3.93 \(\int \frac{\cos (a+b \sqrt{c+d x})}{x} \, dx\)

Optimal. Leaf size=126 \[ \cos \left (a-b \sqrt{c}\right ) \text{CosIntegral}\left (b \left (\sqrt{c+d x}+\sqrt{c}\right )\right )+\cos \left (a+b \sqrt{c}\right ) \text{CosIntegral}\left (b \sqrt{c}-b \sqrt{c+d x}\right )-\sin \left (a-b \sqrt{c}\right ) \text{Si}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )+\sin \left (a+b \sqrt{c}\right ) \text{Si}\left (b \sqrt{c}-b \sqrt{c+d x}\right ) \]

[Out]

Cos[a - b*Sqrt[c]]*CosIntegral[b*(Sqrt[c] + Sqrt[c + d*x])] + Cos[a + b*Sqrt[c]]*CosIntegral[b*Sqrt[c] - b*Sqr
t[c + d*x]] - Sin[a - b*Sqrt[c]]*SinIntegral[b*(Sqrt[c] + Sqrt[c + d*x])] + Sin[a + b*Sqrt[c]]*SinIntegral[b*S
qrt[c] - b*Sqrt[c + d*x]]

________________________________________________________________________________________

Rubi [A]  time = 0.238869, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3432, 3303, 3299, 3302} \[ \cos \left (a-b \sqrt{c}\right ) \text{CosIntegral}\left (b \left (\sqrt{c+d x}+\sqrt{c}\right )\right )+\cos \left (a+b \sqrt{c}\right ) \text{CosIntegral}\left (b \sqrt{c}-b \sqrt{c+d x}\right )-\sin \left (a-b \sqrt{c}\right ) \text{Si}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )+\sin \left (a+b \sqrt{c}\right ) \text{Si}\left (b \sqrt{c}-b \sqrt{c+d x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*Sqrt[c + d*x]]/x,x]

[Out]

Cos[a - b*Sqrt[c]]*CosIntegral[b*(Sqrt[c] + Sqrt[c + d*x])] + Cos[a + b*Sqrt[c]]*CosIntegral[b*Sqrt[c] - b*Sqr
t[c + d*x]] - Sin[a - b*Sqrt[c]]*SinIntegral[b*(Sqrt[c] + Sqrt[c + d*x])] + Sin[a + b*Sqrt[c]]*SinIntegral[b*S
qrt[c] - b*Sqrt[c + d*x]]

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{c+d x}\right )}{x} \, dx &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{d \cos (a+b x)}{2 \left (\sqrt{c}-x\right )}+\frac{d \cos (a+b x)}{2 \left (\sqrt{c}+x\right )}\right ) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=-\operatorname{Subst}\left (\int \frac{\cos (a+b x)}{\sqrt{c}-x} \, dx,x,\sqrt{c+d x}\right )+\operatorname{Subst}\left (\int \frac{\cos (a+b x)}{\sqrt{c}+x} \, dx,x,\sqrt{c+d x}\right )\\ &=\cos \left (a-b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b \sqrt{c}+b x\right )}{\sqrt{c}+x} \, dx,x,\sqrt{c+d x}\right )-\cos \left (a+b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b \sqrt{c}-b x\right )}{\sqrt{c}-x} \, dx,x,\sqrt{c+d x}\right )-\sin \left (a-b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b \sqrt{c}+b x\right )}{\sqrt{c}+x} \, dx,x,\sqrt{c+d x}\right )-\sin \left (a+b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b \sqrt{c}-b x\right )}{\sqrt{c}-x} \, dx,x,\sqrt{c+d x}\right )\\ &=\cos \left (a-b \sqrt{c}\right ) \text{Ci}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )+\cos \left (a+b \sqrt{c}\right ) \text{Ci}\left (b \sqrt{c}-b \sqrt{c+d x}\right )-\sin \left (a-b \sqrt{c}\right ) \text{Si}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )+\sin \left (a+b \sqrt{c}\right ) \text{Si}\left (b \sqrt{c}-b \sqrt{c+d x}\right )\\ \end{align*}

Mathematica [C]  time = 0.66874, size = 145, normalized size = 1.15 \[ \frac{1}{2} e^{-i \left (a+b \sqrt{c}\right )} \left (e^{2 i \left (a+b \sqrt{c}\right )} \text{Ei}\left (i b \left (\sqrt{c+d x}-\sqrt{c}\right )\right )+e^{2 i a} \text{Ei}\left (i b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )+\text{Ei}\left (-i b \left (\sqrt{c+d x}-\sqrt{c}\right )\right )+e^{2 i b \sqrt{c}} \text{Ei}\left (-i b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*Sqrt[c + d*x]]/x,x]

[Out]

(ExpIntegralEi[(-I)*b*(-Sqrt[c] + Sqrt[c + d*x])] + E^((2*I)*(a + b*Sqrt[c]))*ExpIntegralEi[I*b*(-Sqrt[c] + Sq
rt[c + d*x])] + E^((2*I)*b*Sqrt[c])*ExpIntegralEi[(-I)*b*(Sqrt[c] + Sqrt[c + d*x])] + E^((2*I)*a)*ExpIntegralE
i[I*b*(Sqrt[c] + Sqrt[c + d*x])])/(2*E^(I*(a + b*Sqrt[c])))

________________________________________________________________________________________

Maple [B]  time = 0.058, size = 271, normalized size = 2.2 \begin{align*} 2\,{\frac{1}{{b}^{2}} \left ( 1/2\,{\frac{b \left ( a+b\sqrt{c} \right ) \left ({\it Si} \left ( b\sqrt{c}-b\sqrt{dx+c} \right ) \sin \left ( a+b\sqrt{c} \right ) +{\it Ci} \left ( b\sqrt{dx+c}-b\sqrt{c} \right ) \cos \left ( a+b\sqrt{c} \right ) \right ) }{\sqrt{c}}}-1/2\,{\frac{b \left ( a-b\sqrt{c} \right ) \left ( -{\it Si} \left ( b\sqrt{dx+c}+b\sqrt{c} \right ) \sin \left ( a-b\sqrt{c} \right ) +{\it Ci} \left ( b\sqrt{dx+c}+b\sqrt{c} \right ) \cos \left ( a-b\sqrt{c} \right ) \right ) }{\sqrt{c}}}-a{b}^{2} \left ( 1/2\,{\frac{{\it Si} \left ( b\sqrt{c}-b\sqrt{dx+c} \right ) \sin \left ( a+b\sqrt{c} \right ) +{\it Ci} \left ( b\sqrt{dx+c}-b\sqrt{c} \right ) \cos \left ( a+b\sqrt{c} \right ) }{b\sqrt{c}}}-1/2\,{\frac{-{\it Si} \left ( b\sqrt{dx+c}+b\sqrt{c} \right ) \sin \left ( a-b\sqrt{c} \right ) +{\it Ci} \left ( b\sqrt{dx+c}+b\sqrt{c} \right ) \cos \left ( a-b\sqrt{c} \right ) }{b\sqrt{c}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (\sqrt{d x + c} 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)^(1/2))/x,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [C]  time = 1.72537, size = 360, normalized size = 2.86 \begin{align*} \frac{1}{2} \,{\rm Ei}\left (i \, \sqrt{d x + c} b - \sqrt{-b^{2} c}\right ) e^{\left (i \, a + \sqrt{-b^{2} c}\right )} + \frac{1}{2} \,{\rm Ei}\left (i \, \sqrt{d x + c} b + \sqrt{-b^{2} c}\right ) e^{\left (i \, a - \sqrt{-b^{2} c}\right )} + \frac{1}{2} \,{\rm Ei}\left (-i \, \sqrt{d x + c} b - \sqrt{-b^{2} c}\right ) e^{\left (-i \, a + \sqrt{-b^{2} c}\right )} + \frac{1}{2} \,{\rm Ei}\left (-i \, \sqrt{d x + c} b + \sqrt{-b^{2} c}\right ) e^{\left (-i \, a - \sqrt{-b^{2} c}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (a + b \sqrt{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)**(1/2))/x,x)

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (\sqrt{d x + c} 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)^(1/2))/x,x, algorithm="giac")

[Out]

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