\(\int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx\) [712]

Optimal result

Integrand size = 35, antiderivative size = 270 \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\frac {2 b \left (5 A b^2+a^2 (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 d}+\frac {2 \left (7 A b^2+a^2 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^3 d}+\frac {2 b^2 \left (A b^2+a^2 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a^4 (a+b) d}+\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 A b \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 A b^2+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 b \left (5 A b^2+a^2 (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt {\cos (c+d x)}} \] Output:

2/5*b*(5*A*b^2+a^2*(3*A+5*C))*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/a^4/d+ 
2/21*(7*A*b^2+a^2*(5*A+7*C))*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/a^3/d+ 
2*b^2*(A*b^2+C*a^2)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))/a^4/( 
a+b)/d+2/7*A*sin(d*x+c)/a/d/cos(d*x+c)^(7/2)-2/5*A*b*sin(d*x+c)/a^2/d/cos( 
d*x+c)^(5/2)+2/21*(7*A*b^2+a^2*(5*A+7*C))*sin(d*x+c)/a^3/d/cos(d*x+c)^(3/2 
)-2/5*b*(5*A*b^2+a^2*(3*A+5*C))*sin(d*x+c)/a^4/d/cos(d*x+c)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 4.88 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.25 \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\frac {\frac {\left (315 A b^4+10 a^4 (5 A+7 C)+7 a^2 b^2 (19 A+45 C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {8 \left (35 a A b^2+a^3 (22 A+35 C)\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a+b}+\frac {21 \left (5 A b^2+a^2 (3 A+5 C)\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a \sqrt {\sin ^2(c+d x)}}+\frac {-42 b \left (a^2 A+\left (5 A b^2+a^2 (3 A+5 C)\right ) \cos ^2(c+d x)\right ) \sin (c+d x)+5 \left (\left (7 a A b^2+a^3 (5 A+7 C)\right ) \sin (2 (c+d x))+6 a^3 A \tan (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)}}{105 a^4 d} \] Input:

Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*(a + b*Cos[c + d*x])) 
,x]
 

Output:

(((315*A*b^4 + 10*a^4*(5*A + 7*C) + 7*a^2*b^2*(19*A + 45*C))*EllipticPi[(2 
*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(35*a*A*b^2 + a^3*(22*A + 35*C) 
)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d* 
x)/2, 2]))/(a + b) + (21*(5*A*b^2 + a^2*(3*A + 5*C))*(-2*a*b*EllipticE[Arc 
Sin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d 
*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], 
 -1])*Sin[c + d*x])/(a*Sqrt[Sin[c + d*x]^2]) + (-42*b*(a^2*A + (5*A*b^2 + 
a^2*(3*A + 5*C))*Cos[c + d*x]^2)*Sin[c + d*x] + 5*((7*a*A*b^2 + a^3*(5*A + 
 7*C))*Sin[2*(c + d*x)] + 6*a^3*A*Tan[c + d*x]))/Cos[c + d*x]^(5/2))/(105* 
a^4*d)
 

Rubi [A] (verified)

Time = 2.72 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.09, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3535, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 27, 3042, 3119, 3481, 3042, 3120, 3284}

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.

\(\displaystyle \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx\)

\(\Big \downarrow \) 3535

\(\displaystyle \frac {2 \int -\frac {-5 A b \cos ^2(c+d x)-a (5 A+7 C) \cos (c+d x)+7 A b}{2 \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}dx}{7 a}+\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-5 A b \cos ^2(c+d x)-a (5 A+7 C) \cos (c+d x)+7 A b}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}dx}{7 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-5 A b \sin \left (c+d x+\frac {\pi }{2}\right )^2-a (5 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+7 A b}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{7 a}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {-21 A b^2 \cos ^2(c+d x)+4 a A b \cos (c+d x)+5 \left ((5 A+7 C) a^2+7 A b^2\right )}{2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{5 a}+\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-21 A b^2 \cos ^2(c+d x)+4 a A b \cos (c+d x)+5 \left ((5 A+7 C) a^2+7 A b^2\right )}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{5 a}}{7 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-21 A b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2+4 a A b \sin \left (c+d x+\frac {\pi }{2}\right )+5 \left ((5 A+7 C) a^2+7 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{5 a}}{7 a}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {-5 b \left ((5 A+7 C) a^2+7 A b^2\right ) \cos ^2(c+d x)+a \left (28 A b^2-5 a^2 (5 A+7 C)\right ) \cos (c+d x)+21 b \left ((3 A+5 C) a^2+5 A b^2\right )}{2 \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{3 a}+\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}}{7 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-5 b \left ((5 A+7 C) a^2+7 A b^2\right ) \cos ^2(c+d x)+a \left (28 A b^2-5 a^2 (5 A+7 C)\right ) \cos (c+d x)+21 b \left ((3 A+5 C) a^2+5 A b^2\right )}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{3 a}}{5 a}}{7 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-5 b \left ((5 A+7 C) a^2+7 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (28 A b^2-5 a^2 (5 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+21 b \left ((3 A+5 C) a^2+5 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 a}}{5 a}}{7 a}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {21 b^2 \left ((3 A+5 C) a^2+5 A b^2\right ) \cos ^2(c+d x)+4 a b \left ((22 A+35 C) a^2+35 A b^2\right ) \cos (c+d x)+5 \left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right )}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}+\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}}{3 a}}{5 a}}{7 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {21 b^2 \left ((3 A+5 C) a^2+5 A b^2\right ) \cos ^2(c+d x)+4 a b \left ((22 A+35 C) a^2+35 A b^2\right ) \cos (c+d x)+5 \left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}}{3 a}}{5 a}}{7 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {21 b^2 \left ((3 A+5 C) a^2+5 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+4 a b \left ((22 A+35 C) a^2+35 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 \left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{3 a}}{5 a}}{7 a}\)

\(\Big \downarrow \) 3538

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {21 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \int \sqrt {\cos (c+d x)}dx-\frac {\int -\frac {5 \left (a \left ((5 A+7 C) a^2+7 A b^2\right ) \cos (c+d x) b^2+\left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right ) b\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{3 a}}{5 a}}{7 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {21 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \int \sqrt {\cos (c+d x)}dx+\frac {5 \int \frac {a \left ((5 A+7 C) a^2+7 A b^2\right ) \cos (c+d x) b^2+\left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right ) b}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{3 a}}{5 a}}{7 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {21 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {5 \int \frac {a \left ((5 A+7 C) a^2+7 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+\left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right ) b}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{a}}{3 a}}{5 a}}{7 a}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \int \frac {a \left ((5 A+7 C) a^2+7 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+\left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right ) b}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{3 a}}{5 a}}{7 a}\)

\(\Big \downarrow \) 3481

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (a b \left (a^2 (5 A+7 C)+7 A b^2\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+21 b^3 \left (a^2 C+A b^2\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx\right )}{b}+\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{3 a}}{5 a}}{7 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (a b \left (a^2 (5 A+7 C)+7 A b^2\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+21 b^3 \left (a^2 C+A b^2\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx\right )}{b}+\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{3 a}}{5 a}}{7 a}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (21 b^3 \left (a^2 C+A b^2\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {2 a b \left (a^2 (5 A+7 C)+7 A b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}\right )}{b}+\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{3 a}}{5 a}}{7 a}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 A b \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {42 b \left (a^2 (3 A+5 C)+5 A b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {5 \left (\frac {2 a b \left (a^2 (5 A+7 C)+7 A b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {42 b^3 \left (a^2 C+A b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}\right )}{b}}{a}}{3 a}}{5 a}}{7 a}\)

Input:

Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*(a + b*Cos[c + d*x])),x]
 

Output:

(2*A*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - ((14*A*b*Sin[c + d*x])/(5* 
a*d*Cos[c + d*x]^(5/2)) - ((10*(7*A*b^2 + a^2*(5*A + 7*C))*Sin[c + d*x])/( 
3*a*d*Cos[c + d*x]^(3/2)) - (-(((42*b*(5*A*b^2 + a^2*(3*A + 5*C))*Elliptic 
E[(c + d*x)/2, 2])/d + (5*((2*a*b*(7*A*b^2 + a^2*(5*A + 7*C))*EllipticF[(c 
 + d*x)/2, 2])/d + (42*b^3*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + 
d*x)/2, 2])/((a + b)*d)))/b)/a) + (42*b*(5*A*b^2 + a^2*(3*A + 5*C))*Sin[c 
+ d*x])/(a*d*Sqrt[Cos[c + d*x]]))/(3*a))/(5*a))/(7*a)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* 
(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3120

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 
)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3284

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 
2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 
0] && GtQ[c + d, 0]
 

rule 3481

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ 
B/d   Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d   Int[(a + b* 
Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, 
 B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3534

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 
NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

rule 3535

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*S 
in[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Simp[1/((m 
+ 1)*(b*c - a*d)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin 
[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n 
+ 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d 
*(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 
- d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) || 
 !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 
 0])))
 

rule 3538

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d)   Int[Sqrt[a + b*Sin[e + f*x]], x] 
, x] - Simp[1/(b*d)   Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ 
e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre 
eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 
] && NeQ[c^2 - d^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(954\) vs. \(2(255)=510\).

Time = 12.18 (sec) , antiderivative size = 955, normalized size of antiderivative = 3.54

method	result	size
default	\(\text {Expression too large to display}\)	\(955\)

Input:

int((A+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c)),x,method=_RETURNV 
ERBOSE)
 

Output:

-(-(1-2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A/a*(-1/56*co 
s(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos 
(1/2*d*x+1/2*c)^2-1/2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+ 
sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+5/21*(sin(1/2*d*x 
+1/2*c)^2)^(1/2)*(1-2*cos(1/2*d*x+1/2*c)^2)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4 
+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*(A*b 
^2+C*a^2)/a^3*(-1/6*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d* 
x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^ 
(1/2)*(1-2*cos(1/2*d*x+1/2*c)^2)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d* 
x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))-4*(A*b^2+C*a^2)*b 
^3/a^4/(-2*a*b+2*b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(1-2*cos(1/2*d*x+1/2*c) 
^2)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi( 
cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))-2/5*A*b/a^2/(8*sin(1/2*d*x+1/2*c)^6 
-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2*(2 
4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1 
/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin 
(1/2*d*x+1/2*c)^4-24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+12*(2*sin(1/2 
*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+ 
1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/ 
2*c)-3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*El...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\text {Timed out} \] Input:

integrate((A+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c)),x, algorith 
m="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\text {Timed out} \] Input:

integrate((A+C*cos(d*x+c)**2)/cos(d*x+c)**(9/2)/(a+b*cos(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate((A+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c)),x, algorith 
m="maxima")
 

Output:

integrate((C*cos(d*x + c)^2 + A)/((b*cos(d*x + c) + a)*cos(d*x + c)^(9/2)) 
, x)
 

Giac [F]

\[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate((A+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c)),x, algorith 
m="giac")
 

Output:

integrate((C*cos(d*x + c)^2 + A)/((b*cos(d*x + c) + a)*cos(d*x + c)^(9/2)) 
, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^{9/2}\,\left (a+b\,\cos \left (c+d\,x\right )\right )} \,d x \] Input:

int((A + C*cos(c + d*x)^2)/(cos(c + d*x)^(9/2)*(a + b*cos(c + d*x))),x)
 

Output:

int((A + C*cos(c + d*x)^2)/(cos(c + d*x)^(9/2)*(a + b*cos(c + d*x))), x)
 

Reduce [F]

\[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6} b +\cos \left (d x +c \right )^{5} a}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4} b +\cos \left (d x +c \right )^{3} a}d x \right ) c \] Input:

int((A+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c)),x)
 

Output:

int(sqrt(cos(c + d*x))/(cos(c + d*x)**6*b + cos(c + d*x)**5*a),x)*a + int( 
sqrt(cos(c + d*x))/(cos(c + d*x)**4*b + cos(c + d*x)**3*a),x)*c