Integrand size = 18, antiderivative size = 102 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}} \, dx=\sqrt {x}-\frac {3 \sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{8 b^{3/2}}-\frac {3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)}{8 b^{3/2}}+\frac {3 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{4 b} \]
3/4*x^(1/6)*sin(2*a+2*b*x^(1/3))/b-3/8*cos(2*a)*FresnelS(2*x^(1/6)*b^(1/2) /Pi^(1/2))*Pi^(1/2)/b^(3/2)-3/8*FresnelC(2*x^(1/6)*b^(1/2)/Pi^(1/2))*sin(2 *a)*Pi^(1/2)/b^(3/2)+x^(1/2)
Time = 0.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.01 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}} \, dx=\frac {-3 \sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)+2 \sqrt {b} \sqrt [6]{x} \left (4 b \sqrt [3]{x}+3 \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )\right )}{8 b^{3/2}} \]
(-3*Sqrt[Pi]*Cos[2*a]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]] - 3*Sqrt[Pi]* FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a] + 2*Sqrt[b]*x^(1/6)*(4*b*x ^(1/3) + 3*Sin[2*(a + b*x^(1/3))]))/(8*b^(3/2))
Time = 0.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3897, 3042, 3793, 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.
\(\displaystyle \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}} \, dx\) |
\(\Big \downarrow \) 3897 |
\(\displaystyle 3 \int \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 \int \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )^2d\sqrt [3]{x}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle 3 \int \left (\frac {1}{2} \sqrt [6]{x} \cos \left (2 a+2 b \sqrt [3]{x}\right )+\frac {\sqrt [6]{x}}{2}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {\sqrt {\pi } \sin (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{8 b^{3/2}}-\frac {\sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{8 b^{3/2}}+\frac {\sqrt [6]{x} \sin \left (2 a+2 b \sqrt [3]{x}\right )}{4 b}+\frac {\sqrt {x}}{3}\right )\) |
3*(Sqrt[x]/3 - (Sqrt[Pi]*Cos[2*a]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]])/ (8*b^(3/2)) - (Sqrt[Pi]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a])/( 8*b^(3/2)) + (x^(1/6)*Sin[2*a + 2*b*x^(1/3)])/(4*b))
3.1.57.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol ] :> Module[{k = Denominator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*Cos[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]
Time = 0.52 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\sqrt {x}+\frac {3 x^{\frac {1}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {3 \sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )+\sin \left (2 a \right ) \operatorname {C}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )}{8 b^{\frac {3}{2}}}\) | \(67\) |
default | \(\sqrt {x}+\frac {3 x^{\frac {1}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {3 \sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )+\sin \left (2 a \right ) \operatorname {C}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )}{8 b^{\frac {3}{2}}}\) | \(67\) |
x^(1/2)+3/4*x^(1/6)*sin(2*a+2*b*x^(1/3))/b-3/8/b^(3/2)*Pi^(1/2)*(cos(2*a)* FresnelS(2*x^(1/6)*b^(1/2)/Pi^(1/2))+sin(2*a)*FresnelC(2*x^(1/6)*b^(1/2)/P i^(1/2)))
Time = 0.30 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.88 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}} \, dx=-\frac {3 \, \pi \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) + 3 \, \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) - 12 \, b x^{\frac {1}{6}} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right ) - 8 \, b^{2} \sqrt {x}}{8 \, b^{2}} \]
-1/8*(3*pi*sqrt(b/pi)*cos(2*a)*fresnel_sin(2*x^(1/6)*sqrt(b/pi)) + 3*pi*sq rt(b/pi)*fresnel_cos(2*x^(1/6)*sqrt(b/pi))*sin(2*a) - 12*b*x^(1/6)*cos(b*x ^(1/3) + a)*sin(b*x^(1/3) + a) - 8*b^2*sqrt(x))/b^2
\[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}} \, dx=\int \frac {\cos ^{2}{\left (a + b \sqrt [3]{x} \right )}}{\sqrt {x}}\, dx \]
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}} \, dx=-\frac {3 \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i + 1\right ) \, \cos \left (2 \, a\right ) - \left (i - 1\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {2 i \, b} x^{\frac {1}{6}}\right ) + {\left (-\left (i - 1\right ) \, \cos \left (2 \, a\right ) + \left (i + 1\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-2 i \, b} x^{\frac {1}{6}}\right )\right )} b^{\frac {3}{2}} - 64 \, b^{3} \sqrt {x} - 48 \, b^{2} x^{\frac {1}{6}} \sin \left (2 \, b x^{\frac {1}{3}} + 2 \, a\right )}{64 \, b^{3}} \]
-1/64*(3*4^(1/4)*sqrt(2)*sqrt(pi)*(((I + 1)*cos(2*a) - (I - 1)*sin(2*a))*e rf(sqrt(2*I*b)*x^(1/6)) + (-(I - 1)*cos(2*a) + (I + 1)*sin(2*a))*erf(sqrt( -2*I*b)*x^(1/6)))*b^(3/2) - 64*b^3*sqrt(x) - 48*b^2*x^(1/6)*sin(2*b*x^(1/3 ) + 2*a))/b^3
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.22 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}} \, dx=\sqrt {x} - \frac {3 i \, x^{\frac {1}{6}} e^{\left (2 i \, b x^{\frac {1}{3}} + 2 i \, a\right )}}{8 \, b} + \frac {3 i \, x^{\frac {1}{6}} e^{\left (-2 i \, b x^{\frac {1}{3}} - 2 i \, a\right )}}{8 \, b} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-i \, \sqrt {b} x^{\frac {1}{6}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (2 i \, a\right )}}{16 \, b^{\frac {3}{2}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {b} x^{\frac {1}{6}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-2 i \, a\right )}}{16 \, b^{\frac {3}{2}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]
sqrt(x) - 3/8*I*x^(1/6)*e^(2*I*b*x^(1/3) + 2*I*a)/b + 3/8*I*x^(1/6)*e^(-2* I*b*x^(1/3) - 2*I*a)/b - 3/16*sqrt(pi)*erf(-I*sqrt(b)*x^(1/6)*(I*b/abs(b) + 1))*e^(2*I*a)/(b^(3/2)*(I*b/abs(b) + 1)) - 3/16*sqrt(pi)*erf(I*sqrt(b)*x ^(1/6)*(-I*b/abs(b) + 1))*e^(-2*I*a)/(b^(3/2)*(-I*b/abs(b) + 1))
Timed out. \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}} \, dx=\int \frac {{\cos \left (a+b\,x^{1/3}\right )}^2}{\sqrt {x}} \,d x \]