\(\int \frac {\cos (a+b \sqrt {c+d x})}{x^2} \, dx\) [94]

Optimal result

Integrand size = 18, antiderivative size = 184 \[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=-\frac {\cos \left (a+b \sqrt {c+d x}\right )}{x}+\frac {b d \operatorname {CosIntegral}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right ) \sin \left (a-b \sqrt {c}\right )}{2 \sqrt {c}}-\frac {b d \operatorname {CosIntegral}\left (b \sqrt {c}-b \sqrt {c+d x}\right ) \sin \left (a+b \sqrt {c}\right )}{2 \sqrt {c}}+\frac {b d \cos \left (a-b \sqrt {c}\right ) \text {Si}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}+\frac {b d \cos \left (a+b \sqrt {c}\right ) \text {Si}\left (b \sqrt {c}-b \sqrt {c+d x}\right )}{2 \sqrt {c}} \] Output:

-cos(a+b*(d*x+c)^(1/2))/x+1/2*b*d*Ci(b*(c^(1/2)+(d*x+c)^(1/2)))*sin(a-b*c^ 
(1/2))/c^(1/2)-1/2*b*d*Ci(b*c^(1/2)-b*(d*x+c)^(1/2))*sin(a+b*c^(1/2))/c^(1 
/2)+1/2*b*d*cos(a-b*c^(1/2))*Si(b*(c^(1/2)+(d*x+c)^(1/2)))/c^(1/2)+1/2*b*d 
*cos(a+b*c^(1/2))*Si(b*c^(1/2)-b*(d*x+c)^(1/2))/c^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.46 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.30 \[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\frac {i \left (e^{-i a} \left (2 i \sqrt {c} e^{-i b \sqrt {c+d x}}-b d e^{-i b \sqrt {c}} x \operatorname {ExpIntegralEi}\left (-i b \left (-\sqrt {c}+\sqrt {c+d x}\right )\right )+b d e^{i b \sqrt {c}} x \operatorname {ExpIntegralEi}\left (-i b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )\right )+e^{i \left (a-b \sqrt {c}\right )} \left (2 i \sqrt {c} e^{i b \left (\sqrt {c}+\sqrt {c+d x}\right )}+b d e^{2 i b \sqrt {c}} x \operatorname {ExpIntegralEi}\left (i b \left (-\sqrt {c}+\sqrt {c+d x}\right )\right )-b d x \operatorname {ExpIntegralEi}\left (i b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )\right )\right )}{4 \sqrt {c} x} \] Input:

Integrate[Cos[a + b*Sqrt[c + d*x]]/x^2,x]
 

Output:

((I/4)*((((2*I)*Sqrt[c])/E^(I*b*Sqrt[c + d*x]) - (b*d*x*ExpIntegralEi[(-I) 
*b*(-Sqrt[c] + Sqrt[c + d*x])])/E^(I*b*Sqrt[c]) + b*d*E^(I*b*Sqrt[c])*x*Ex 
pIntegralEi[(-I)*b*(Sqrt[c] + Sqrt[c + d*x])])/E^(I*a) + E^(I*(a - b*Sqrt[ 
c]))*((2*I)*Sqrt[c]*E^(I*b*(Sqrt[c] + Sqrt[c + d*x])) + b*d*E^((2*I)*b*Sqr 
t[c])*x*ExpIntegralEi[I*b*(-Sqrt[c] + Sqrt[c + d*x])] - b*d*x*ExpIntegralE 
i[I*b*(Sqrt[c] + Sqrt[c + d*x])])))/(Sqrt[c]*x)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3913, 27, 3823, 3814, 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.

\(\displaystyle \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx\)

\(\Big \downarrow \) 3913

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3823

\(\displaystyle 2 d \left (\frac {1}{2} b \int -\frac {\sin \left (a+b \sqrt {c+d x}\right )}{d x}d\sqrt {c+d x}-\frac {\cos \left (a+b \sqrt {c+d x}\right )}{2 d x}\right )\)

\(\Big \downarrow \) 3814

\(\displaystyle 2 d \left (\frac {1}{2} b \int \left (\frac {\sin \left (a+b \sqrt {c+d x}\right )}{2 \sqrt {c} \left (\sqrt {c}-\sqrt {c+d x}\right )}+\frac {\sin \left (a+b \sqrt {c+d x}\right )}{2 \sqrt {c} \left (\sqrt {c}+\sqrt {c+d x}\right )}\right )d\sqrt {c+d x}-\frac {\cos \left (a+b \sqrt {c+d x}\right )}{2 d x}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 d \left (\frac {1}{2} b \left (\frac {\sin \left (a-b \sqrt {c}\right ) \operatorname {CosIntegral}\left (\sqrt {c} b+\sqrt {c+d x} b\right )}{2 \sqrt {c}}-\frac {\sin \left (a+b \sqrt {c}\right ) \operatorname {CosIntegral}\left (b \sqrt {c}-b \sqrt {c+d x}\right )}{2 \sqrt {c}}+\frac {\cos \left (a+b \sqrt {c}\right ) \text {Si}\left (b \sqrt {c}-b \sqrt {c+d x}\right )}{2 \sqrt {c}}+\frac {\cos \left (a-b \sqrt {c}\right ) \text {Si}\left (\sqrt {c} b+\sqrt {c+d x} b\right )}{2 \sqrt {c}}\right )-\frac {\cos \left (a+b \sqrt {c+d x}\right )}{2 d x}\right )\)

Input:

Int[Cos[a + b*Sqrt[c + d*x]]/x^2,x]
 

Output:

2*d*(-1/2*Cos[a + b*Sqrt[c + d*x]]/(d*x) + (b*((CosIntegral[b*Sqrt[c] + b* 
Sqrt[c + d*x]]*Sin[a - b*Sqrt[c]])/(2*Sqrt[c]) - (CosIntegral[b*Sqrt[c] - 
b*Sqrt[c + d*x]]*Sin[a + b*Sqrt[c]])/(2*Sqrt[c]) + (Cos[a + b*Sqrt[c]]*Sin 
Integral[b*Sqrt[c] - b*Sqrt[c + d*x]])/(2*Sqrt[c]) + (Cos[a - b*Sqrt[c]]*S 
inIntegral[b*Sqrt[c] + b*Sqrt[c + d*x]])/(2*Sqrt[c])))/2)
 

Defintions of rubi rules used

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 2009

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

rule 3814

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int 
[ExpandIntegrand[Sin[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, 
 x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
 

rule 3823

Int[Cos[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_ 
), x_Symbol] :> Simp[e^m*(a + b*x^n)^(p + 1)*(Cos[c + d*x]/(b*n*(p + 1))), 
x] + Simp[d*(e^m/(b*n*(p + 1)))   Int[(a + b*x^n)^(p + 1)*Sin[c + d*x], x], 
 x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, -1] && EqQ[m, n - 1] && ( 
IntegerQ[n] || GtQ[e, 0])
 

rule 3913

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.)*((g_ 
.) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[1/(n*f)   Subst[Int[ExpandIntegra 
nd[(a + b*Cos[c + d*x])^p, x^(1/n - 1)*(g - e*(h/f) + h*(x^(1/n)/f))^m, x], 
 x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p 
, 0] && IntegerQ[1/n]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(713\) vs. \(2(142)=284\).

Time = 1.19 (sec) , antiderivative size = 714, normalized size of antiderivative = 3.88

method	result	size
derivativedivides	\(\frac {2 d \left (\frac {\cos \left (a +\sqrt {d x +c}\, b \right ) \left (-\frac {a \,b^{2} \left (a +\sqrt {d x +c}\, b \right )}{2 c}+\frac {b^{2} \left (-b^{2} c +a^{2}\right )}{2 c}\right )}{-b^{2} c +a^{2}-2 \left (a +\sqrt {d x +c}\, b \right ) a +\left (a +\sqrt {d x +c}\, b \right )^{2}}-\frac {a b \left (\sin \left (a +b \sqrt {c}\right ) \operatorname {Si}\left (b \sqrt {c}-\sqrt {d x +c}\, b \right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b -b \sqrt {c}\right ) \cos \left (a +b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}+\frac {a b \left (-\operatorname {Si}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}+\frac {b \left (-b^{2} c +a^{2}-\left (a +b \sqrt {c}\right ) a \right ) \left (-\operatorname {Si}\left (b \sqrt {c}-\sqrt {d x +c}\, b \right ) \cos \left (a +b \sqrt {c}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b -b \sqrt {c}\right ) \sin \left (a +b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}-\frac {b \left (-b^{2} c +a^{2}-\left (a -b \sqrt {c}\right ) a \right ) \left (\operatorname {Si}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}-a \,b^{4} \left (\frac {\cos \left (a +\sqrt {d x +c}\, b \right ) \left (-\frac {a +\sqrt {d x +c}\, b}{2 c \,b^{2}}+\frac {a}{2 c \,b^{2}}\right )}{-b^{2} c +a^{2}-2 \left (a +\sqrt {d x +c}\, b \right ) a +\left (a +\sqrt {d x +c}\, b \right )^{2}}-\frac {\sin \left (a +b \sqrt {c}\right ) \operatorname {Si}\left (b \sqrt {c}-\sqrt {d x +c}\, b \right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b -b \sqrt {c}\right ) \cos \left (a +b \sqrt {c}\right )}{4 c^{\frac {3}{2}} b^{3}}+\frac {-\operatorname {Si}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )}{4 c^{\frac {3}{2}} b^{3}}-\frac {-\operatorname {Si}\left (b \sqrt {c}-\sqrt {d x +c}\, b \right ) \cos \left (a +b \sqrt {c}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b -b \sqrt {c}\right ) \sin \left (a +b \sqrt {c}\right )}{4 c \,b^{2}}-\frac {\operatorname {Si}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )}{4 c \,b^{2}}\right )\right )}{b^{2}}\)	\(714\)
default	\(\frac {2 d \left (\frac {\cos \left (a +\sqrt {d x +c}\, b \right ) \left (-\frac {a \,b^{2} \left (a +\sqrt {d x +c}\, b \right )}{2 c}+\frac {b^{2} \left (-b^{2} c +a^{2}\right )}{2 c}\right )}{-b^{2} c +a^{2}-2 \left (a +\sqrt {d x +c}\, b \right ) a +\left (a +\sqrt {d x +c}\, b \right )^{2}}-\frac {a b \left (\sin \left (a +b \sqrt {c}\right ) \operatorname {Si}\left (b \sqrt {c}-\sqrt {d x +c}\, b \right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b -b \sqrt {c}\right ) \cos \left (a +b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}+\frac {a b \left (-\operatorname {Si}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}+\frac {b \left (-b^{2} c +a^{2}-\left (a +b \sqrt {c}\right ) a \right ) \left (-\operatorname {Si}\left (b \sqrt {c}-\sqrt {d x +c}\, b \right ) \cos \left (a +b \sqrt {c}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b -b \sqrt {c}\right ) \sin \left (a +b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}-\frac {b \left (-b^{2} c +a^{2}-\left (a -b \sqrt {c}\right ) a \right ) \left (\operatorname {Si}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}-a \,b^{4} \left (\frac {\cos \left (a +\sqrt {d x +c}\, b \right ) \left (-\frac {a +\sqrt {d x +c}\, b}{2 c \,b^{2}}+\frac {a}{2 c \,b^{2}}\right )}{-b^{2} c +a^{2}-2 \left (a +\sqrt {d x +c}\, b \right ) a +\left (a +\sqrt {d x +c}\, b \right )^{2}}-\frac {\sin \left (a +b \sqrt {c}\right ) \operatorname {Si}\left (b \sqrt {c}-\sqrt {d x +c}\, b \right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b -b \sqrt {c}\right ) \cos \left (a +b \sqrt {c}\right )}{4 c^{\frac {3}{2}} b^{3}}+\frac {-\operatorname {Si}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )}{4 c^{\frac {3}{2}} b^{3}}-\frac {-\operatorname {Si}\left (b \sqrt {c}-\sqrt {d x +c}\, b \right ) \cos \left (a +b \sqrt {c}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b -b \sqrt {c}\right ) \sin \left (a +b \sqrt {c}\right )}{4 c \,b^{2}}-\frac {\operatorname {Si}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b +b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )}{4 c \,b^{2}}\right )\right )}{b^{2}}\)	\(714\)

Input:

int(cos(a+(d*x+c)^(1/2)*b)/x^2,x,method=_RETURNVERBOSE)
 

Output:

2*d/b^2*(cos(a+(d*x+c)^(1/2)*b)*(-1/2*a*b^2/c*(a+(d*x+c)^(1/2)*b)+1/2*b^2* 
(-b^2*c+a^2)/c)/(-b^2*c+a^2-2*(a+(d*x+c)^(1/2)*b)*a+(a+(d*x+c)^(1/2)*b)^2) 
-1/4*a*b/c^(3/2)*(sin(a+b*c^(1/2))*Si(b*c^(1/2)-(d*x+c)^(1/2)*b)+Ci((d*x+c 
)^(1/2)*b-b*c^(1/2))*cos(a+b*c^(1/2)))+1/4*a*b/c^(3/2)*(-Si((d*x+c)^(1/2)* 
b+b*c^(1/2))*sin(a-b*c^(1/2))+Ci((d*x+c)^(1/2)*b+b*c^(1/2))*cos(a-b*c^(1/2 
)))+1/4*b*(-b^2*c+a^2-(a+b*c^(1/2))*a)/c^(3/2)*(-Si(b*c^(1/2)-(d*x+c)^(1/2 
)*b)*cos(a+b*c^(1/2))+Ci((d*x+c)^(1/2)*b-b*c^(1/2))*sin(a+b*c^(1/2)))-1/4* 
b*(-b^2*c+a^2-(a-b*c^(1/2))*a)/c^(3/2)*(Si((d*x+c)^(1/2)*b+b*c^(1/2))*cos( 
a-b*c^(1/2))+Ci((d*x+c)^(1/2)*b+b*c^(1/2))*sin(a-b*c^(1/2)))-a*b^4*(cos(a+ 
(d*x+c)^(1/2)*b)*(-1/2/c/b^2*(a+(d*x+c)^(1/2)*b)+1/2*a/c/b^2)/(-b^2*c+a^2- 
2*(a+(d*x+c)^(1/2)*b)*a+(a+(d*x+c)^(1/2)*b)^2)-1/4/c^(3/2)/b^3*(sin(a+b*c^ 
(1/2))*Si(b*c^(1/2)-(d*x+c)^(1/2)*b)+Ci((d*x+c)^(1/2)*b-b*c^(1/2))*cos(a+b 
*c^(1/2)))+1/4/c^(3/2)/b^3*(-Si((d*x+c)^(1/2)*b+b*c^(1/2))*sin(a-b*c^(1/2) 
)+Ci((d*x+c)^(1/2)*b+b*c^(1/2))*cos(a-b*c^(1/2)))-1/4/c/b^2*(-Si(b*c^(1/2) 
-(d*x+c)^(1/2)*b)*cos(a+b*c^(1/2))+Ci((d*x+c)^(1/2)*b-b*c^(1/2))*sin(a+b*c 
^(1/2)))-1/4/c/b^2*(Si((d*x+c)^(1/2)*b+b*c^(1/2))*cos(a-b*c^(1/2))+Ci((d*x 
+c)^(1/2)*b+b*c^(1/2))*sin(a-b*c^(1/2)))))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.14 \[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\frac {\sqrt {-b^{2} c} d x {\rm Ei}\left (i \, \sqrt {d x + c} b - \sqrt {-b^{2} c}\right ) e^{\left (i \, a + \sqrt {-b^{2} c}\right )} - \sqrt {-b^{2} c} d x {\rm Ei}\left (i \, \sqrt {d x + c} b + \sqrt {-b^{2} c}\right ) e^{\left (i \, a - \sqrt {-b^{2} c}\right )} + \sqrt {-b^{2} c} d x {\rm Ei}\left (-i \, \sqrt {d x + c} b - \sqrt {-b^{2} c}\right ) e^{\left (-i \, a + \sqrt {-b^{2} c}\right )} - \sqrt {-b^{2} c} d x {\rm Ei}\left (-i \, \sqrt {d x + c} b + \sqrt {-b^{2} c}\right ) e^{\left (-i \, a - \sqrt {-b^{2} c}\right )} - 4 \, c \cos \left (\sqrt {d x + c} b + a\right )}{4 \, c x} \] Input:

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

Output:

1/4*(sqrt(-b^2*c)*d*x*Ei(I*sqrt(d*x + c)*b - sqrt(-b^2*c))*e^(I*a + sqrt(- 
b^2*c)) - sqrt(-b^2*c)*d*x*Ei(I*sqrt(d*x + c)*b + sqrt(-b^2*c))*e^(I*a - s 
qrt(-b^2*c)) + sqrt(-b^2*c)*d*x*Ei(-I*sqrt(d*x + c)*b - sqrt(-b^2*c))*e^(- 
I*a + sqrt(-b^2*c)) - sqrt(-b^2*c)*d*x*Ei(-I*sqrt(d*x + c)*b + sqrt(-b^2*c 
))*e^(-I*a - sqrt(-b^2*c)) - 4*c*cos(sqrt(d*x + c)*b + a))/(c*x)
 

Sympy [F]

\[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\int \frac {\cos {\left (a + b \sqrt {c + d x} \right )}}{x^{2}}\, dx \] Input:

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

Output:

Integral(cos(a + b*sqrt(c + d*x))/x**2, x)
 

Maxima [F]

\[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\int { \frac {\cos \left (\sqrt {d x + c} b + a\right )}{x^{2}} \,d x } \] Input:

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

Output:

integrate(cos(sqrt(d*x + c)*b + a)/x^2, x)
 

Giac [F]

\[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\int { \frac {\cos \left (\sqrt {d x + c} b + a\right )}{x^{2}} \,d x } \] Input:

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

Output:

integrate(cos(sqrt(d*x + c)*b + a)/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\int \frac {\cos \left (a+b\,\sqrt {c+d\,x}\right )}{x^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\int \frac {\cos \left (\sqrt {d x +c}\, b +a \right )}{x^{2}}d x \] Input:

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

Output:

int(cos(sqrt(c + d*x)*b + a)/x**2,x)