\(\int \frac {\cos (a+b \sqrt [3]{c+d x})}{x^2} \, dx\) [99]

Optimal result

Integrand size = 18, antiderivative size = 332 \[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x}-\frac {b d \operatorname {CosIntegral}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right ) \sin \left (a+b \sqrt [3]{c}\right )}{3 c^{2/3}}+\frac {\sqrt [3]{-1} b d \operatorname {CosIntegral}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right ) \sin \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right )}{3 c^{2/3}}-\frac {(-1)^{2/3} b d \operatorname {CosIntegral}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right ) \sin \left (a+(-1)^{2/3} b \sqrt [3]{c}\right )}{3 c^{2/3}}+\frac {b d \cos \left (a+b \sqrt [3]{c}\right ) \text {Si}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}+\frac {(-1)^{2/3} b d \cos \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Si}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}+\frac {\sqrt [3]{-1} b d \cos \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Si}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right )}{3 c^{2/3}} \]

[Out]

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

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3513, 3423, 3414, 3384, 3380, 3383} \[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=-\frac {b d \sin \left (a+b \sqrt [3]{c}\right ) \operatorname {CosIntegral}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}+\frac {\sqrt [3]{-1} b d \sin \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \operatorname {CosIntegral}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right )}{3 c^{2/3}}-\frac {(-1)^{2/3} b d \sin \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \operatorname {CosIntegral}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}+\frac {b d \cos \left (a+b \sqrt [3]{c}\right ) \text {Si}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}+\frac {(-1)^{2/3} b d \cos \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Si}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}+\frac {\sqrt [3]{-1} b d \cos \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Si}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right )}{3 c^{2/3}}-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \]

[In]

Int[Cos[a + b*(c + d*x)^(1/3)]/x^2,x]

[Out]

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

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 3414

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 3423

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

Mathematica [C] (verified)

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

Time = 0.77 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.42 \[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x}-\frac {1}{6} i b d \text {RootSum}\left [c-\text {$\#$1}^3\&,\frac {e^{-i a-i b \text {$\#$1}} \operatorname {ExpIntegralEi}\left (-i b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ]+\frac {1}{6} i b d \text {RootSum}\left [c-\text {$\#$1}^3\&,\frac {e^{i a+i b \text {$\#$1}} \operatorname {ExpIntegralEi}\left (i b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ] \]

[In]

Integrate[Cos[a + b*(c + d*x)^(1/3)]/x^2,x]

[Out]

-(Cos[a + b*(c + d*x)^(1/3)]/x) - (I/6)*b*d*RootSum[c - #1^3 & , (E^((-I)*a - I*b*#1)*ExpIntegralEi[(-I)*b*((c
+ d*x)^(1/3) - #1)])/#1^2 & ] + (I/6)*b*d*RootSum[c - #1^3 & , (E^(I*a + I*b*#1)*ExpIntegralEi[I*b*((c + d*x)
^(1/3) - #1)])/#1^2 & ]

Maple [C] (verified)

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

Time = 1.33 (sec) , antiderivative size = 931, normalized size of antiderivative = 2.80

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(931\)
default	\(\text {Expression too large to display}\)	\(931\)

[In]

int(cos(a+b*(d*x+c)^(1/3))/x^2,x,method=_RETURNVERBOSE)

[Out]

3*d/b^3*(b^6*a^2*(cos(a+b*(d*x+c)^(1/3))*(1/3/c/b^3*(a+b*(d*x+c)^(1/3))-1/3*a/b^3/c)/(b^3*c+a^3-3*a^2*(a+b*(d*
x+c)^(1/3))+3*a*(a+b*(d*x+c)^(1/3))^2-(a+b*(d*x+c)^(1/3))^3)-2/9/c/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)*cos(_R1)),_R1=RootOf(-b^3*c+_Z^3-3*_Z^2*a+3*_Z*a^2-a^3))+1/9
/c/b^3*sum(1/(-_RR1+a)*(-Si(-b*(d*x+c)^(1/3)+_RR1-a)*cos(_RR1)+Ci(b*(d*x+c)^(1/3)-_RR1+a)*sin(_RR1)),_RR1=Root
Of(-b^3*c+_Z^3-3*_Z^2*a+3*_Z*a^2-a^3)))+cos(a+b*(d*x+c)^(1/3))*(-2/3*a*b^3/c*(a+b*(d*x+c)^(1/3))^2+2/3*a^2*b^3
/c*(a+b*(d*x+c)^(1/3)))/(b^3*c+a^3-3*a^2*(a+b*(d*x+c)^(1/3))+3*a*(a+b*(d*x+c)^(1/3))^2-(a+b*(d*x+c)^(1/3))^3)+
2/9*a*b^3/c*sum((_R1+a)/(_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))-2/9*a*b^3/c*sum(_RR1/(-_RR1+a)*(-Si(-b*(d*x+c)^(1/3)+_RR
1-a)*cos(_RR1)+Ci(b*(d*x+c)^(1/3)-_RR1+a)*sin(_RR1)),_RR1=RootOf(-b^3*c+_Z^3-3*_Z^2*a+3*_Z*a^2-a^3))+cos(a+b*(
d*x+c)^(1/3))*(2/3*a*b^3/c*(a+b*(d*x+c)^(1/3))^2-a^2*b^3/c*(a+b*(d*x+c)^(1/3))+1/3*b^3*(b^3*c+a^3)/c)/(b^3*c+a
^3-3*a^2*(a+b*(d*x+c)^(1/3))+3*a*(a+b*(d*x+c)^(1/3))^2-(a+b*(d*x+c)^(1/3))^3)-2/9*a*b^3/c*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))-1/9*b^3/c*sum((b^3*c+2*_RR1^2*a-3*_RR1*a^2+a^3)/(_RR1^2-2*_RR1*a+a^2)*(-Si(-b*(d*x+c)^(1/3)+
_RR1-a)*cos(_RR1)+Ci(b*(d*x+c)^(1/3)-_RR1+a)*sin(_RR1)),_RR1=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 = 0.31 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.22 \[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=-\frac {2 \, \left (i \, b^{3} c\right )^{\frac {1}{3}} d x {\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 )} + 2 \, \left (-i \, b^{3} c\right )^{\frac {1}{3}} d x {\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 )} - \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} d x + d x\right )} {\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 )} - \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} d x + d x\right )} {\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 )} - \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} d x + d x\right )} {\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 )} - \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} d x + d x\right )} {\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 )} + 12 \, c \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{12 \, c x} \]

[In]

integrate(cos(a+b*(d*x+c)^(1/3))/x^2,x, algorithm="fricas")

[Out]

-1/12*(2*(I*b^3*c)^(1/3)*d*x*Ei(I*(d*x + c)^(1/3)*b + (I*b^3*c)^(1/3))*e^(I*a - (I*b^3*c)^(1/3)) + 2*(-I*b^3*c
)^(1/3)*d*x*Ei(-I*(d*x + c)^(1/3)*b + (-I*b^3*c)^(1/3))*e^(-I*a - (-I*b^3*c)^(1/3)) - (I*b^3*c)^(1/3)*(I*sqrt(
3)*d*x + d*x)*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) - (-I*b^3*c)^(1/3)*(I*sqrt(3)*d*x + d*x)*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) - (I*b^3*c)^(1/3)*(-I*sqrt(3)*d*x + d*x)*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) - (-I*b^3*c)^(
1/3)*(-I*sqrt(3)*d*x + d*x)*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) + 12*c*cos((d*x + c)^(1/3)*b + a))/(c*x)

Sympy [F]

\[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int \frac {\cos {\left (a + b \sqrt [3]{c + d x} \right )}}{x^{2}}\, dx \]

[In]

integrate(cos(a+b*(d*x+c)**(1/3))/x**2,x)

[Out]

Integral(cos(a + b*(c + d*x)**(1/3))/x**2, x)

Maxima [F]

\[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int { \frac {\cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x^{2}} \,d x } \]

[In]

integrate(cos(a+b*(d*x+c)^(1/3))/x^2,x, algorithm="maxima")

[Out]

integrate(cos((d*x + c)^(1/3)*b + a)/x^2, x)

Giac [F]

\[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int { \frac {\cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x^{2}} \,d x } \]

[In]

integrate(cos(a+b*(d*x+c)^(1/3))/x^2,x, algorithm="giac")

[Out]

integrate(cos((d*x + c)^(1/3)*b + a)/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int \frac {\cos \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{x^2} \,d x \]

[In]

int(cos(a + b*(c + d*x)^(1/3))/x^2,x)

[Out]

int(cos(a + b*(c + d*x)^(1/3))/x^2, x)