\(\int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx\) [669]

Optimal result

Integrand size = 27, antiderivative size = 389 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx=-\frac {2 \left (2 a^4 C-3 b^4 (3 A+5 C)-a^2 b^2 (23 A+19 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^2 \left (a^2-b^2\right )^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {4 a \left (4 A b^2-\left (a^2-5 b^2\right ) C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{5 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {4 a \left (4 A b^2-a^2 C+5 b^2 C\right ) \sin (c+d x)}{15 b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}+\frac {2 \left (2 a^4 C-3 b^4 (3 A+5 C)-a^2 b^2 (23 A+19 C)\right ) \sin (c+d x)}{15 b \left (a^2-b^2\right )^3 d \sqrt {a+b \cos (c+d x)}} \] Output:

-2/15*(2*a^4*C-3*b^4*(3*A+5*C)-a^2*b^2*(23*A+19*C))*(a+b*cos(d*x+c))^(1/2) 
*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))/b^2/(a^2-b^2)^3/d/( 
(a+b*cos(d*x+c))/(a+b))^(1/2)-4/15*a*(4*A*b^2-(a^2-5*b^2)*C)*((a+b*cos(d*x 
+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1/2))/b 
^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^(1/2)-2/5*(A*b^2+C*a^2)*sin(d*x+c)/b/(a^ 
2-b^2)/d/(a+b*cos(d*x+c))^(5/2)-4/15*a*(4*A*b^2-C*a^2+5*C*b^2)*sin(d*x+c)/ 
b/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^(3/2)+2/15*(2*a^4*C-3*b^4*(3*A+5*C)-a^2*b 
^2*(23*A+19*C))*sin(d*x+c)/b/(a^2-b^2)^3/d/(a+b*cos(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 3.14 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.81 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx=\frac {2 \left (\frac {\left (\frac {a+b \cos (c+d x)}{a+b}\right )^{5/2} \left (\left (-2 a^4 C+3 b^4 (3 A+5 C)+a^2 b^2 (23 A+19 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+2 a (a-b) \left (-4 A b^2+\left (a^2-5 b^2\right ) C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )}{(a-b)^3}+\frac {b \left (68 a^4 A b^2+13 a^2 A b^4+15 A b^6-2 a^6 C+48 a^4 b^2 C+35 a^2 b^4 C+15 b^6 C-4 a b \left (3 a^4 C-5 b^4 (A+2 C)-a^2 b^2 (27 A+25 C)\right ) \cos (c+d x)+\left (-2 a^4 b^2 C+3 b^6 (3 A+5 C)+a^2 b^4 (23 A+19 C)\right ) \cos (2 (c+d x))\right ) \sin (c+d x)}{2 \left (-a^2+b^2\right )^3}\right )}{15 b^2 d (a+b \cos (c+d x))^{5/2}} \] Input:

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

Output:

(2*((((a + b*Cos[c + d*x])/(a + b))^(5/2)*((-2*a^4*C + 3*b^4*(3*A + 5*C) + 
 a^2*b^2*(23*A + 19*C))*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*a*(a - b 
)*(-4*A*b^2 + (a^2 - 5*b^2)*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a 
- b)^3 + (b*(68*a^4*A*b^2 + 13*a^2*A*b^4 + 15*A*b^6 - 2*a^6*C + 48*a^4*b^2 
*C + 35*a^2*b^4*C + 15*b^6*C - 4*a*b*(3*a^4*C - 5*b^4*(A + 2*C) - a^2*b^2* 
(27*A + 25*C))*Cos[c + d*x] + (-2*a^4*b^2*C + 3*b^6*(3*A + 5*C) + a^2*b^4* 
(23*A + 19*C))*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*(-a^2 + b^2)^3)))/(15*b^ 
2*d*(a + b*Cos[c + d*x])^(5/2))
 

Rubi [A] (verified)

Time = 1.97 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.06, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3501, 27, 3042, 3233, 27, 3042, 3233, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3501

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

\(\displaystyle \frac {\frac {\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {2 \int -\frac {a b \left ((15 A+7 C) a^2+b^2 (17 A+25 C)\right )-\left (2 C a^4-b^2 (23 A+19 C) a^2-3 b^4 (3 A+5 C)\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {a b \left ((15 A+7 C) a^2+b^2 (17 A+25 C)\right )-\left (2 C a^4-b^2 (23 A+19 C) a^2-3 b^4 (3 A+5 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {a b \left ((15 A+7 C) a^2+b^2 (17 A+25 C)\right )+\left (-2 C a^4+b^2 (23 A+19 C) a^2+3 b^4 (3 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {\left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {-\frac {4 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a^2-b^2}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

Input:

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

Output:

(-2*(A*b^2 + a^2*C)*Sin[c + d*x])/(5*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^ 
(5/2)) + ((-4*a*(4*A*b^2 - (a^2 - 5*b^2)*C)*Sin[c + d*x])/(3*(a^2 - b^2)*d 
*(a + b*Cos[c + d*x])^(3/2)) + (((-2*(2*a^4*C - 3*b^4*(3*A + 5*C) - a^2*b^ 
2*(23*A + 19*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a 
+ b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*(a^2 - b^2)*(4*A*b^ 
2 - a^2*C + 5*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x 
)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]))/(a^2 - b^2) + (2*(2*a 
^4*C - 3*b^4*(3*A + 5*C) - a^2*b^2*(23*A + 19*C))*Sin[c + d*x])/((a^2 - b^ 
2)*d*Sqrt[a + b*Cos[c + d*x]]))/(3*(a^2 - b^2)))/(5*b*(a^2 - b^2))
 

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 3132

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

rule 3134

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

rule 3140

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

rule 3142

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

rule 3231

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

rule 3233

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

rule 3501

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1308\) vs. \(2(374)=748\).

Time = 16.95 (sec) , antiderivative size = 1309, normalized size of antiderivative = 3.37

method	result	size
default	\(\text {Expression too large to display}\)	\(1309\)
parts	\(\text {Expression too large to display}\)	\(1917\)

Input:

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

Output:

-(-(-2*b*cos(1/2*d*x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*C/b^2/s 
in(1/2*d*x+1/2*c)^2/(2*b*sin(1/2*d*x+1/2*c)^2-a-b)/(a^2-b^2)*(-2*b*sin(1/2 
*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*b*sin(1/2*d*x+1/2*c)^2* 
cos(1/2*d*x+1/2*c)+(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Ell 
ipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2) 
*a-(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d 
*x+1/2*c),(-2*b/(a-b))^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*b)+2*(A*b^2+C*a 
^2)/b^2*(1/20/b^2/(a-b)/(a+b)*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/2*c)^ 
4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2+1/2/b*(a-b))^3+4 
/15*a/b/(a-b)^2/(a+b)^2*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b 
)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2+1/2/b*(a-b))^2+2/15*b* 
sin(1/2*d*x+1/2*c)^2/(a-b)^3/(a+b)^3*cos(1/2*d*x+1/2*c)*(23*a^2+9*b^2)/(-( 
-2*b*cos(1/2*d*x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)+(15*a^2-8*a*b+9 
*b^2)/(15*a^5+15*a^4*b-30*a^3*b^2-30*a^2*b^3+15*a*b^4+15*b^5)*(sin(1/2*d*x 
+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/ 
2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2 
*c),(-2*b/(a-b))^(1/2))-1/15*(23*a^2+9*b^2)/(a-b)^2/(a+b)^3*(sin(1/2*d*x+1 
/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2* 
d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2* 
c),(-2*b/(a-b))^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 1361, normalized size of antiderivative = 3.50 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

2/45*(sqrt(1/2)*(-4*I*C*a^8 + I*(A + 17*C)*a^6*b^2 - 3*I*(11*A + 15*C)*a^4 
*b^4 + (-4*I*C*a^5*b^3 + I*(A + 17*C)*a^3*b^5 - 3*I*(11*A + 15*C)*a*b^7)*c 
os(d*x + c)^3 + 3*(-4*I*C*a^6*b^2 + I*(A + 17*C)*a^4*b^4 - 3*I*(11*A + 15* 
C)*a^2*b^6)*cos(d*x + c)^2 + 3*(-4*I*C*a^7*b + I*(A + 17*C)*a^5*b^3 - 3*I* 
(11*A + 15*C)*a^3*b^5)*cos(d*x + c))*sqrt(b)*weierstrassPInverse(4/3*(4*a^ 
2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b 
*sin(d*x + c) + 2*a)/b) + sqrt(1/2)*(4*I*C*a^8 - I*(A + 17*C)*a^6*b^2 + 3* 
I*(11*A + 15*C)*a^4*b^4 + (4*I*C*a^5*b^3 - I*(A + 17*C)*a^3*b^5 + 3*I*(11* 
A + 15*C)*a*b^7)*cos(d*x + c)^3 + 3*(4*I*C*a^6*b^2 - I*(A + 17*C)*a^4*b^4 
+ 3*I*(11*A + 15*C)*a^2*b^6)*cos(d*x + c)^2 + 3*(4*I*C*a^7*b - I*(A + 17*C 
)*a^5*b^3 + 3*I*(11*A + 15*C)*a^3*b^5)*cos(d*x + c))*sqrt(b)*weierstrassPI 
nverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos( 
d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) + 3*sqrt(1/2)*(-2*I*C*a^7*b + I*(2 
3*A + 19*C)*a^5*b^3 + 3*I*(3*A + 5*C)*a^3*b^5 + (-2*I*C*a^4*b^4 + I*(23*A 
+ 19*C)*a^2*b^6 + 3*I*(3*A + 5*C)*b^8)*cos(d*x + c)^3 + 3*(-2*I*C*a^5*b^3 
+ I*(23*A + 19*C)*a^3*b^5 + 3*I*(3*A + 5*C)*a*b^7)*cos(d*x + c)^2 + 3*(-2* 
I*C*a^6*b^2 + I*(23*A + 19*C)*a^4*b^4 + 3*I*(3*A + 5*C)*a^2*b^6)*cos(d*x + 
 c))*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b 
^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b 
^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b)) + 3*sqrt...
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}}{\cos \left (d x +c \right )^{4} b^{4}+4 \cos \left (d x +c \right )^{3} a \,b^{3}+6 \cos \left (d x +c \right )^{2} a^{2} b^{2}+4 \cos \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{4} b^{4}+4 \cos \left (d x +c \right )^{3} a \,b^{3}+6 \cos \left (d x +c \right )^{2} a^{2} b^{2}+4 \cos \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) c \] Input:

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

Output:

int(sqrt(cos(c + d*x)*b + a)/(cos(c + d*x)**4*b**4 + 4*cos(c + d*x)**3*a*b 
**3 + 6*cos(c + d*x)**2*a**2*b**2 + 4*cos(c + d*x)*a**3*b + a**4),x)*a + i 
nt((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2)/(cos(c + d*x)**4*b**4 + 4*co 
s(c + d*x)**3*a*b**3 + 6*cos(c + d*x)**2*a**2*b**2 + 4*cos(c + d*x)*a**3*b 
 + a**4),x)*c