\(\int \frac {\cosh (a+b \sqrt {c+d x})}{x} \, dx\) [62]

Optimal result

Integrand size = 18, antiderivative size = 126 \[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\cosh \left (a-b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )+\cosh \left (a+b \sqrt {c}\right ) \text {Chi}\left (b \sqrt {c}-b \sqrt {c+d x}\right )+\sinh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )-\sinh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \sqrt {c}-b \sqrt {c+d x}\right ) \] Output:

cosh(a-b*c^(1/2))*Chi(b*(c^(1/2)+(d*x+c)^(1/2)))+cosh(a+b*c^(1/2))*Chi(b*c 
^(1/2)-b*(d*x+c)^(1/2))+sinh(a-b*c^(1/2))*Shi(b*(c^(1/2)+(d*x+c)^(1/2)))-s 
inh(a+b*c^(1/2))*Shi(b*c^(1/2)-b*(d*x+c)^(1/2))
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.01 \[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\frac {1}{2} e^{-a-b \sqrt {c}} \left (\operatorname {ExpIntegralEi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )+e^{2 \left (a+b \sqrt {c}\right )} \operatorname {ExpIntegralEi}\left (b \left (-\sqrt {c}+\sqrt {c+d x}\right )\right )+e^{2 b \sqrt {c}} \operatorname {ExpIntegralEi}\left (-b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )+e^{2 a} \operatorname {ExpIntegralEi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )\right ) \] Input:

Integrate[Cosh[a + b*Sqrt[c + d*x]]/x,x]
 

Output:

(E^(-a - b*Sqrt[c])*(ExpIntegralEi[b*(Sqrt[c] - Sqrt[c + d*x])] + E^(2*(a 
+ b*Sqrt[c]))*ExpIntegralEi[b*(-Sqrt[c] + Sqrt[c + d*x])] + E^(2*b*Sqrt[c] 
)*ExpIntegralEi[-(b*(Sqrt[c] + Sqrt[c + d*x]))] + E^(2*a)*ExpIntegralEi[b* 
(Sqrt[c] + Sqrt[c + d*x])]))/2
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5888, 25, 7267, 5816, 2009}

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.

\(\displaystyle \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx\)

\(\Big \downarrow \) 5888

\(\displaystyle \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{d x}d(c+d x)\)

\(\Big \downarrow \) 25

\(\displaystyle -\int -\frac {\cosh \left (a+b \sqrt {c+d x}\right )}{d x}d(c+d x)\)

\(\Big \downarrow \) 7267

\(\displaystyle -2 \int -\frac {\sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{d x}d\sqrt {c+d x}\)

\(\Big \downarrow \) 5816

\(\displaystyle -2 \int \left (\frac {\cosh \left (a+b \sqrt {c+d x}\right )}{2 \left (-c+\sqrt {c}-d x\right )}-\frac {\cosh \left (a+b \sqrt {c+d x}\right )}{2 \left (\sqrt {c}+\sqrt {c+d x}\right )}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (-\frac {1}{2} \cosh \left (a+b \sqrt {c}\right ) \text {Chi}\left (b \sqrt {c}-b \sqrt {c+d x}\right )-\frac {1}{2} \cosh \left (a-b \sqrt {c}\right ) \text {Chi}\left (\sqrt {c} b+\sqrt {c+d x} b\right )+\frac {1}{2} \sinh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \sqrt {c}-b \sqrt {c+d x}\right )-\frac {1}{2} \sinh \left (a-b \sqrt {c}\right ) \text {Shi}\left (\sqrt {c} b+\sqrt {c+d x} b\right )\right )\)

Input:

Int[Cosh[a + b*Sqrt[c + d*x]]/x,x]
 

Output:

-2*(-1/2*(Cosh[a + b*Sqrt[c]]*CoshIntegral[b*Sqrt[c] - b*Sqrt[c + d*x]]) - 
 (Cosh[a - b*Sqrt[c]]*CoshIntegral[b*Sqrt[c] + b*Sqrt[c + d*x]])/2 + (Sinh 
[a + b*Sqrt[c]]*SinhIntegral[b*Sqrt[c] - b*Sqrt[c + d*x]])/2 - (Sinh[a - b 
*Sqrt[c]]*SinhIntegral[b*Sqrt[c] + b*Sqrt[c + d*x]])/2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5816

Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Sy 
mbol] :> Int[ExpandIntegrand[Cosh[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Fr 
eeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (EqQ[n, 
2] || EqQ[p, -1])
 

rule 5888

Int[((a_.) + Cosh[(c_.) + (d_.)*(u_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo 
l] :> Simp[1/Coefficient[u, x, 1]^(m + 1)   Subst[Int[(x - Coefficient[u, x 
, 0])^m*(a + b*Cosh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n, p 
}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

Maple [F]

\[\int \frac {\cosh \left (a +b \sqrt {d x +c}\right )}{x}d x\]

Input:

int(cosh(a+b*(d*x+c)^(1/2))/x,x)
 

Output:

int(cosh(a+b*(d*x+c)^(1/2))/x,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (102) = 204\).

Time = 0.09 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.72 \[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) + {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \cosh \left (a + \sqrt {b^{2} c}\right ) + \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) + {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \cosh \left (-a + \sqrt {b^{2} c}\right ) + \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) - {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \sinh \left (a + \sqrt {b^{2} c}\right ) - \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) - {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \sinh \left (-a + \sqrt {b^{2} c}\right ) \] Input:

integrate(cosh(a+b*(d*x+c)^(1/2))/x,x, algorithm="fricas")
 

Output:

1/2*(Ei(sqrt(d*x + c)*b - sqrt(b^2*c)) + Ei(-sqrt(d*x + c)*b + sqrt(b^2*c) 
))*cosh(a + sqrt(b^2*c)) + 1/2*(Ei(sqrt(d*x + c)*b + sqrt(b^2*c)) + Ei(-sq 
rt(d*x + c)*b - sqrt(b^2*c)))*cosh(-a + sqrt(b^2*c)) + 1/2*(Ei(sqrt(d*x + 
c)*b - sqrt(b^2*c)) - Ei(-sqrt(d*x + c)*b + sqrt(b^2*c)))*sinh(a + sqrt(b^ 
2*c)) - 1/2*(Ei(sqrt(d*x + c)*b + sqrt(b^2*c)) - Ei(-sqrt(d*x + c)*b - sqr 
t(b^2*c)))*sinh(-a + sqrt(b^2*c))
 

Sympy [F]

\[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\int \frac {\cosh {\left (a + b \sqrt {c + d x} \right )}}{x}\, dx \] Input:

integrate(cosh(a+b*(d*x+c)**(1/2))/x,x)
 

Output:

Integral(cosh(a + b*sqrt(c + d*x))/x, x)
 

Maxima [F]

\[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\int { \frac {\cosh \left (\sqrt {d x + c} b + a\right )}{x} \,d x } \] Input:

integrate(cosh(a+b*(d*x+c)^(1/2))/x,x, algorithm="maxima")
 

Output:

integrate(cosh(sqrt(d*x + c)*b + a)/x, x)
 

Giac [F]

\[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\int { \frac {\cosh \left (\sqrt {d x + c} b + a\right )}{x} \,d x } \] Input:

integrate(cosh(a+b*(d*x+c)^(1/2))/x,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate(cosh(sqrt(d*x + c)*b + a)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,\sqrt {c+d\,x}\right )}{x} \,d x \] Input:

int(cosh(a + b*(c + d*x)^(1/2))/x,x)
 

Output:

int(cosh(a + b*(c + d*x)^(1/2))/x, x)
 

Reduce [F]

\[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\int \frac {\cosh \left (\sqrt {d x +c}\, b +a \right )}{x}d x \] Input:

int(cosh(a+b*(d*x+c)^(1/2))/x,x)
 

Output:

int(cosh(sqrt(c + d*x)*b + a)/x,x)