3.26 \(\int \frac{\cos (a+b x^2)}{\sqrt{x}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{e^{i a} \sqrt{x} \text{Gamma}\left (\frac{1}{4},-i b x^2\right )}{4 \sqrt [4]{-i b x^2}}-\frac{e^{-i a} \sqrt{x} \text{Gamma}\left (\frac{1}{4},i b x^2\right )}{4 \sqrt [4]{i b x^2}} \]

[Out]

-(E^(I*a)*Sqrt[x]*Gamma[1/4, (-I)*b*x^2])/(4*((-I)*b*x^2)^(1/4)) - (Sqrt[x]*Gamma[1/4, I*b*x^2])/(4*E^(I*a)*(I
*b*x^2)^(1/4))

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Rubi [A]  time = 0.0653032, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3390, 2218} \[ -\frac{e^{i a} \sqrt{x} \text{Gamma}\left (\frac{1}{4},-i b x^2\right )}{4 \sqrt [4]{-i b x^2}}-\frac{e^{-i a} \sqrt{x} \text{Gamma}\left (\frac{1}{4},i b x^2\right )}{4 \sqrt [4]{i b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x^2]/Sqrt[x],x]

[Out]

-(E^(I*a)*Sqrt[x]*Gamma[1/4, (-I)*b*x^2])/(4*((-I)*b*x^2)^(1/4)) - (Sqrt[x]*Gamma[1/4, I*b*x^2])/(4*E^(I*a)*(I
*b*x^2)^(1/4))

Rule 3390

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[1/2, Int[(e*x)^m*E^(-(c*I) - d*I*x^n),
x], x] + Dist[1/2, Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{\cos \left (a+b x^2\right )}{\sqrt{x}} \, dx &=\frac{1}{2} \int \frac{e^{-i a-i b x^2}}{\sqrt{x}} \, dx+\frac{1}{2} \int \frac{e^{i a+i b x^2}}{\sqrt{x}} \, dx\\ &=-\frac{e^{i a} \sqrt{x} \Gamma \left (\frac{1}{4},-i b x^2\right )}{4 \sqrt [4]{-i b x^2}}-\frac{e^{-i a} \sqrt{x} \Gamma \left (\frac{1}{4},i b x^2\right )}{4 \sqrt [4]{i b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0714116, size = 89, normalized size = 1.1 \[ -\frac{\sqrt{x} \left (\sqrt [4]{-i b x^2} (\cos (a)-i \sin (a)) \text{Gamma}\left (\frac{1}{4},i b x^2\right )+\sqrt [4]{i b x^2} (\cos (a)+i \sin (a)) \text{Gamma}\left (\frac{1}{4},-i b x^2\right )\right )}{4 \sqrt [4]{b^2 x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x^2]/Sqrt[x],x]

[Out]

-(Sqrt[x]*(((-I)*b*x^2)^(1/4)*Gamma[1/4, I*b*x^2]*(Cos[a] - I*Sin[a]) + (I*b*x^2)^(1/4)*Gamma[1/4, (-I)*b*x^2]
*(Cos[a] + I*Sin[a])))/(4*(b^2*x^4)^(1/4))

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Maple [C]  time = 0.078, size = 338, normalized size = 4.2 \begin{align*}{\frac{\cos \left ( a \right ) \sqrt{\pi }\sqrt [4]{2}}{4} \left ( 6\,{\frac{{2}^{3/4}\sqrt [8]{{b}^{2}}\sin \left ( b{x}^{2} \right ) }{\sqrt{\pi }{x}^{3/2}b} \left ({\frac{8\,{b}^{2}{x}^{4}}{27}}+2/3 \right ) }+4\,{\frac{{2}^{3/4}\sqrt [8]{{b}^{2}} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{\sqrt{\pi }{x}^{3/2}b}}-{\frac{16\,{b}^{2}{2}^{3/4}\sin \left ( b{x}^{2} \right ) }{9\,\sqrt{\pi }}{x}^{{\frac{9}{2}}}\sqrt [8]{{b}^{2}}{\it LommelS1} \left ({\frac{7}{4}},{\frac{3}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{7}{4}}}}-4\,{\frac{{x}^{9/2}\sqrt [8]{{b}^{2}}{b}^{2}{2}^{3/4} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ){\it LommelS1} \left ( 3/4,1/2,b{x}^{2} \right ) }{\sqrt{\pi } \left ( b{x}^{2} \right ) ^{11/4}}} \right ){\frac{1}{\sqrt [8]{{b}^{2}}}}}-{\frac{\sin \left ( a \right ) \sqrt{\pi }\sqrt [4]{2}}{4} \left ({\frac{4\,{2}^{3/4}\sin \left ( b{x}^{2} \right ) }{5\,\sqrt{\pi }}\sqrt{x}\sqrt [4]{b}}-{\frac{16\,{2}^{3/4} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{5\,\sqrt{\pi }}\sqrt{x}\sqrt [4]{b}}-{\frac{4\,{2}^{3/4}\sin \left ( b{x}^{2} \right ) }{5\,\sqrt{\pi }}{x}^{{\frac{9}{2}}}{b}^{{\frac{9}{4}}}{\it LommelS1} \left ({\frac{3}{4}},{\frac{3}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{7}{4}}}}+{\frac{16\,{2}^{3/4} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{5\,\sqrt{\pi }}{x}^{{\frac{9}{2}}}{b}^{{\frac{9}{4}}}{\it LommelS1} \left ({\frac{7}{4}},{\frac{1}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{11}{4}}}} \right ){\frac{1}{\sqrt [4]{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*cos(a)*Pi^(1/2)*2^(1/4)/(b^2)^(1/8)*(6/Pi^(1/2)/x^(3/2)*2^(3/4)*(b^2)^(1/8)*(8/27*b^2*x^4+2/3)*sin(b*x^2)/
b+4/Pi^(1/2)/x^(3/2)*2^(3/4)*(b^2)^(1/8)/b*(cos(b*x^2)*b*x^2-sin(b*x^2))-16/9/Pi^(1/2)*x^(9/2)*(b^2)^(1/8)*b^2
*2^(3/4)/(b*x^2)^(7/4)*sin(b*x^2)*LommelS1(7/4,3/2,b*x^2)-4/Pi^(1/2)*x^(9/2)*(b^2)^(1/8)*b^2*2^(3/4)/(b*x^2)^(
11/4)*(cos(b*x^2)*b*x^2-sin(b*x^2))*LommelS1(3/4,1/2,b*x^2))-1/4*sin(a)*Pi^(1/2)*2^(1/4)/b^(1/4)*(4/5/Pi^(1/2)
*x^(1/2)*2^(3/4)*b^(1/4)*sin(b*x^2)-16/5/Pi^(1/2)*x^(1/2)*2^(3/4)*b^(1/4)*(cos(b*x^2)*b*x^2-sin(b*x^2))-4/5/Pi
^(1/2)*x^(9/2)*b^(9/4)*2^(3/4)/(b*x^2)^(7/4)*sin(b*x^2)*LommelS1(3/4,3/2,b*x^2)+16/5/Pi^(1/2)*x^(9/2)*b^(9/4)*
2^(3/4)/(b*x^2)^(11/4)*(cos(b*x^2)*b*x^2-sin(b*x^2))*LommelS1(7/4,1/2,b*x^2))

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Maxima [B]  time = 1.44696, size = 359, normalized size = 4.43 \begin{align*} -\frac{{\left ({\left ({\left (\Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \cos \left (\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) -{\left (i \, \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \sin \left (\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) -{\left (-i \, \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + i \, \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) -{\left ({\left (i \, \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \cos \left (\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) +{\left (i \, \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \sin \left (\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) -{\left (\Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )} \sqrt{x}}{8 \, \left (x^{2}{\left | b \right |}\right )^{\frac{1}{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(((gamma(1/4, I*b*x^2) + gamma(1/4, -I*b*x^2))*cos(1/8*pi + 1/4*arctan2(0, b)) + (gamma(1/4, I*b*x^2) + g
amma(1/4, -I*b*x^2))*cos(-1/8*pi + 1/4*arctan2(0, b)) - (I*gamma(1/4, I*b*x^2) - I*gamma(1/4, -I*b*x^2))*sin(1
/8*pi + 1/4*arctan2(0, b)) - (-I*gamma(1/4, I*b*x^2) + I*gamma(1/4, -I*b*x^2))*sin(-1/8*pi + 1/4*arctan2(0, b)
))*cos(a) - ((I*gamma(1/4, I*b*x^2) - I*gamma(1/4, -I*b*x^2))*cos(1/8*pi + 1/4*arctan2(0, b)) + (I*gamma(1/4,
I*b*x^2) - I*gamma(1/4, -I*b*x^2))*cos(-1/8*pi + 1/4*arctan2(0, b)) + (gamma(1/4, I*b*x^2) + gamma(1/4, -I*b*x
^2))*sin(1/8*pi + 1/4*arctan2(0, b)) - (gamma(1/4, I*b*x^2) + gamma(1/4, -I*b*x^2))*sin(-1/8*pi + 1/4*arctan2(
0, b)))*sin(a))*sqrt(x)/(x^2*abs(b))^(1/4)

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Fricas [A]  time = 1.71051, size = 132, normalized size = 1.63 \begin{align*} \frac{i \, \left (i \, b\right )^{\frac{3}{4}} e^{\left (-i \, a\right )} \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) - i \, \left (-i \, b\right )^{\frac{3}{4}} e^{\left (i \, a\right )} \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(I*(I*b)^(3/4)*e^(-I*a)*gamma(1/4, I*b*x^2) - I*(-I*b)^(3/4)*e^(I*a)*gamma(1/4, -I*b*x^2))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(a + b*x**2)/sqrt(x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x^{2} + a\right )}{\sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(b*x^2 + a)/sqrt(x), x)