 b x)-\frac {\text {Chi}(b x) \sinh (b x)}{x}-\frac {\sinh (2 b x)}{2 x} \] Input:

Integrate[(CoshIntegral[b*x]*Sinh[b*x])/x^2,x]
 

Output:

(b*CoshIntegral[b*x]^2)/2 + b*CoshIntegral[2*b*x] - (CoshIntegral[b*x]*Sin 
h[b*x])/x - Sinh[2*b*x]/(2*x)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.23, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {7105, 27, 5971, 27, 3042, 26, 3778, 3042, 3782, 7237}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx\)

\(\Big \downarrow \) 7105

\(\displaystyle b \int \frac {\cosh (b x) \text {Chi}(b x)}{x}dx+b \int \frac {\cosh (b x) \sinh (b x)}{b x^2}dx-\frac {\text {Chi}(b x) \sinh (b x)}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle b \int \frac {\cosh (b x) \text {Chi}(b x)}{x}dx+\int \frac {\cosh (b x) \sinh (b x)}{x^2}dx-\frac {\text {Chi}(b x) \sinh (b x)}{x}\)

\(\Big \downarrow \) 5971

\(\displaystyle b \int \frac {\cosh (b x) \text {Chi}(b x)}{x}dx+\int \frac {\sinh (2 b x)}{2 x^2}dx-\frac {\text {Chi}(b x) \sinh (b x)}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle b \int \frac {\cosh (b x) \text {Chi}(b x)}{x}dx+\frac {1}{2} \int \frac {\sinh (2 b x)}{x^2}dx-\frac {\text {Chi}(b x) \sinh (b x)}{x}\)

\(\Big \downarrow \) 3042

\(\displaystyle b \int \frac {\cosh (b x) \text {Chi}(b x)}{x}dx+\frac {1}{2} \int -\frac {i \sin (2 i b x)}{x^2}dx-\frac {\text {Chi}(b x) \sinh (b x)}{x}\)

\(\Big \downarrow \) 26

\(\displaystyle b \int \frac {\cosh (b x) \text {Chi}(b x)}{x}dx-\frac {1}{2} i \int \frac {\sin (2 i b x)}{x^2}dx-\frac {\text {Chi}(b x) \sinh (b x)}{x}\)

\(\Big \downarrow \) 3778

\(\displaystyle b \int \frac {\cosh (b x) \text {Chi}(b x)}{x}dx-\frac {1}{2} i \left (2 i b \int \frac {\cosh (2 b x)}{x}dx-\frac {i \sinh (2 b x)}{x}\right )-\frac {\text {Chi}(b x) \sinh (b x)}{x}\)

\(\Big \downarrow \) 3042

\(\displaystyle b \int \frac {\cosh (b x) \text {Chi}(b x)}{x}dx-\frac {1}{2} i \left (2 i b \int \frac {\sin \left (2 i b x+\frac {\pi }{2}\right )}{x}dx-\frac {i \sinh (2 b x)}{x}\right )-\frac {\text {Chi}(b x) \sinh (b x)}{x}\)

\(\Big \downarrow \) 3782

\(\displaystyle b \int \frac {\cosh (b x) \text {Chi}(b x)}{x}dx-\frac {\text {Chi}(b x) \sinh (b x)}{x}-\frac {1}{2} i \left (2 i b \text {Chi}(2 b x)-\frac {i \sinh (2 b x)}{x}\right )\)

\(\Big \downarrow \) 7237

\(\displaystyle \frac {1}{2} b \text {Chi}(b x)^2-\frac {\text {Chi}(b x) \sinh (b x)}{x}-\frac {1}{2} i \left (2 i b \text {Chi}(2 b x)-\frac {i \sinh (2 b x)}{x}\right )\)

Input:

Int[(CoshIntegral[b*x]*Sinh[b*x])/x^2,x]
 

Output:

(b*CoshIntegral[b*x]^2)/2 - (CoshIntegral[b*x]*Sinh[b*x])/x - (I/2)*((2*I) 
*b*CoshIntegral[2*b*x] - (I*Sinh[2*b*x])/x)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3778

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c 
 + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1))   Int[( 
c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 
1]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> 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 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 7105

Int[CoshIntegral[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_)*Sinh[(a_.) + 
 (b_.)*(x_)], x_Symbol] :> Simp[(e + f*x)^(m + 1)*Sinh[a + b*x]*(CoshIntegr 
al[c + d*x]/(f*(m + 1))), x] + (-Simp[b/(f*(m + 1))   Int[(e + f*x)^(m + 1) 
*Cosh[a + b*x]*CoshIntegral[c + d*x], x], x] - Simp[d/(f*(m + 1))   Int[(e 
+ f*x)^(m + 1)*Sinh[a + b*x]*(Cosh[c + d*x]/(c + d*x)), x], x]) /; FreeQ[{a 
, b, c, d, e, f}, x] && ILtQ[m, -1]
 

rule 7237

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si 
mp[q*(y^(m + 1)/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
 

Maple [F]

\[\int \frac {\operatorname {Chi}\left (b x \right ) \sinh \left (b x \right )}{x^{2}}d x\]

Input:

int(Chi(b*x)*sinh(b*x)/x^2,x)
 

Output:

int(Chi(b*x)*sinh(b*x)/x^2,x)
 

Fricas [F]

\[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x\right ) \sinh \left (b x\right )}{x^{2}} \,d x } \] Input:

integrate(Chi(b*x)*sinh(b*x)/x^2,x, algorithm="fricas")
 

Output:

integral(cosh_integral(b*x)*sinh(b*x)/x^2, x)
 

Sympy [F]

\[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\int \frac {\sinh {\left (b x \right )} \operatorname {Chi}\left (b x\right )}{x^{2}}\, dx \] Input:

integrate(Chi(b*x)*sinh(b*x)/x**2,x)
 

Output:

Integral(sinh(b*x)*Chi(b*x)/x**2, x)
 

Maxima [F]

\[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x\right ) \sinh \left (b x\right )}{x^{2}} \,d x } \] Input:

integrate(Chi(b*x)*sinh(b*x)/x^2,x, algorithm="maxima")
 

Output:

integrate(Chi(b*x)*sinh(b*x)/x^2, x)
                                                                                    
                                                                                    
 

Giac [F]

\[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x\right ) \sinh \left (b x\right )}{x^{2}} \,d x } \] Input:

integrate(Chi(b*x)*sinh(b*x)/x^2,x, algorithm="giac")
 

Output:

integrate(Chi(b*x)*sinh(b*x)/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\int \frac {\mathrm {coshint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right )}{x^2} \,d x \] Input:

int((coshint(b*x)*sinh(b*x))/x^2,x)
 

Output:

int((coshint(b*x)*sinh(b*x))/x^2, x)
 

Reduce [F]

\[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\int \frac {\chi \left (b x \right ) \sinh \left (b x \right )}{x^{2}}d x \] Input:

int(Chi(b*x)*sinh(b*x)/x^2,x)
 

Output:

int((chi(b*x)*sinh(b*x))/x**2,x)