Integrand size = 10, antiderivative size = 153 \[ \int \cos ^3\left (a+b x^2\right ) \, dx=\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)}{4 \sqrt {b}} \]
1/24*cos(3*a)*FresnelC(x*b^(1/2)*6^(1/2)/Pi^(1/2))*6^(1/2)*Pi^(1/2)/b^(1/2 )-1/24*FresnelS(x*b^(1/2)*6^(1/2)/Pi^(1/2))*sin(3*a)*6^(1/2)*Pi^(1/2)/b^(1 /2)+3/8*cos(a)*FresnelC(x*b^(1/2)*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)/b^(1/ 2)-3/8*FresnelS(x*b^(1/2)*2^(1/2)/Pi^(1/2))*sin(a)*2^(1/2)*Pi^(1/2)/b^(1/2 )
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.76 \[ \int \cos ^3\left (a+b x^2\right ) \, dx=\frac {\sqrt {\frac {\pi }{6}} \left (3 \sqrt {3} \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )+\cos (3 a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )-3 \sqrt {3} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)-\operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)\right )}{4 \sqrt {b}} \]
(Sqrt[Pi/6]*(3*Sqrt[3]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x] + Cos[3*a]*Fr esnelC[Sqrt[b]*Sqrt[6/Pi]*x] - 3*Sqrt[3]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*Si n[a] - FresnelS[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a]))/(4*Sqrt[b])
Time = 0.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3839, 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.
\(\displaystyle \int \cos ^3\left (a+b x^2\right ) \, dx\) |
\(\Big \downarrow \) 3839 |
\(\displaystyle \int \left (\frac {3}{4} \cos \left (a+b x^2\right )+\frac {1}{4} \cos \left (3 a+3 b x^2\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}}-\frac {3 \sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} \sin (3 a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}}\) |
(3*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x])/(4*Sqrt[b]) + (Sqrt[P i/6]*Cos[3*a]*FresnelC[Sqrt[b]*Sqrt[6/Pi]*x])/(4*Sqrt[b]) - (3*Sqrt[Pi/2]* FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/(4*Sqrt[b]) - (Sqrt[Pi/6]*FresnelS[ Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a])/(4*Sqrt[b])
3.1.18.3.1 Defintions of rubi rules used
Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_), x_Sy mbol] :> Int[ExpandTrigReduce[(a + b*Cos[c + d*(e + f*x)^n])^p, x], x] /; F reeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 1] && IGtQ[n, 1]
Time = 0.43 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.66
method | result | size |
default | \(\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{8 \sqrt {b}}+\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (3 a \right ) \operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )-\sin \left (3 a \right ) \operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )\right )}{24 \sqrt {b}}\) | \(101\) |
risch | \(\frac {{\mathrm e}^{-3 i a} \sqrt {\pi }\, \sqrt {3}\, \operatorname {erf}\left (\sqrt {3}\, \sqrt {i b}\, x \right )}{48 \sqrt {i b}}+\frac {3 \,{\mathrm e}^{-i a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {i b}\, x \right )}{16 \sqrt {i b}}+\frac {{\mathrm e}^{3 i a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-3 i b}\, x \right )}{16 \sqrt {-3 i b}}+\frac {3 \,{\mathrm e}^{i a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-i b}\, x \right )}{16 \sqrt {-i b}}\) | \(108\) |
3/8*2^(1/2)*Pi^(1/2)/b^(1/2)*(cos(a)*FresnelC(x*b^(1/2)*2^(1/2)/Pi^(1/2))- sin(a)*FresnelS(x*b^(1/2)*2^(1/2)/Pi^(1/2)))+1/24*2^(1/2)*Pi^(1/2)*3^(1/2) /b^(1/2)*(cos(3*a)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*b^(1/2)*x)-sin(3*a)*F resnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*b^(1/2)*x))
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.79 \[ \int \cos ^3\left (a+b x^2\right ) \, dx=\frac {\sqrt {6} \pi \sqrt {\frac {b}{\pi }} \cos \left (3 \, a\right ) \operatorname {C}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) + 9 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) - \sqrt {6} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (3 \, a\right ) - 9 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right )}{24 \, b} \]
1/24*(sqrt(6)*pi*sqrt(b/pi)*cos(3*a)*fresnel_cos(sqrt(6)*x*sqrt(b/pi)) + 9 *sqrt(2)*pi*sqrt(b/pi)*cos(a)*fresnel_cos(sqrt(2)*x*sqrt(b/pi)) - sqrt(6)* pi*sqrt(b/pi)*fresnel_sin(sqrt(6)*x*sqrt(b/pi))*sin(3*a) - 9*sqrt(2)*pi*sq rt(b/pi)*fresnel_sin(sqrt(2)*x*sqrt(b/pi))*sin(a))/b
Time = 0.52 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.84 \[ \int \cos ^3\left (a+b x^2\right ) \, dx=\frac {3 \sqrt {2} \sqrt {\pi } \left (- \sin {\left (a \right )} S\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right ) + \cos {\left (a \right )} C\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right )\right ) \sqrt {\frac {1}{b}}}{8} + \frac {\sqrt {6} \sqrt {\pi } \left (- \sin {\left (3 a \right )} S\left (\frac {\sqrt {6} \sqrt {b} x}{\sqrt {\pi }}\right ) + \cos {\left (3 a \right )} C\left (\frac {\sqrt {6} \sqrt {b} x}{\sqrt {\pi }}\right )\right ) \sqrt {\frac {1}{b}}}{24} \]
3*sqrt(2)*sqrt(pi)*(-sin(a)*fresnels(sqrt(2)*sqrt(b)*x/sqrt(pi)) + cos(a)* fresnelc(sqrt(2)*sqrt(b)*x/sqrt(pi)))*sqrt(1/b)/8 + sqrt(6)*sqrt(pi)*(-sin (3*a)*fresnels(sqrt(6)*sqrt(b)*x/sqrt(pi)) + cos(3*a)*fresnelc(sqrt(6)*sqr t(b)*x/sqrt(pi)))*sqrt(1/b)/24
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.73 \[ \int \cos ^3\left (a+b x^2\right ) \, dx=-\frac {9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (3 \, a\right ) + \left (i + 1\right ) \, \sin \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3 i \, b} x\right ) + {\left (-\left (i + 1\right ) \, \cos \left (3 \, a\right ) - \left (i - 1\right ) \, \sin \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-3 i \, b} x\right )\right )} b^{\frac {3}{2}} - 9 \, \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i - 1\right ) \, \cos \left (a\right ) - \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {i \, b} x\right ) + {\left (\left (i + 1\right ) \, \cos \left (a\right ) + \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-i \, b} x\right )\right )} b^{\frac {3}{2}}}{96 \, b^{2}} \]
-1/96*(9^(1/4)*sqrt(2)*sqrt(pi)*(((I - 1)*cos(3*a) + (I + 1)*sin(3*a))*erf (sqrt(3*I*b)*x) + (-(I + 1)*cos(3*a) - (I - 1)*sin(3*a))*erf(sqrt(-3*I*b)* x))*b^(3/2) - 9*sqrt(2)*sqrt(pi)*((-(I - 1)*cos(a) - (I + 1)*sin(a))*erf(s qrt(I*b)*x) + ((I + 1)*cos(a) + (I - 1)*sin(a))*erf(sqrt(-I*b)*x))*b^(3/2) )/b^2
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.21 \[ \int \cos ^3\left (a+b x^2\right ) \, dx=\frac {i \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {6} \sqrt {b} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (3 i \, a\right )}}{48 \, \sqrt {b} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} + \frac {3 i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {2} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{16 \, {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {3 i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {2} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{16 \, {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {i \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {6} \sqrt {b} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-3 i \, a\right )}}{48 \, \sqrt {b} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]
1/48*I*sqrt(6)*sqrt(pi)*erf(-1/2*I*sqrt(6)*sqrt(b)*x*(I*b/abs(b) + 1))*e^( 3*I*a)/(sqrt(b)*(I*b/abs(b) + 1)) + 3/16*I*sqrt(2)*sqrt(pi)*erf(-1/2*I*sqr t(2)*x*(I*b/abs(b) + 1)*sqrt(abs(b)))*e^(I*a)/((I*b/abs(b) + 1)*sqrt(abs(b ))) - 3/16*I*sqrt(2)*sqrt(pi)*erf(1/2*I*sqrt(2)*x*(-I*b/abs(b) + 1)*sqrt(a bs(b)))*e^(-I*a)/((-I*b/abs(b) + 1)*sqrt(abs(b))) - 1/48*I*sqrt(6)*sqrt(pi )*erf(1/2*I*sqrt(6)*sqrt(b)*x*(-I*b/abs(b) + 1))*e^(-3*I*a)/(sqrt(b)*(-I*b /abs(b) + 1))
Timed out. \[ \int \cos ^3\left (a+b x^2\right ) \, dx=\int {\cos \left (b\,x^2+a\right )}^3 \,d x \]