Integrand size = 18, antiderivative size = 158 \[ \int \sqrt {c+d x} \cos ^2(a+b x) \, dx=\frac {(c+d x)^{3/2}}{3 d}-\frac {\sqrt {d} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{8 b^{3/2}}-\frac {\sqrt {d} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{8 b^{3/2}}+\frac {\sqrt {c+d x} \sin (2 a+2 b x)}{4 b} \]
1/3*(d*x+c)^(3/2)/d-1/8*cos(2*a-2*b*c/d)*FresnelS(2*b^(1/2)*(d*x+c)^(1/2)/ d^(1/2)/Pi^(1/2))*d^(1/2)*Pi^(1/2)/b^(3/2)-1/8*FresnelC(2*b^(1/2)*(d*x+c)^ (1/2)/d^(1/2)/Pi^(1/2))*sin(2*a-2*b*c/d)*d^(1/2)*Pi^(1/2)/b^(3/2)+1/4*sin( 2*b*x+2*a)*(d*x+c)^(1/2)/b
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.95 \[ \int \sqrt {c+d x} \cos ^2(a+b x) \, dx=\frac {16 (c+d x)^2+\frac {3 \sqrt {2} d^2 e^{2 i \left (a-\frac {b c}{d}\right )} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {2 i b (c+d x)}{d}\right )}{b^2}+\frac {3 \sqrt {2} d^2 e^{-2 i \left (a-\frac {b c}{d}\right )} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},\frac {2 i b (c+d x)}{d}\right )}{b^2}}{48 d \sqrt {c+d x}} \]
(16*(c + d*x)^2 + (3*Sqrt[2]*d^2*E^((2*I)*(a - (b*c)/d))*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[3/2, ((-2*I)*b*(c + d*x))/d])/b^2 + (3*Sqrt[2]*d^2*Sqrt[(I *b*(c + d*x))/d]*Gamma[3/2, ((2*I)*b*(c + d*x))/d])/(b^2*E^((2*I)*(a - (b* c)/d))))/(48*d*Sqrt[c + d*x])
Time = 0.47 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {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.
\(\displaystyle \int \sqrt {c+d x} \cos ^2(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {c+d x} \sin \left (a+b x+\frac {\pi }{2}\right )^2dx\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \int \left (\frac {1}{2} \sqrt {c+d x} \cos (2 a+2 b x)+\frac {1}{2} \sqrt {c+d x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt {\pi } \sqrt {d} \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{8 b^{3/2}}-\frac {\sqrt {\pi } \sqrt {d} \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{8 b^{3/2}}+\frac {\sqrt {c+d x} \sin (2 a+2 b x)}{4 b}+\frac {(c+d x)^{3/2}}{3 d}\) |
(c + d*x)^(3/2)/(3*d) - (Sqrt[d]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2 *Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(8*b^(3/2)) - (Sqrt[d]*Sqrt[P i]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c )/d])/(8*b^(3/2)) + (Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(4*b)
3.1.50.3.1 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]))
Time = 1.04 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 b c}{d}\right )}{4 b}-\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 b c}{d}\right ) \operatorname {S}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 a d -2 b c}{d}\right ) \operatorname {C}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}}{d}\) | \(150\) |
default | \(\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 b c}{d}\right )}{4 b}-\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 b c}{d}\right ) \operatorname {S}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 a d -2 b c}{d}\right ) \operatorname {C}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}}{d}\) | \(150\) |
2/d*(1/6*(d*x+c)^(3/2)+1/8/b*d*(d*x+c)^(1/2)*sin(2*b*(d*x+c)/d+2*(a*d-b*c) /d)-1/16/b*d*Pi^(1/2)/(b/d)^(1/2)*(cos(2*(a*d-b*c)/d)*FresnelS(2/Pi^(1/2)/ (b/d)^(1/2)*b*(d*x+c)^(1/2)/d)+sin(2*(a*d-b*c)/d)*FresnelC(2/Pi^(1/2)/(b/d )^(1/2)*b*(d*x+c)^(1/2)/d)))
Time = 0.31 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.94 \[ \int \sqrt {c+d x} \cos ^2(a+b x) \, dx=-\frac {3 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 3 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 4 \, {\left (2 \, b^{2} d x + 3 \, b d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 2 \, b^{2} c\right )} \sqrt {d x + c}}{24 \, b^{2} d} \]
-1/24*(3*pi*d^2*sqrt(b/(pi*d))*cos(-2*(b*c - a*d)/d)*fresnel_sin(2*sqrt(d* x + c)*sqrt(b/(pi*d))) + 3*pi*d^2*sqrt(b/(pi*d))*fresnel_cos(2*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-2*(b*c - a*d)/d) - 4*(2*b^2*d*x + 3*b*d*cos(b*x + a)*sin(b*x + a) + 2*b^2*c)*sqrt(d*x + c))/(b^2*d)
\[ \int \sqrt {c+d x} \cos ^2(a+b x) \, dx=\int \sqrt {c + d x} \cos ^{2}{\left (a + b x \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.45 \[ \int \sqrt {c+d x} \cos ^2(a+b x) \, dx=\frac {\sqrt {2} {\left (\frac {32 \, \sqrt {2} {\left (d x + c\right )}^{\frac {3}{2}} b^{2}}{d} + 24 \, \sqrt {2} \sqrt {d x + c} b \sin \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 3 \, {\left (\left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {2 i \, b}{d}}\right ) - 3 \, {\left (-\left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {2 i \, b}{d}}\right )\right )}}{192 \, b^{2}} \]
1/192*sqrt(2)*(32*sqrt(2)*(d*x + c)^(3/2)*b^2/d + 24*sqrt(2)*sqrt(d*x + c) *b*sin(2*((d*x + c)*b - b*c + a*d)/d) - 3*((I + 1)*4^(1/4)*sqrt(pi)*d*(b^2 /d^2)^(1/4)*cos(-2*(b*c - a*d)/d) - (I - 1)*4^(1/4)*sqrt(pi)*d*(b^2/d^2)^( 1/4)*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(2*I*b/d)) - 3*(-(I - 1) *4^(1/4)*sqrt(pi)*d*(b^2/d^2)^(1/4)*cos(-2*(b*c - a*d)/d) + (I + 1)*4^(1/4 )*sqrt(pi)*d*(b^2/d^2)^(1/4)*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt (-2*I*b/d)))/b^2
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 436, normalized size of antiderivative = 2.76 \[ \int \sqrt {c+d x} \cos ^2(a+b x) \, dx=-\frac {12 \, {\left (-\frac {i \, \sqrt {\pi } d \operatorname {erf}\left (-\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (i \, b c - i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} + \frac {i \, \sqrt {\pi } d \operatorname {erf}\left (\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, b c + i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} - 4 \, \sqrt {d x + c}\right )} c + \frac {3 i \, \sqrt {\pi } {\left (4 \, b c - i \, d\right )} d \operatorname {erf}\left (-\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (i \, b c - i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} - \frac {3 i \, \sqrt {\pi } {\left (4 \, b c + i \, d\right )} d \operatorname {erf}\left (\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, b c + i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} - 16 \, {\left (d x + c\right )}^{\frac {3}{2}} + 48 \, \sqrt {d x + c} c - \frac {6 i \, \sqrt {d x + c} d e^{\left (-\frac {2 \, {\left (i \, {\left (d x + c\right )} b - i \, b c + i \, a d\right )}}{d}\right )}}{b} + \frac {6 i \, \sqrt {d x + c} d e^{\left (-\frac {2 \, {\left (-i \, {\left (d x + c\right )} b + i \, b c - i \, a d\right )}}{d}\right )}}{b}}{48 \, d} \]
-1/48*(12*(-I*sqrt(pi)*d*erf(-I*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^ 2) + 1)/d)*e^(-2*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)) + I*sqrt(pi)*d*erf(I*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d) *e^(-2*(-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)) - 4*sqrt (d*x + c))*c + 3*I*sqrt(pi)*(4*b*c - I*d)*d*erf(-I*sqrt(b*d)*sqrt(d*x + c) *(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-2*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/s qrt(b^2*d^2) + 1)*b) - 3*I*sqrt(pi)*(4*b*c + I*d)*d*erf(I*sqrt(b*d)*sqrt(d *x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-2*(-I*b*c + I*a*d)/d)/(sqrt(b*d) *(-I*b*d/sqrt(b^2*d^2) + 1)*b) - 16*(d*x + c)^(3/2) + 48*sqrt(d*x + c)*c - 6*I*sqrt(d*x + c)*d*e^(-2*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b + 6*I*sqrt (d*x + c)*d*e^(-2*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b)/d
Timed out. \[ \int \sqrt {c+d x} \cos ^2(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^2\,\sqrt {c+d\,x} \,d x \]