Integrand size = 43, antiderivative size = 184 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\frac {(4 A+3 C) x \sqrt {\cos (c+d x)}}{8 \sqrt {b \cos (c+d x)}}+\frac {B \sqrt {\cos (c+d x)} \sin (c+d x)}{d \sqrt {b \cos (c+d x)}}+\frac {(4 A+3 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 d \sqrt {b \cos (c+d x)}}+\frac {C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{4 d \sqrt {b \cos (c+d x)}}-\frac {B \sqrt {\cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {b \cos (c+d x)}} \]
1/8*(4*A+3*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/d/(b*cos(d*x+c))^(1/2)+1/4*C*cos (d*x+c)^(7/2)*sin(d*x+c)/d/(b*cos(d*x+c))^(1/2)+1/8*(4*A+3*C)*x*cos(d*x+c) ^(1/2)/(b*cos(d*x+c))^(1/2)+B*sin(d*x+c)*cos(d*x+c)^(1/2)/d/(b*cos(d*x+c)) ^(1/2)-1/3*B*sin(d*x+c)^3*cos(d*x+c)^(1/2)/d/(b*cos(d*x+c))^(1/2)
Time = 0.78 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.50 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sqrt {\cos (c+d x)} (48 A c+36 c C+48 A d x+36 C d x+72 B \sin (c+d x)+24 (A+C) \sin (2 (c+d x))+8 B \sin (3 (c+d x))+3 C \sin (4 (c+d x)))}{96 d \sqrt {b \cos (c+d x)}} \]
Integrate[(Cos[c + d*x]^(5/2)*(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]]*(48*A*c + 36*c*C + 48*A*d*x + 36*C*d*x + 72*B*Sin[c + d*x] + 24*(A + C)*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d*Sqrt[b*Cos[c + d*x]])
Time = 0.51 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.60, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2031, 3042, 3502, 3042, 3227, 3042, 3113, 2009, 3115, 24}
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.
\(\displaystyle \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx\) |
\(\Big \downarrow \) 2031 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \int \cos ^2(c+d x) \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )dx}{\sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )dx}{\sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 3502 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \left (\frac {1}{4} \int \cos ^2(c+d x) (4 A+3 C+4 B \cos (c+d x))dx+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{\sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \left (\frac {1}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (4 A+3 C+4 B \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{\sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \left (\frac {1}{4} \left ((4 A+3 C) \int \cos ^2(c+d x)dx+4 B \int \cos ^3(c+d x)dx\right )+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{\sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \left (\frac {1}{4} \left ((4 A+3 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+4 B \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{\sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \left (\frac {1}{4} \left ((4 A+3 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {4 B \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{\sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \left (\frac {1}{4} \left ((4 A+3 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {4 B \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{\sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \left (\frac {1}{4} \left ((4 A+3 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {4 B \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{\sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \left (\frac {1}{4} \left ((4 A+3 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {4 B \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{\sqrt {b \cos (c+d x)}}\) |
(Sqrt[Cos[c + d*x]]*((C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + ((4*A + 3*C)* (x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) - (4*B*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d)/4))/Sqrt[b*Cos[c + d*x]]
3.4.15.3.1 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]
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Time = 9.50 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.62
method | result | size |
default | \(\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \left (6 C \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+8 B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+12 A \sin \left (d x +c \right ) \cos \left (d x +c \right )+9 C \cos \left (d x +c \right ) \sin \left (d x +c \right )+12 A \left (d x +c \right )+16 B \sin \left (d x +c \right )+9 C \left (d x +c \right )\right )}{24 d \sqrt {\cos \left (d x +c \right ) b}}\) | \(114\) |
parts | \(\frac {A \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )+d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{2 d \sqrt {\cos \left (d x +c \right ) b}}+\frac {B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{3 d \sqrt {\cos \left (d x +c \right ) b}}+\frac {C \left (2 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )+3 d x +3 c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{8 d \sqrt {\cos \left (d x +c \right ) b}}\) | \(146\) |
risch | \(\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) x \left (8 A +6 C \right )}{16 \sqrt {\cos \left (d x +c \right ) b}}+\frac {3 B \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{4 d \sqrt {\cos \left (d x +c \right ) b}}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) C \sin \left (4 d x +4 c \right )}{32 \sqrt {\cos \left (d x +c \right ) b}\, d}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) B \sin \left (3 d x +3 c \right )}{12 \sqrt {\cos \left (d x +c \right ) b}\, d}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \left (A +C \right ) \sin \left (2 d x +2 c \right )}{4 \sqrt {\cos \left (d x +c \right ) b}\, d}\) | \(161\) |
int(cos(d*x+c)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(cos(d*x+c)*b)^(1/2), x,method=_RETURNVERBOSE)
1/24/d*cos(d*x+c)^(1/2)*(6*C*cos(d*x+c)^3*sin(d*x+c)+8*B*sin(d*x+c)*cos(d* x+c)^2+12*A*sin(d*x+c)*cos(d*x+c)+9*C*cos(d*x+c)*sin(d*x+c)+12*A*(d*x+c)+1 6*B*sin(d*x+c)+9*C*(d*x+c))/(cos(d*x+c)*b)^(1/2)
Time = 0.33 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.53 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\left [-\frac {3 \, {\left (4 \, A + 3 \, C\right )} \sqrt {-b} \cos \left (d x + c\right ) \log \left (2 \, b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right ) - 2 \, {\left (6 \, C \cos \left (d x + c\right )^{3} + 8 \, B \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, A + 3 \, C\right )} \cos \left (d x + c\right ) + 16 \, B\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{48 \, b d \cos \left (d x + c\right )}, \frac {3 \, {\left (4 \, A + 3 \, C\right )} \sqrt {b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right ) \cos \left (d x + c\right ) + {\left (6 \, C \cos \left (d x + c\right )^{3} + 8 \, B \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, A + 3 \, C\right )} \cos \left (d x + c\right ) + 16 \, B\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{24 \, b d \cos \left (d x + c\right )}\right ] \]
integrate(cos(d*x+c)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(b*cos(d*x+c))^ (1/2),x, algorithm="fricas")
[-1/48*(3*(4*A + 3*C)*sqrt(-b)*cos(d*x + c)*log(2*b*cos(d*x + c)^2 + 2*sqr t(b*cos(d*x + c))*sqrt(-b)*sqrt(cos(d*x + c))*sin(d*x + c) - b) - 2*(6*C*c os(d*x + c)^3 + 8*B*cos(d*x + c)^2 + 3*(4*A + 3*C)*cos(d*x + c) + 16*B)*sq rt(b*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c))/(b*d*cos(d*x + c)), 1/ 24*(3*(4*A + 3*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) + (6*C*cos(d*x + c)^3 + 8*B*cos(d*x + c)^2 + 3*(4*A + 3*C)*cos(d*x + c) + 16*B)*sqrt(b*cos(d*x + c))*sqrt(cos(d* x + c))*sin(d*x + c))/(b*d*cos(d*x + c))]
Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\text {Timed out} \]
Time = 0.55 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.63 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A}{\sqrt {b}} + \frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )\right )} C}{\sqrt {b}} + \frac {8 \, B {\left (\sin \left (3 \, d x + 3 \, c\right ) + 9 \, \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )}}{\sqrt {b}}}{96 \, d} \]
integrate(cos(d*x+c)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(b*cos(d*x+c))^ (1/2),x, algorithm="maxima")
1/96*(24*(2*d*x + 2*c + sin(2*d*x + 2*c))*A/sqrt(b) + 3*(12*d*x + 12*c + s in(4*d*x + 4*c) + 8*sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))))* C/sqrt(b) + 8*B*(sin(3*d*x + 3*c) + 9*sin(1/3*arctan2(sin(3*d*x + 3*c), co s(3*d*x + 3*c))))/sqrt(b))/d
\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
integrate(cos(d*x+c)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(b*cos(d*x+c))^ (1/2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)^(5/2)/sqrt( b*cos(d*x + c)), x)
Time = 3.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (24\,A\,\sin \left (c+d\,x\right )+24\,C\,\sin \left (c+d\,x\right )+24\,A\,\sin \left (3\,c+3\,d\,x\right )+80\,B\,\sin \left (2\,c+2\,d\,x\right )+8\,B\,\sin \left (4\,c+4\,d\,x\right )+27\,C\,\sin \left (3\,c+3\,d\,x\right )+3\,C\,\sin \left (5\,c+5\,d\,x\right )+96\,A\,d\,x\,\cos \left (c+d\,x\right )+72\,C\,d\,x\,\cos \left (c+d\,x\right )\right )}{96\,b\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]