Integrand size = 35, antiderivative size = 122 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{5/2}} \, dx=\frac {(4 A+3 C) x \sqrt {\cos (c+d x)}}{8 b^2 \sqrt {b \cos (c+d x)}}+\frac {(4 A+3 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 b^2 d \sqrt {b \cos (c+d x)}}+\frac {C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{4 b^2 d \sqrt {b \cos (c+d x)}} \]
1/8*(4*A+3*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/b^2/d/(b*cos(d*x+c))^(1/2)+1/4*C *cos(d*x+c)^(7/2)*sin(d*x+c)/b^2/d/(b*cos(d*x+c))^(1/2)+1/8*(4*A+3*C)*x*co s(d*x+c)^(1/2)/b^2/(b*cos(d*x+c))^(1/2)
Time = 0.95 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.57 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{5/2}} \, dx=\frac {\sqrt {\cos (c+d x)} (4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x)))}{32 b^2 d \sqrt {b \cos (c+d x)}} \]
(Sqrt[Cos[c + d*x]]*(4*(4*A + 3*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + C*Sin[4*(c + d*x)]))/(32*b^2*d*Sqrt[b*Cos[c + d*x]])
Time = 0.33 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.70, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2031, 3042, 3493, 3042, 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.
\(\displaystyle \int \frac {\cos ^{\frac {9}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 2031 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \int \cos ^2(c+d x) \left (C \cos ^2(c+d x)+A\right )dx}{b^2 \sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx}{b^2 \sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 3493 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \left (\frac {1}{4} (4 A+3 C) \int \cos ^2(c+d x)dx+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{b^2 \sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \left (\frac {1}{4} (4 A+3 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{b^2 \sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \left (\frac {1}{4} (4 A+3 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{b^2 \sqrt {b \cos (c+d x)}}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \left (\frac {1}{4} (4 A+3 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{b^2 \sqrt {b \cos (c+d x)}}\) |
(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^2*Sqrt[b*Cos[c + d*x]])
3.2.32.3.1 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[((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_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*( x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f *(m + 2))), x] + Simp[(A*(m + 2) + C*(m + 1))/(m + 2) Int[(b*Sin[e + f*x] )^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && !LtQ[m, -1]
Time = 7.73 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \left (2 C \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 A \sin \left (d x +c \right ) \cos \left (d x +c \right )+3 C \cos \left (d x +c \right ) \sin \left (d x +c \right )+4 A \left (d x +c \right )+3 C \left (d x +c \right )\right )}{8 b^{2} d \sqrt {\cos \left (d x +c \right ) b}}\) | \(91\) |
risch | \(\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) x \left (8 A +6 C \right )}{16 b^{2} \sqrt {\cos \left (d x +c \right ) b}}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) C \sin \left (4 d x +4 c \right )}{32 b^{2} \sqrt {\cos \left (d x +c \right ) b}\, d}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \left (A +C \right ) \sin \left (2 d x +2 c \right )}{4 b^{2} \sqrt {\cos \left (d x +c \right ) b}\, d}\) | \(107\) |
parts | \(\frac {A \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )+d x +c \right )}{2 d \,b^{2} \sqrt {\cos \left (d x +c \right ) b}}+\frac {C \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (2 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )+3 d x +3 c \right )}{8 d \,b^{2} \sqrt {\cos \left (d x +c \right ) b}}\) | \(112\) |
1/8/b^2/d*cos(d*x+c)^(1/2)*(2*C*cos(d*x+c)^3*sin(d*x+c)+4*A*sin(d*x+c)*cos (d*x+c)+3*C*cos(d*x+c)*sin(d*x+c)+4*A*(d*x+c)+3*C*(d*x+c))/(cos(d*x+c)*b)^ (1/2)
Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.70 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{5/2}} \, dx=\left [\frac {2 \, {\left (2 \, C \cos \left (d x + c\right )^{2} + 4 \, A + 3 \, C\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - {\left (4 \, A + 3 \, C\right )} \sqrt {-b} \log \left (2 \, b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{16 \, b^{3} d}, \frac {{\left (2 \, C \cos \left (d x + c\right )^{2} + 4 \, A + 3 \, C\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + {\left (4 \, A + 3 \, C\right )} \sqrt {b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right )}{8 \, b^{3} d}\right ] \]
[1/16*(2*(2*C*cos(d*x + c)^2 + 4*A + 3*C)*sqrt(b*cos(d*x + c))*sqrt(cos(d* x + c))*sin(d*x + c) - (4*A + 3*C)*sqrt(-b)*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))/(b^3*d), 1/8*((2*C*cos(d*x + c)^2 + 4*A + 3*C)*sqrt(b*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + (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))))/(b^3*d)]
Timed out. \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Time = 0.43 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.61 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{5/2}} \, dx=\frac {\frac {8 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A}{b^{\frac {5}{2}}} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )\right )} C}{b^{\frac {5}{2}}}}{32 \, d} \]
1/32*(8*(2*d*x + 2*c + sin(2*d*x + 2*c))*A/b^(5/2) + (12*d*x + 12*c + sin( 4*d*x + 4*c) + 8*sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))))*C/b ^(5/2))/d
\[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {9}{2}}}{\left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
Time = 2.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{5/2}} \, dx=\frac {\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (8\,A\,\sin \left (c+d\,x\right )+8\,C\,\sin \left (c+d\,x\right )+8\,A\,\sin \left (3\,c+3\,d\,x\right )+9\,C\,\sin \left (3\,c+3\,d\,x\right )+C\,\sin \left (5\,c+5\,d\,x\right )+32\,A\,d\,x\,\cos \left (c+d\,x\right )+24\,C\,d\,x\,\cos \left (c+d\,x\right )\right )}{32\,b^3\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]