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

Optimal. Leaf size=81 \[ -\frac{e^{i a} x^{3/2} \text{Gamma}\left (\frac{3}{4},-i b x^2\right )}{4 \left (-i b x^2\right )^{3/4}}-\frac{e^{-i a} x^{3/2} \text{Gamma}\left (\frac{3}{4},i b x^2\right )}{4 \left (i b x^2\right )^{3/4}} \]

[Out]

-(E^(I*a)*x^(3/2)*Gamma[3/4, (-I)*b*x^2])/(4*((-I)*b*x^2)^(3/4)) - (x^(3/2)*Gamma[3/4, I*b*x^2])/(4*E^(I*a)*(I
*b*x^2)^(3/4))

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Rubi [A]  time = 0.0629466, 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} \[ -\frac{e^{i a} x^{3/2} \text{Gamma}\left (\frac{3}{4},-i b x^2\right )}{4 \left (-i b x^2\right )^{3/4}}-\frac{e^{-i a} x^{3/2} \text{Gamma}\left (\frac{3}{4},i b x^2\right )}{4 \left (i b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*Cos[a + b*x^2],x]

[Out]

-(E^(I*a)*x^(3/2)*Gamma[3/4, (-I)*b*x^2])/(4*((-I)*b*x^2)^(3/4)) - (x^(3/2)*Gamma[3/4, I*b*x^2])/(4*E^(I*a)*(I
*b*x^2)^(3/4))

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 \sqrt{x} \cos \left (a+b x^2\right ) \, dx &=\frac{1}{2} \int e^{-i a-i b x^2} \sqrt{x} \, dx+\frac{1}{2} \int e^{i a+i b x^2} \sqrt{x} \, dx\\ &=-\frac{e^{i a} x^{3/2} \Gamma \left (\frac{3}{4},-i b x^2\right )}{4 \left (-i b x^2\right )^{3/4}}-\frac{e^{-i a} x^{3/2} \Gamma \left (\frac{3}{4},i b x^2\right )}{4 \left (i b x^2\right )^{3/4}}\\ \end{align*}

Mathematica [A]  time = 0.0898811, size = 89, normalized size = 1.1 \[ -\frac{x^{3/2} \left (\left (-i b x^2\right )^{3/4} (\cos (a)-i \sin (a)) \text{Gamma}\left (\frac{3}{4},i b x^2\right )+\left (i b x^2\right )^{3/4} (\cos (a)+i \sin (a)) \text{Gamma}\left (\frac{3}{4},-i b x^2\right )\right )}{4 \left (b^2 x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*Cos[a + b*x^2],x]

[Out]

-(x^(3/2)*(((-I)*b*x^2)^(3/4)*Gamma[3/4, I*b*x^2]*(Cos[a] - I*Sin[a]) + (I*b*x^2)^(3/4)*Gamma[3/4, (-I)*b*x^2]
*(Cos[a] + I*Sin[a])))/(4*(b^2*x^4)^(3/4))

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Maple [C]  time = 0.066, size = 290, normalized size = 3.6 \begin{align*}{\frac{{2}^{{\frac{3}{4}}}\cos \left ( a \right ) \sqrt{\pi }}{4} \left ({\frac{4\,\sqrt [4]{2}\sin \left ( b{x}^{2} \right ) }{3\,\sqrt{\pi }b} \left ({b}^{2} \right ) ^{{\frac{3}{8}}}{\frac{1}{\sqrt{x}}}}+{\frac{4\,\sqrt [4]{2} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{3\,\sqrt{\pi }b} \left ({b}^{2} \right ) ^{{\frac{3}{8}}}{\frac{1}{\sqrt{x}}}}-{\frac{\sqrt [4]{2}b\sin \left ( b{x}^{2} \right ) }{3\,\sqrt{\pi }}{x}^{{\frac{7}{2}}} \left ({b}^{2} \right ) ^{{\frac{3}{8}}}{\it LommelS1} \left ({\frac{1}{4}},{\frac{3}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{5}{4}}}}-{\frac{4\,\sqrt [4]{2}b \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{3\,\sqrt{\pi }}{x}^{{\frac{7}{2}}} \left ({b}^{2} \right ) ^{{\frac{3}{8}}}{\it LommelS1} \left ({\frac{5}{4}},{\frac{1}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{9}{4}}}} \right ) \left ({b}^{2} \right ) ^{-{\frac{3}{8}}}}-{\frac{{2}^{{\frac{3}{4}}}\sin \left ( a \right ) \sqrt{\pi }}{4} \left ({\frac{4\,\sqrt [4]{2}\sin \left ( b{x}^{2} \right ) }{7\,\sqrt{\pi }}{x}^{{\frac{3}{2}}}{b}^{{\frac{3}{4}}}}-{\frac{4\,\sqrt [4]{2}\sin \left ( b{x}^{2} \right ) }{7\,\sqrt{\pi }}{x}^{{\frac{7}{2}}}{b}^{{\frac{7}{4}}}{\it LommelS1} \left ({\frac{5}{4}},{\frac{3}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{5}{4}}}}-{\frac{\sqrt [4]{2} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{\sqrt{\pi }}{x}^{{\frac{7}{2}}}{b}^{{\frac{7}{4}}}{\it LommelS1} \left ({\frac{1}{4}},{\frac{1}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{9}{4}}}} \right ){b}^{-{\frac{3}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.6082, size = 365, normalized size = 4.51 \begin{align*} -\frac{\left (x^{2}{\left | b \right |}\right )^{\frac{1}{4}}{\left ({\left ({\left (\Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \cos \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) -{\left (i \, \Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \sin \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) -{\left (-i \, \Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) + i \, \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) -{\left ({\left (i \, \Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \cos \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) +{\left (i \, \Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \sin \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) -{\left (\Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )}}{8 \, \sqrt{x}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(x^2*abs(b))^(1/4)*(((gamma(3/4, I*b*x^2) + gamma(3/4, -I*b*x^2))*cos(3/8*pi + 3/4*arctan2(0, b)) + (gamm
a(3/4, I*b*x^2) + gamma(3/4, -I*b*x^2))*cos(-3/8*pi + 3/4*arctan2(0, b)) - (I*gamma(3/4, I*b*x^2) - I*gamma(3/
4, -I*b*x^2))*sin(3/8*pi + 3/4*arctan2(0, b)) - (-I*gamma(3/4, I*b*x^2) + I*gamma(3/4, -I*b*x^2))*sin(-3/8*pi
+ 3/4*arctan2(0, b)))*cos(a) - ((I*gamma(3/4, I*b*x^2) - I*gamma(3/4, -I*b*x^2))*cos(3/8*pi + 3/4*arctan2(0, b
)) + (I*gamma(3/4, I*b*x^2) - I*gamma(3/4, -I*b*x^2))*cos(-3/8*pi + 3/4*arctan2(0, b)) + (gamma(3/4, I*b*x^2)
+ gamma(3/4, -I*b*x^2))*sin(3/8*pi + 3/4*arctan2(0, b)) - (gamma(3/4, I*b*x^2) + gamma(3/4, -I*b*x^2))*sin(-3/
8*pi + 3/4*arctan2(0, b)))*sin(a))/(sqrt(x)*abs(b))

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Fricas [A]  time = 1.7082, size = 132, normalized size = 1.63 \begin{align*} \frac{i \, \left (i \, b\right )^{\frac{1}{4}} e^{\left (-i \, a\right )} \Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) - i \, \left (-i \, b\right )^{\frac{1}{4}} e^{\left (i \, a\right )} \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x)*cos(a + b*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x)*cos(b*x^2 + a), x)