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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3513, 3384, 3380, 3383} \[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\cos \left (a+b \sqrt [3]{c}\right ) \operatorname {CosIntegral}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \operatorname {CosIntegral}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \operatorname {CosIntegral}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right )+\sin \left (a+b \sqrt [3]{c}\right ) \text {Si}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\sin \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Si}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\sin \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Si}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right ) \]

[In]

Int[Cos[a + b*(c + d*x)^(1/3)]/x,x]

[Out]

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

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3513

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.)*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {3 \text {Subst}\left (\int \left (-\frac {d \cos (a+b x)}{3 \left (\sqrt [3]{c}-x\right )}-\frac {d \cos (a+b x)}{3 \left (-\sqrt [3]{-1} \sqrt [3]{c}-x\right )}-\frac {d \cos (a+b x)}{3 \left ((-1)^{2/3} \sqrt [3]{c}-x\right )}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d} \\ & = -\text {Subst}\left (\int \frac {\cos (a+b x)}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\text {Subst}\left (\int \frac {\cos (a+b x)}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\text {Subst}\left (\int \frac {\cos (a+b x)}{(-1)^{2/3} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right ) \\ & = -\left (\cos \left (a+b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {\cos \left (b \sqrt [3]{c}-b x\right )}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )\right )-\cos \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {\cos \left (\sqrt [3]{-1} b \sqrt [3]{c}+b x\right )}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\cos \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {\cos \left ((-1)^{2/3} b \sqrt [3]{c}-b x\right )}{(-1)^{2/3} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\sin \left (a+b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {\sin \left (b \sqrt [3]{c}-b x\right )}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )+\sin \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {\sin \left (\sqrt [3]{-1} b \sqrt [3]{c}+b x\right )}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\sin \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {\sin \left ((-1)^{2/3} b \sqrt [3]{c}-b x\right )}{(-1)^{2/3} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right ) \\ & = \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 ) \\ \end{align*}

Mathematica [C] (verified)

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

Time = 11.08 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.04 \[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\frac {1}{2} \left (\text {RootSum}\left [c-\text {$\#$1}^3\&,\cos (a+b \text {$\#$1}) \operatorname {CosIntegral}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-i \operatorname {CosIntegral}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right ) \sin (a+b \text {$\#$1})-i \cos (a+b \text {$\#$1}) \text {Si}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\sin (a+b \text {$\#$1}) \text {Si}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]+\text {RootSum}\left [c-\text {$\#$1}^3\&,\cos (a+b \text {$\#$1}) \operatorname {CosIntegral}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+i \operatorname {CosIntegral}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right ) \sin (a+b \text {$\#$1})+i \cos (a+b \text {$\#$1}) \text {Si}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\sin (a+b \text {$\#$1}) \text {Si}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]\right ) \]

[In]

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

[Out]

(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*CosInte
gral[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

Maple [C] (verified)

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

Time = 1.10 (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	\(\frac {a^{2} b^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-b^{3} c +\textit {\_Z}^{3}-3 a \,\textit {\_Z}^{2}+3 a^{2} \textit {\_Z} -a^{3}\right )}{\sum }\frac {\operatorname {Si}\left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \sin \left (\textit {\_R1} \right )+\operatorname {Ci}\left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \cos \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )-2 a \,b^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-b^{3} c +\textit {\_Z}^{3}-3 a \,\textit {\_Z}^{2}+3 a^{2} \textit {\_Z} -a^{3}\right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {Si}\left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \sin \left (\textit {\_R1} \right )+\operatorname {Ci}\left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \cos \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )+b^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-b^{3} c +\textit {\_Z}^{3}-3 a \,\textit {\_Z}^{2}+3 a^{2} \textit {\_Z} -a^{3}\right )}{\sum }\frac {\textit {\_R1}^{2} \left (\operatorname {Si}\left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \sin \left (\textit {\_R1} \right )+\operatorname {Ci}\left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \cos \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )}{b^{3}}\)	\(279\)

[In]

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

[Out]

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)*sin(_R1)+Ci(b*(d*x+c)^(1/3)-_R1+a)*co
s(_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=Ro
otOf(-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 = 0.28 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.23 \[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\frac {1}{2} \, {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, a\right )} + \frac {1}{2} \, {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, a\right )} + \frac {1}{2} \, {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, a\right )} + \frac {1}{2} \, {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, a\right )} + \frac {1}{2} \, {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \left (i \, b^{3} c\right )^{\frac {1}{3}}\right ) e^{\left (i \, a - \left (i \, b^{3} c\right )^{\frac {1}{3}}\right )} + \frac {1}{2} \, {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \left (-i \, b^{3} c\right )^{\frac {1}{3}}\right ) e^{\left (-i \, a - \left (-i \, b^{3} c\right )^{\frac {1}{3}}\right )} \]

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(cos(a + b*(c + d*x)^(1/3))/x,x)

[Out]

int(cos(a + b*(c + d*x)^(1/3))/x, x)