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

Optimal result

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

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

Mathematica [C] (verified)

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

Time = 0.82 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.42 \[ \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x}-\frac {1}{6} i b d \text {RootSum}\left [c-\text {$\#$1}^3\&,\frac {e^{-i a-i b \text {$\#$1}} \operatorname {ExpIntegralEi}\left (-i b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ]+\frac {1}{6} i b d \text {RootSum}\left [c-\text {$\#$1}^3\&,\frac {e^{i a+i b \text {$\#$1}} \operatorname {ExpIntegralEi}\left (i b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ] \] Input:

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

Output:

-(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 & ]
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.01, 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.

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

\(\Big \downarrow \) 3913

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3823

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

\(\Big \downarrow \) 3814

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 [C] (verified)

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

Time = 1.14 (sec) , antiderivative size = 931, normalized size of antiderivative = 2.80

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

Input:

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

Output:

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=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+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)+_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))+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)*si...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (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} \] Input:

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

Output:

-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*sq 
rt(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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