\(\int \frac {\cos (a+b \sqrt [3]{c+d x})}{x} \, dx\) [98]

Optimal result

Integrand size = 18, antiderivative size = 234 \[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\cos \left (a+b \sqrt [3]{c}\right ) \operatorname {CosIntegral}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \operatorname {CosIntegral}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \operatorname {CosIntegral}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right )+\sin \left (a+b \sqrt [3]{c}\right ) \text {Si}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\sin \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Si}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\sin \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Si}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 11.09 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.04 \[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\frac {1}{2} \left (\text {RootSum}\left [c-\text {$\#$1}^3\&,\cos (a+b \text {$\#$1}) \operatorname {CosIntegral}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-i \operatorname {CosIntegral}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right ) \sin (a+b \text {$\#$1})-i \cos (a+b \text {$\#$1}) \text {Si}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\sin (a+b \text {$\#$1}) \text {Si}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]+\text {RootSum}\left [c-\text {$\#$1}^3\&,\cos (a+b \text {$\#$1}) \operatorname {CosIntegral}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+i \operatorname {CosIntegral}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right ) \sin (a+b \text {$\#$1})+i \cos (a+b \text {$\#$1}) \text {Si}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\sin (a+b \text {$\#$1}) \text {Si}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]\right ) \] Input:

Integrate[Cos[a + b*(c + d*x)^(1/3)]/x,x]
 

Output:

(RootSum[c - #1^3 & , Cos[a + b*#1]*CosIntegral[b*((c + d*x)^(1/3) - #1)] 
- I*CosIntegral[b*((c + d*x)^(1/3) - #1)]*Sin[a + b*#1] - I*Cos[a + b*#1]* 
SinIntegral[b*((c + d*x)^(1/3) - #1)] - Sin[a + b*#1]*SinIntegral[b*((c + 
d*x)^(1/3) - #1)] & ] + RootSum[c - #1^3 & , Cos[a + b*#1]*CosIntegral[b*( 
(c + d*x)^(1/3) - #1)] + I*CosIntegral[b*((c + d*x)^(1/3) - #1)]*Sin[a + b 
*#1] + I*Cos[a + b*#1]*SinIntegral[b*((c + d*x)^(1/3) - #1)] - Sin[a + b*# 
1]*SinIntegral[b*((c + d*x)^(1/3) - #1)] & ])/2
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3913, 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.

\(\displaystyle \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx\)

\(\Big \downarrow \) 3913

\(\displaystyle \frac {3 \int \left (-\frac {d \cos \left (a+b \sqrt [3]{c+d x}\right )}{3 \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )}+\frac {d \cos \left (a+b \sqrt [3]{c+d x}\right )}{3 \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )}+\frac {\sqrt [3]{-1} d \cos \left (a+b \sqrt [3]{c+d x}\right )}{3 \left (\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{c+d x}\right )}\right )d\sqrt [3]{c+d x}}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \left (\frac {1}{3} d \cos \left (a+b \sqrt [3]{c}\right ) \operatorname {CosIntegral}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\frac {1}{3} d \cos \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \operatorname {CosIntegral}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\frac {1}{3} d \cos \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \operatorname {CosIntegral}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right )+\frac {1}{3} d \sin \left (a+b \sqrt [3]{c}\right ) \text {Si}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\frac {1}{3} d \sin \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Si}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\frac {1}{3} d \sin \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Si}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right )\right )}{d}\)

Input:

Int[Cos[a + b*(c + d*x)^(1/3)]/x,x]
 

Output:

(3*((d*Cos[a + b*c^(1/3)]*CosIntegral[b*c^(1/3) - b*(c + d*x)^(1/3)])/3 + 
(d*Cos[a + (-1)^(2/3)*b*c^(1/3)]*CosIntegral[(-1)^(2/3)*b*c^(1/3) - b*(c + 
 d*x)^(1/3)])/3 + (d*Cos[a - (-1)^(1/3)*b*c^(1/3)]*CosIntegral[(-1)^(1/3)* 
b*c^(1/3) + b*(c + d*x)^(1/3)])/3 + (d*Sin[a + b*c^(1/3)]*SinIntegral[b*c^ 
(1/3) - b*(c + d*x)^(1/3)])/3 + (d*Sin[a + (-1)^(2/3)*b*c^(1/3)]*SinIntegr 
al[(-1)^(2/3)*b*c^(1/3) - b*(c + d*x)^(1/3)])/3 - (d*Sin[a - (-1)^(1/3)*b* 
c^(1/3)]*SinIntegral[(-1)^(1/3)*b*c^(1/3) + b*(c + d*x)^(1/3)])/3))/d
 

Defintions of rubi rules used

rule 2009

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

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 [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.95 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(\frac {a^{2} b^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-b^{3} c +\textit {\_Z}^{3}-3 a \,\textit {\_Z}^{2}+3 a^{2} \textit {\_Z} -a^{3}\right )}{\sum }\frac {\operatorname {Si}\left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \sin \left (\textit {\_R1} \right )+\operatorname {Ci}\left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \cos \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )-2 a \,b^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-b^{3} c +\textit {\_Z}^{3}-3 a \,\textit {\_Z}^{2}+3 a^{2} \textit {\_Z} -a^{3}\right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {Si}\left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \sin \left (\textit {\_R1} \right )+\operatorname {Ci}\left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \cos \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )+b^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-b^{3} c +\textit {\_Z}^{3}-3 a \,\textit {\_Z}^{2}+3 a^{2} \textit {\_Z} -a^{3}\right )}{\sum }\frac {\textit {\_R1}^{2} \left (\operatorname {Si}\left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \sin \left (\textit {\_R1} \right )+\operatorname {Ci}\left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \cos \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )}{b^{3}}\)	\(279\)
default	\(\frac {a^{2} b^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-b^{3} c +\textit {\_Z}^{3}-3 a \,\textit {\_Z}^{2}+3 a^{2} \textit {\_Z} -a^{3}\right )}{\sum }\frac {\operatorname {Si}\left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \sin \left (\textit {\_R1} \right )+\operatorname {Ci}\left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \cos \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )-2 a \,b^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-b^{3} c +\textit {\_Z}^{3}-3 a \,\textit {\_Z}^{2}+3 a^{2} \textit {\_Z} -a^{3}\right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {Si}\left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \sin \left (\textit {\_R1} \right )+\operatorname {Ci}\left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \cos \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )+b^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-b^{3} c +\textit {\_Z}^{3}-3 a \,\textit {\_Z}^{2}+3 a^{2} \textit {\_Z} -a^{3}\right )}{\sum }\frac {\textit {\_R1}^{2} \left (\operatorname {Si}\left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \sin \left (\textit {\_R1} \right )+\operatorname {Ci}\left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \cos \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )}{b^{3}}\)	\(279\)

Input:

int(cos(a+b*(d*x+c)^(1/3))/x,x,method=_RETURNVERBOSE)
 

Output:

3/b^3*(1/3*a^2*b^3*sum(1/(_R1^2-2*_R1*a+a^2)*(Si(-b*(d*x+c)^(1/3)+_R1-a)*s 
in(_R1)+Ci(b*(d*x+c)^(1/3)-_R1+a)*cos(_R1)),_R1=RootOf(-b^3*c+_Z^3-3*_Z^2* 
a+3*_Z*a^2-a^3))-2/3*a*b^3*sum(_R1/(_R1^2-2*_R1*a+a^2)*(Si(-b*(d*x+c)^(1/3 
)+_R1-a)*sin(_R1)+Ci(b*(d*x+c)^(1/3)-_R1+a)*cos(_R1)),_R1=RootOf(-b^3*c+_Z 
^3-3*_Z^2*a+3*_Z*a^2-a^3))+1/3*b^3*sum(_R1^2/(_R1^2-2*_R1*a+a^2)*(Si(-b*(d 
*x+c)^(1/3)+_R1-a)*sin(_R1)+Ci(b*(d*x+c)^(1/3)-_R1+a)*cos(_R1)),_R1=RootOf 
(-b^3*c+_Z^3-3*_Z^2*a+3*_Z*a^2-a^3)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.23 \[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\frac {1}{2} \, {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, a\right )} + \frac {1}{2} \, {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, a\right )} + \frac {1}{2} \, {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, a\right )} + \frac {1}{2} \, {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, a\right )} + \frac {1}{2} \, {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \left (i \, b^{3} c\right )^{\frac {1}{3}}\right ) e^{\left (i \, a - \left (i \, b^{3} c\right )^{\frac {1}{3}}\right )} + \frac {1}{2} \, {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \left (-i \, b^{3} c\right )^{\frac {1}{3}}\right ) e^{\left (-i \, a - \left (-i \, b^{3} c\right )^{\frac {1}{3}}\right )} \] Input:

integrate(cos(a+b*(d*x+c)^(1/3))/x,x, algorithm="fricas")
 

Output:

1/2*Ei(I*(d*x + c)^(1/3)*b + 1/2*(I*b^3*c)^(1/3)*(-I*sqrt(3) - 1))*e^(1/2* 
(I*b^3*c)^(1/3)*(I*sqrt(3) + 1) + I*a) + 1/2*Ei(-I*(d*x + c)^(1/3)*b + 1/2 
*(-I*b^3*c)^(1/3)*(-I*sqrt(3) - 1))*e^(1/2*(-I*b^3*c)^(1/3)*(I*sqrt(3) + 1 
) - I*a) + 1/2*Ei(I*(d*x + c)^(1/3)*b + 1/2*(I*b^3*c)^(1/3)*(I*sqrt(3) - 1 
))*e^(1/2*(I*b^3*c)^(1/3)*(-I*sqrt(3) + 1) + I*a) + 1/2*Ei(-I*(d*x + c)^(1 
/3)*b + 1/2*(-I*b^3*c)^(1/3)*(I*sqrt(3) - 1))*e^(1/2*(-I*b^3*c)^(1/3)*(-I* 
sqrt(3) + 1) - I*a) + 1/2*Ei(I*(d*x + c)^(1/3)*b + (I*b^3*c)^(1/3))*e^(I*a 
 - (I*b^3*c)^(1/3)) + 1/2*Ei(-I*(d*x + c)^(1/3)*b + (-I*b^3*c)^(1/3))*e^(- 
I*a - (-I*b^3*c)^(1/3))
 

Sympy [F]

\[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int \frac {\cos {\left (a + b \sqrt [3]{c + d x} \right )}}{x}\, dx \] Input:

integrate(cos(a+b*(d*x+c)**(1/3))/x,x)
                                                                                    
                                                                                    
 

Output:

Integral(cos(a + b*(c + d*x)**(1/3))/x, x)
 

Maxima [F]

\[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int { \frac {\cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x} \,d x } \] Input:

integrate(cos(a+b*(d*x+c)^(1/3))/x,x, algorithm="maxima")
 

Output:

integrate(cos((d*x + c)^(1/3)*b + a)/x, x)
 

Giac [F]

\[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int { \frac {\cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x} \,d x } \] Input:

integrate(cos(a+b*(d*x+c)^(1/3))/x,x, algorithm="giac")
 

Output:

integrate(cos((d*x + c)^(1/3)*b + a)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int \frac {\cos \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{x} \,d x \] Input:

int(cos(a + b*(c + d*x)^(1/3))/x,x)
 

Output:

int(cos(a + b*(c + d*x)^(1/3))/x, x)
 

Reduce [F]

\[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int \frac {\cos \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right )}{x}d x \] Input:

int(cos(a+b*(d*x+c)^(1/3))/x,x)
 

Output:

int(cos((c + d*x)**(1/3)*b + a)/x,x)