\(\int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+C \cos ^2(c+d x))}{a+b \cos (c+d x)} \, dx\) [704]

Optimal result

Integrand size = 35, antiderivative size = 299 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {2 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^5 d}-\frac {2 a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 b^6 d}+\frac {2 a^4 \left (A b^2+a^2 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b^6 (a+b) d}-\frac {2 a \left (7 A b^2+7 a^2 C+5 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^4 d}+\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 b^3 d}-\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d} \] Output:

2/15*(15*a^4*C+3*a^2*b^2*(5*A+3*C)+b^4*(9*A+7*C))*EllipticE(sin(1/2*d*x+1/ 
2*c),2^(1/2))/b^5/d-2/21*a*(21*a^4*C+7*a^2*b^2*(3*A+C)+b^4*(7*A+5*C))*Inve 
rseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/b^6/d+2*a^4*(A*b^2+C*a^2)*EllipticPi(si 
n(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))/b^6/(a+b)/d-2/21*a*(7*A*b^2+7*C*a^2+5* 
C*b^2)*cos(d*x+c)^(1/2)*sin(d*x+c)/b^4/d+2/45*(9*a^2*C+b^2*(9*A+7*C))*cos( 
d*x+c)^(3/2)*sin(d*x+c)/b^3/d-2/7*a*C*cos(d*x+c)^(5/2)*sin(d*x+c)/b^2/d+2/ 
9*C*cos(d*x+c)^(7/2)*sin(d*x+c)/b/d
 

Mathematica [A] (warning: unable to verify)

Time = 4.52 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.20 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {\sqrt {\cos (c+d x)} \left (7 b \left (36 A b^2+36 a^2 C+43 b^2 C\right ) \cos (c+d x)-5 \left (84 a A b^2+84 a^3 C+78 a b^2 C+18 a b^2 C \cos (2 (c+d x))-7 b^3 C \cos (3 (c+d x))\right )\right ) \sin (c+d x)+6 \left (\frac {\left (35 a^4 C+7 b^4 (9 A+7 C)+a^2 b^2 (35 A+13 C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {8 a \left (7 A b^2+7 a^2 C+6 b^2 C\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a+b}+\frac {7 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b^2 \sqrt {\sin ^2(c+d x)}}\right )}{630 b^4 d} \] Input:

Integrate[(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]) 
,x]
 

Output:

(Sqrt[Cos[c + d*x]]*(7*b*(36*A*b^2 + 36*a^2*C + 43*b^2*C)*Cos[c + d*x] - 5 
*(84*a*A*b^2 + 84*a^3*C + 78*a*b^2*C + 18*a*b^2*C*Cos[2*(c + d*x)] - 7*b^3 
*C*Cos[3*(c + d*x)]))*Sin[c + d*x] + 6*(((35*a^4*C + 7*b^4*(9*A + 7*C) + a 
^2*b^2*(35*A + 13*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + 
 (8*a*(7*A*b^2 + 7*a^2*C + 6*b^2*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a 
*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (7*(15*a^4*C + 3*a^ 
2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*(-2*a*b*EllipticE[ArcSin[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*b^2*Sqrt[Sin[c + d*x]^2])))/(630*b^4*d)
 

Rubi [A] (verified)

Time = 2.72 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.10, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3529, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 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.

\(\displaystyle \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3529

\(\displaystyle \frac {2 \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (-9 a C \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+7 a C\right )}{2 (a+b \cos (c+d x))}dx}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (-9 a C \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+7 a C\right )}{a+b \cos (c+d x)}dx}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (-9 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (9 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+7 a C\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {2 \int -\frac {\cos ^{\frac {3}{2}}(c+d x) \left (45 C a^2-4 b C \cos (c+d x) a-7 \left (9 C a^2+b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{2 (a+b \cos (c+d x))}dx}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (45 C a^2-4 b C \cos (c+d x) a-7 \left (9 C a^2+b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)}dx}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (45 C a^2-4 b C \sin \left (c+d x+\frac {\pi }{2}\right ) a-7 \left (9 C a^2+b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {-\frac {\frac {2 \int -\frac {3 \sqrt {\cos (c+d x)} \left (-15 a \left (7 C a^2+7 A b^2+5 b^2 C\right ) \cos ^2(c+d x)+b \left (-12 C a^2+63 A b^2+49 b^2 C\right ) \cos (c+d x)+7 a \left (9 C a^2+b^2 (9 A+7 C)\right )\right )}{2 (a+b \cos (c+d x))}dx}{5 b}-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {3 \int \frac {\sqrt {\cos (c+d x)} \left (-15 a \left (7 C a^2+7 A b^2+5 b^2 C\right ) \cos ^2(c+d x)+b \left (-12 C a^2+63 A b^2+49 b^2 C\right ) \cos (c+d x)+7 a \left (9 C a^2+b^2 (9 A+7 C)\right )\right )}{a+b \cos (c+d x)}dx}{5 b}-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {3 \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (-15 a \left (7 C a^2+7 A b^2+5 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (-12 C a^2+63 A b^2+49 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+7 a \left (9 C a^2+b^2 (9 A+7 C)\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b}-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {-\frac {-\frac {3 \left (\frac {2 \int -\frac {3 \left (5 \left (7 C a^2+7 A b^2+5 b^2 C\right ) a^2-4 b \left (7 C a^2+7 A b^2+6 b^2 C\right ) \cos (c+d x) a-7 \left (15 C a^4+3 b^2 (5 A+3 C) a^2+b^4 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}-\frac {10 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}\right )}{5 b}-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\int \frac {5 \left (7 C a^2+7 A b^2+5 b^2 C\right ) a^2-4 b \left (7 C a^2+7 A b^2+6 b^2 C\right ) \cos (c+d x) a-7 \left (15 C a^4+3 b^2 (5 A+3 C) a^2+b^4 (9 A+7 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}-\frac {10 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}\right )}{5 b}-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\int \frac {5 \left (7 C a^2+7 A b^2+5 b^2 C\right ) a^2-4 b \left (7 C a^2+7 A b^2+6 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-7 \left (15 C a^4+3 b^2 (5 A+3 C) a^2+b^4 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {10 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}\right )}{5 b}-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3538

\(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {-\frac {7 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)}dx}{b}-\frac {\int -\frac {5 \left (b \left (7 C a^2+7 A b^2+5 b^2 C\right ) a^2+\left (21 C a^4+7 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right ) \cos (c+d x) a\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{b}-\frac {10 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}\right )}{5 b}-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\frac {5 \int \frac {b \left (7 C a^2+7 A b^2+5 b^2 C\right ) a^2+\left (21 C a^4+7 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right ) \cos (c+d x) a}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}-\frac {7 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)}dx}{b}}{b}-\frac {10 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}\right )}{5 b}-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\frac {5 \int \frac {b \left (7 C a^2+7 A b^2+5 b^2 C\right ) a^2+\left (21 C a^4+7 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {7 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{b}-\frac {10 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}\right )}{5 b}-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\frac {5 \int \frac {b \left (7 C a^2+7 A b^2+5 b^2 C\right ) a^2+\left (21 C a^4+7 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {14 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{b}-\frac {10 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}\right )}{5 b}-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3481

\(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\frac {5 \left (\frac {a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b}-\frac {21 a^4 \left (a^2 C+A b^2\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}\right )}{b}-\frac {14 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{b}-\frac {10 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}\right )}{5 b}-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\frac {5 \left (\frac {a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {21 a^4 \left (a^2 C+A b^2\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}\right )}{b}-\frac {14 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{b}-\frac {10 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}\right )}{5 b}-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\frac {5 \left (\frac {2 a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d}-\frac {21 a^4 \left (a^2 C+A b^2\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}\right )}{b}-\frac {14 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{b}-\frac {10 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}\right )}{5 b}-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {-\frac {-\frac {14 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}-\frac {3 \left (-\frac {10 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}-\frac {\frac {5 \left (\frac {2 a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d}-\frac {42 a^4 \left (a^2 C+A b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b d (a+b)}\right )}{b}-\frac {14 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{b}\right )}{5 b}}{7 b}-\frac {18 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d}\)

Input:

Int[(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]
 

Output:

(2*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*b*d) + ((-18*a*C*Cos[c + d*x]^(5/ 
2)*Sin[c + d*x])/(7*b*d) - ((-14*(9*a^2*C + b^2*(9*A + 7*C))*Cos[c + d*x]^ 
(3/2)*Sin[c + d*x])/(5*b*d) - (3*(-(((-14*(15*a^4*C + 3*a^2*b^2*(5*A + 3*C 
) + b^4*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(b*d) + (5*((2*a*(21*a^4*C 
 + 7*a^2*b^2*(3*A + C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(b*d) 
 - (42*a^4*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b*( 
a + b)*d)))/b)/b) - (10*a*(7*A*b^2 + 7*a^2*C + 5*b^2*C)*Sqrt[Cos[c + d*x]] 
*Sin[c + d*x])/(b*d)))/(5*b))/(7*b))/(9*b)
 

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 3528

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[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x 
])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + 
n + 2))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* 
d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a 
*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
 n + 2))*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] && GtQ[ 
m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
 

rule 3529

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[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 
1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2))   Int[(a + b*Sin[e + f*x 
])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*( 
n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* 
(a*d*m - b*c*(m + 1))*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 
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. \(1553\) vs. \(2(284)=568\).

Time = 34.51 (sec) , antiderivative size = 1554, normalized size of antiderivative = 5.20

method	result	size
default	\(\text {Expression too large to display}\)	\(1554\)

Input:

int(cos(d*x+c)^(7/2)*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c)),x,method=_RETURNV 
ERBOSE)
 

Output:

-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*((-720*C*a^ 
2*b^4+2960*C*a*b^5-2240*C*b^6)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-5 
04*A*a*b^5+504*A*b^6-504*C*a^3*b^3+1584*C*a^2*b^4-3152*C*a*b^5+2072*C*b^6) 
*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(-420*A*a^2*b^4+924*A*a*b^5-504*A 
*b^6-420*C*a^4*b^2+924*C*a^3*b^3-1344*C*a^2*b^4+1792*C*a*b^5-952*C*b^6)*si 
n(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(210*A*a^2*b^4-336*A*a*b^5+126*A*b^6 
+210*C*a^4*b^2-336*C*a^3*b^3+366*C*a^2*b^4-408*C*a*b^5+168*C*b^6)*sin(1/2* 
d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-75*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1 
/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b^4+75* 
C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF( 
cos(1/2*d*x+1/2*c),2^(1/2))*a*b^5-315*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*si 
n(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^5*b+10 
5*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Elliptic 
F(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^5+315*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(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))*a^2*b^ 
4-315*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Elli 
pticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^6+147*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*( 
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))*b^6+ 
189*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellipt 
icE(cos(1/2*d*x+1/2*c),2^(1/2))*b^6+315*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)^(7/2)*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c)),x, algorith 
m="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**(7/2)*(A+C*cos(d*x+c)**2)/(a+b*cos(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \] Input:

integrate(cos(d*x+c)^(7/2)*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c)),x, algorith 
m="maxima")
 

Output:

integrate((C*cos(d*x + c)^2 + A)*cos(d*x + c)^(7/2)/(b*cos(d*x + c) + a), 
x)
 

Giac [F]

\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \] Input:

integrate(cos(d*x+c)^(7/2)*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c)),x, algorith 
m="giac")
 

Output:

integrate((C*cos(d*x + c)^2 + A)*cos(d*x + c)^(7/2)/(b*cos(d*x + c) + a), 
x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{7/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{a+b\,\cos \left (c+d\,x\right )} \,d x \] Input:

int((cos(c + d*x)^(7/2)*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x)),x)
 

Output:

int((cos(c + d*x)^(7/2)*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}}{\cos \left (d x +c \right ) b +a}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\cos \left (d x +c \right ) b +a}d x \right ) a \] Input:

int(cos(d*x+c)^(7/2)*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c)),x)
 

Output:

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