Integrand size = 43, antiderivative size = 123 \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\frac {A x \sqrt {\cos (c+d x)}}{\sqrt {b \cos (c+d x)}}+\frac {C x \sqrt {\cos (c+d x)}}{2 \sqrt {b \cos (c+d x)}}+\frac {B \sqrt {\cos (c+d x)} \sin (c+d x)}{d \sqrt {b \cos (c+d x)}}+\frac {C \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt {b \cos (c+d x)}} \]
1/2*C*cos(d*x+c)^(3/2)*sin(d*x+c)/d/(b*cos(d*x+c))^(1/2)+A*x*cos(d*x+c)^(1 /2)/(b*cos(d*x+c))^(1/2)+1/2*C*x*cos(d*x+c)^(1/2)/(b*cos(d*x+c))^(1/2)+B*s in(d*x+c)*cos(d*x+c)^(1/2)/d/(b*cos(d*x+c))^(1/2)
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.50 \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sqrt {\cos (c+d x)} (2 (2 A+C) (c+d x)+4 B \sin (c+d x)+C \sin (2 (c+d x)))}{4 d \sqrt {b \cos (c+d x)}} \]
Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqr t[b*Cos[c + d*x]],x]
(Sqrt[Cos[c + d*x]]*(2*(2*A + C)*(c + d*x) + 4*B*Sin[c + d*x] + C*Sin[2*(c + d*x)]))/(4*d*Sqrt[b*Cos[c + d*x]])
Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.52, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {2031, 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.
\(\displaystyle \int \frac {\sqrt {\cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx\) |
\(\Big \downarrow \) 2031 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \int \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )dx}{\sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \left (A x+\frac {B \sin (c+d x)}{d}+\frac {C \sin (c+d x) \cos (c+d x)}{2 d}+\frac {C x}{2}\right )}{\sqrt {b \cos (c+d x)}}\) |
(Sqrt[Cos[c + d*x]]*(A*x + (C*x)/2 + (B*Sin[c + d*x])/d + (C*Cos[c + d*x]* Sin[c + d*x])/(2*d)))/Sqrt[b*Cos[c + d*x]]
3.4.17.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[a^(m + 1/ 2)*b^(n - 1/2)*(Sqrt[b*v]/Sqrt[a*v]) Int[v^(m + n)*Fx, x], x] /; FreeQ[{a , b, m}, x] && !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]
Time = 10.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.51
method | result | size |
default | \(\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \left (C \cos \left (d x +c \right ) \sin \left (d x +c \right )+2 A \left (d x +c \right )+2 B \sin \left (d x +c \right )+C \left (d x +c \right )\right )}{2 d \sqrt {\cos \left (d x +c \right ) b}}\) | \(63\) |
risch | \(\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) x \left (4 A +2 C \right )}{4 \sqrt {\cos \left (d x +c \right ) b}}+\frac {B \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{d \sqrt {\cos \left (d x +c \right ) b}}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) C \sin \left (2 d x +2 c \right )}{4 \sqrt {\cos \left (d x +c \right ) b}\, d}\) | \(92\) |
parts | \(\frac {A \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (d x +c \right )}{d \sqrt {\cos \left (d x +c \right ) b}}+\frac {B \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{d \sqrt {\cos \left (d x +c \right ) b}}+\frac {C \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )+d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{2 d \sqrt {\cos \left (d x +c \right ) b}}\) | \(101\) |
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2)/(cos(d*x+c)*b)^(1/2), x,method=_RETURNVERBOSE)
1/2/d*cos(d*x+c)^(1/2)*(C*cos(d*x+c)*sin(d*x+c)+2*A*(d*x+c)+2*B*sin(d*x+c) +C*(d*x+c))/(cos(d*x+c)*b)^(1/2)
Time = 0.30 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\left [-\frac {{\left (2 \, A + C\right )} \sqrt {-b} \cos \left (d x + c\right ) \log \left (2 \, b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right ) - 2 \, {\left (C \cos \left (d x + c\right ) + 2 \, B\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{4 \, b d \cos \left (d x + c\right )}, \frac {{\left (2 \, A + C\right )} \sqrt {b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right ) \cos \left (d x + c\right ) + {\left (C \cos \left (d x + c\right ) + 2 \, B\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, b d \cos \left (d x + c\right )}\right ] \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2)/(b*cos(d*x+c))^ (1/2),x, algorithm="fricas")
[-1/4*((2*A + C)*sqrt(-b)*cos(d*x + c)*log(2*b*cos(d*x + c)^2 + 2*sqrt(b*c os(d*x + c))*sqrt(-b)*sqrt(cos(d*x + c))*sin(d*x + c) - b) - 2*(C*cos(d*x + c) + 2*B)*sqrt(b*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c))/(b*d*cos (d*x + c)), 1/2*((2*A + C)*sqrt(b)*arctan(sqrt(b*cos(d*x + c))*sin(d*x + c )/(sqrt(b)*cos(d*x + c)^(3/2)))*cos(d*x + c) + (C*cos(d*x + c) + 2*B)*sqrt (b*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c))/(b*d*cos(d*x + c))]
Time = 15.79 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\begin {cases} \frac {A x \sqrt {\cos {\left (c + d x \right )}}}{\sqrt {b \cos {\left (c + d x \right )}}} + \frac {B \sin {\left (c + d x \right )} \sqrt {\cos {\left (c + d x \right )}}}{d \sqrt {b \cos {\left (c + d x \right )}}} + \frac {C x \sin ^{2}{\left (c + d x \right )} \sqrt {\cos {\left (c + d x \right )}}}{2 \sqrt {b \cos {\left (c + d x \right )}}} + \frac {C x \cos ^{\frac {5}{2}}{\left (c + d x \right )}}{2 \sqrt {b \cos {\left (c + d x \right )}}} + \frac {C \sin {\left (c + d x \right )} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}{2 d \sqrt {b \cos {\left (c + d x \right )}}} & \text {for}\: d \neq 0 \\\frac {x \left (A + B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \sqrt {\cos {\left (c \right )}}}{\sqrt {b \cos {\left (c \right )}}} & \text {otherwise} \end {cases} \]
Piecewise((A*x*sqrt(cos(c + d*x))/sqrt(b*cos(c + d*x)) + B*sin(c + d*x)*sq rt(cos(c + d*x))/(d*sqrt(b*cos(c + d*x))) + C*x*sin(c + d*x)**2*sqrt(cos(c + d*x))/(2*sqrt(b*cos(c + d*x))) + C*x*cos(c + d*x)**(5/2)/(2*sqrt(b*cos( c + d*x))) + C*sin(c + d*x)*cos(c + d*x)**(3/2)/(2*d*sqrt(b*cos(c + d*x))) , Ne(d, 0)), (x*(A + B*cos(c) + C*cos(c)**2)*sqrt(cos(c))/sqrt(b*cos(c)), True))
Time = 0.47 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C}{\sqrt {b}} + \frac {8 \, A \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{\sqrt {b}} + \frac {4 \, B \sin \left (d x + c\right )}{\sqrt {b}}}{4 \, d} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2)/(b*cos(d*x+c))^ (1/2),x, algorithm="maxima")
1/4*((2*d*x + 2*c + sin(2*d*x + 2*c))*C/sqrt(b) + 8*A*arctan(sin(d*x + c)/ (cos(d*x + c) + 1))/sqrt(b) + 4*B*sin(d*x + c)/sqrt(b))/d
\[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {\cos \left (d x + c\right )}}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2)/(b*cos(d*x+c))^ (1/2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sqrt(cos(d*x + c))/sqrt( b*cos(d*x + c)), x)
Time = 1.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (C\,\sin \left (c+d\,x\right )+4\,B\,\sin \left (2\,c+2\,d\,x\right )+C\,\sin \left (3\,c+3\,d\,x\right )+8\,A\,d\,x\,\cos \left (c+d\,x\right )+4\,C\,d\,x\,\cos \left (c+d\,x\right )\right )}{4\,b\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]