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

Optimal. Leaf size=234 \[ \cos \left (a+b \sqrt [3]{c}\right ) \text{CosIntegral}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text{CosIntegral}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text{CosIntegral}\left (b \sqrt [3]{c+d x}+\sqrt [3]{-1} b \sqrt [3]{c}\right )+\sin \left (a+b \sqrt [3]{c}\right ) \text{Si}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\sin \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text{Si}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\sin \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text{Si}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right ) \]

[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)]

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Rubi [A]  time = 0.468404, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3432, 3303, 3299, 3302} \[ \cos \left (a+b \sqrt [3]{c}\right ) \text{CosIntegral}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text{CosIntegral}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text{CosIntegral}\left (b \sqrt [3]{c+d x}+\sqrt [3]{-1} b \sqrt [3]{c}\right )+\sin \left (a+b \sqrt [3]{c}\right ) \text{Si}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\sin \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text{Si}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\sin \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text{Si}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right ) \]

Antiderivative was successfully verified.

[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 3432

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]

Rule 3303

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 3299

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 3302

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]

Rubi steps

\begin{align*} \int \frac{\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx &=\frac{3 \operatorname{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}\\ &=-\operatorname{Subst}\left (\int \frac{\cos (a+b x)}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\operatorname{Subst}\left (\int \frac{\cos (a+b x)}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \text{Ci}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text{Ci}\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 ) \text{Ci}\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]  time = 0.433259, size = 243, normalized size = 1.04 \[ \frac{1}{2} \left (\text{RootSum}\left [c-\text{$\#$1}^3\& ,-i \sin (\text{$\#$1} b+a) \text{CosIntegral}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\cos (\text{$\#$1} b+a) \text{CosIntegral}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )-\sin (\text{$\#$1} b+a) \text{Si}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )-i \cos (\text{$\#$1} b+a) \text{Si}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )\& \right ]+\text{RootSum}\left [c-\text{$\#$1}^3\& ,i \sin (\text{$\#$1} b+a) \text{CosIntegral}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\cos (\text{$\#$1} b+a) \text{CosIntegral}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )-\sin (\text{$\#$1} b+a) \text{Si}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+i \cos (\text{$\#$1} b+a) \text{Si}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )\& \right ]\right ) \]

Warning: Unable to verify antiderivative.

[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

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Maple [C]  time = 0.149, size = 279, normalized size = 1.2 \begin{align*} 3\,{\frac{1}{{b}^{3}} \left ( 1/3\,{b}^{3}\sum _{{\it \_R1}={\it RootOf} \left ( -{b}^{3}c+{{\it \_Z}}^{3}-3\,a{{\it \_Z}}^{2}+3\,{a}^{2}{\it \_Z}-{a}^{3} \right ) }{\frac{{{\it \_R1}}^{2} \left ({\it Si} \left ( -b\sqrt [3]{dx+c}+{\it \_R1}-a \right ) \sin \left ({\it \_R1} \right ) +{\it Ci} \left ( b\sqrt [3]{dx+c}-{\it \_R1}+a \right ) \cos \left ({\it \_R1} \right ) \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,a+{a}^{2}}}-2/3\,a{b}^{3}\sum _{{\it \_R1}={\it RootOf} \left ( -{b}^{3}c+{{\it \_Z}}^{3}-3\,a{{\it \_Z}}^{2}+3\,{a}^{2}{\it \_Z}-{a}^{3} \right ) }{\frac{{\it \_R1}\, \left ({\it Si} \left ( -b\sqrt [3]{dx+c}+{\it \_R1}-a \right ) \sin \left ({\it \_R1} \right ) +{\it Ci} \left ( b\sqrt [3]{dx+c}-{\it \_R1}+a \right ) \cos \left ({\it \_R1} \right ) \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,a+{a}^{2}}}+1/3\,{a}^{2}{b}^{3}\sum _{{\it \_R1}={\it RootOf} \left ( -{b}^{3}c+{{\it \_Z}}^{3}-3\,a{{\it \_Z}}^{2}+3\,{a}^{2}{\it \_Z}-{a}^{3} \right ) }{\frac{{\it Si} \left ( -b\sqrt [3]{dx+c}+{\it \_R1}-a \right ) \sin \left ({\it \_R1} \right ) +{\it Ci} \left ( b\sqrt [3]{dx+c}-{\it \_R1}+a \right ) \cos \left ({\it \_R1} \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,a+{a}^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/b^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)*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*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)*cos(_R1)),_R1=Ro
otOf(-b^3*c+_Z^3-3*_Z^2*a+3*_Z*a^2-a^3)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [C]  time = 1.77086, size = 833, normalized size = 3.56 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (a + b \sqrt [3]{c + d x} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)