\(\int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}} \, dx\) [1449]

Optimal result

Integrand size = 37, antiderivative size = 618 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}} \, dx=-\frac {2 \left (A b^4-3 a^4 C+a^2 b^2 (3 A+7 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3 a^2 b^2 \sqrt {a+b} \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}}-\frac {2 \left (A b^3+3 a^3 C+a^2 b C-3 a b^2 (A+2 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3 a (a-b) b^2 (a+b)^{3/2} d \sqrt {\sec (c+d x)}}-\frac {2 \sqrt {a+b} C \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{b^3 d \sqrt {\sec (c+d x)}}-\frac {2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}}+\frac {2 \left (A b^4-3 a^4 C+a^2 b^2 (3 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}} \] Output:

-2/3*(A*b^4-3*a^4*C+a^2*b^2*(3*A+7*C))*cos(d*x+c)^(1/2)*csc(d*x+c)*Ellipti 
cE((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2 
))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/b^2/( 
a+b)^(1/2)/(a^2-b^2)/d/sec(d*x+c)^(1/2)-2/3*(A*b^3+3*a^3*C+a^2*b*C-3*a*b^2 
*(A+2*C))*cos(d*x+c)^(1/2)*csc(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2)/(a+ 
b)^(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^( 
1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a/(a-b)/b^2/(a+b)^(3/2)/d/sec(d*x+c)^( 
1/2)-2*(a+b)^(1/2)*C*cos(d*x+c)^(1/2)*csc(d*x+c)*EllipticPi((a+b*cos(d*x+c 
))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(a+b)/b,(-(a+b)/(a-b))^(1/2))*(a*(1- 
sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/b^3/d/sec(d*x+c)^( 
1/2)-2/3*(A*b^2+C*a^2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^(3/2)/sec 
(d*x+c)^(1/2)+2/3*(A*b^4-3*a^4*C+a^2*b^2*(3*A+7*C))*sec(d*x+c)^(1/2)*sin(d 
*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1576\) vs. \(2(618)=1236\).

Time = 15.95 (sec) , antiderivative size = 1576, normalized size of antiderivative = 2.55 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}} \, dx =\text {Too large to display} \] Input:

Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + 
d*x]]),x]
 

Output:

(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(-3*a^2*A*b^2 - A*b^4 + 3 
*a^4*C - 7*a^2*b^2*C)*Sin[c + d*x])/(3*a*b^2*(a^2 - b^2)^2) - (2*(a*A*b^2* 
Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x 
])^2) + (4*(a^2*A*b^2*Sin[c + d*x] + A*b^4*Sin[c + d*x] - 2*a^4*C*Sin[c + 
d*x] + 4*a^2*b^2*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(a + b*Cos[c + d*x 
]))))/d + (2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(3*a^3*A*b^2*Tan[(c + d*x 
)/2] + 3*a^2*A*b^3*Tan[(c + d*x)/2] + a*A*b^4*Tan[(c + d*x)/2] + A*b^5*Tan 
[(c + d*x)/2] - 3*a^5*C*Tan[(c + d*x)/2] - 3*a^4*b*C*Tan[(c + d*x)/2] + 7* 
a^3*b^2*C*Tan[(c + d*x)/2] + 7*a^2*b^3*C*Tan[(c + d*x)/2] - 6*a^2*A*b^3*Ta 
n[(c + d*x)/2]^3 - 2*A*b^5*Tan[(c + d*x)/2]^3 + 6*a^4*b*C*Tan[(c + d*x)/2] 
^3 - 14*a^2*b^3*C*Tan[(c + d*x)/2]^3 - 3*a^3*A*b^2*Tan[(c + d*x)/2]^5 + 3* 
a^2*A*b^3*Tan[(c + d*x)/2]^5 - a*A*b^4*Tan[(c + d*x)/2]^5 + A*b^5*Tan[(c + 
 d*x)/2]^5 + 3*a^5*C*Tan[(c + d*x)/2]^5 - 3*a^4*b*C*Tan[(c + d*x)/2]^5 - 7 
*a^3*b^2*C*Tan[(c + d*x)/2]^5 + 7*a^2*b^3*C*Tan[(c + d*x)/2]^5 + 6*a^5*C*E 
llipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c 
+ d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a 
 + b)] - 12*a^3*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a 
 + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b 
*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a*b^4*C*EllipticPi[-1, ArcSin[Tan[(c + d 
*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a...
 

Rubi [A] (verified)

Time = 2.88 (sec) , antiderivative size = 602, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {3042, 4709, 3042, 3527, 27, 3042, 3530, 3042, 3288, 3472, 3042, 3477, 3042, 3295, 3473}

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.

\(\displaystyle \int \frac {A+C \cos ^2(c+d x)}{\sqrt {\sec (c+d x)} (a+b \cos (c+d x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+C \cos (c+d x)^2}{\sqrt {\sec (c+d x)} (a+b \cos (c+d x))^{5/2}}dx\)

\(\Big \downarrow \) 4709

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3527

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3530

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3288

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\int \frac {b \left (C a^2+A b^2\right )-3 a \left (A b^2-\left (a^2-2 b^2\right ) C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{b}+\frac {6 C \sqrt {a+b} \left (a^2-b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b^2 d}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\right )\)

\(\Big \downarrow \) 3472

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\frac {\int \frac {-3 C a^4+b^2 (3 A+7 C) a^2+2 b \left (2 A b^2-\left (a^2-3 b^2\right ) C\right ) \cos (c+d x) a+A b^4}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a^2-b^2}-\frac {2 \left (-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}}{b}+\frac {6 C \sqrt {a+b} \left (a^2-b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b^2 d}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\frac {\int \frac {-3 C a^4+b^2 (3 A+7 C) a^2+2 b \left (2 A b^2-\left (a^2-3 b^2\right ) C\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+A b^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {2 \left (-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}}{b}+\frac {6 C \sqrt {a+b} \left (a^2-b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b^2 d}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\right )\)

\(\Big \downarrow \) 3477

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\frac {\left (-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+(a-b) \left (3 a^3 C+a^2 b C-3 a b^2 (A+2 C)+A b^3\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx}{a^2-b^2}-\frac {2 \left (-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}}{b}+\frac {6 C \sqrt {a+b} \left (a^2-b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b^2 d}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\frac {\left (-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+(a-b) \left (3 a^3 C+a^2 b C-3 a b^2 (A+2 C)+A b^3\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {2 \left (-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}}{b}+\frac {6 C \sqrt {a+b} \left (a^2-b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b^2 d}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\right )\)

\(\Big \downarrow \) 3295

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\frac {\left (-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (a-b) \sqrt {a+b} \left (3 a^3 C+a^2 b C-3 a b^2 (A+2 C)+A b^3\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{a^2-b^2}-\frac {2 \left (-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}}{b}+\frac {6 C \sqrt {a+b} \left (a^2-b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b^2 d}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\right )\)

\(\Big \downarrow \) 3473

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {\frac {6 C \sqrt {a+b} \left (a^2-b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b^2 d}+\frac {\frac {\frac {2 (a-b) \sqrt {a+b} \left (-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d}+\frac {2 (a-b) \sqrt {a+b} \left (3 a^3 C+a^2 b C-3 a b^2 (A+2 C)+A b^3\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{a^2-b^2}-\frac {2 \left (-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}}{b}}{3 b \left (a^2-b^2\right )}\right )\)

Input:

Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) 
,x]
 

Output:

Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d* 
x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - ((6*Sqr 
t[a + b]*(a^2 - b^2)*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + 
b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqr 
t[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b 
^2*d) + (((2*(a - b)*Sqrt[a + b]*(A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 7*C))*C 
ot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Co 
s[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sq 
rt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) + (2*(a - b)*Sqrt[a + b]*(A*b^ 
3 + 3*a^3*C + a^2*b*C - 3*a*b^2*(A + 2*C))*Cot[c + d*x]*EllipticF[ArcSin[S 
qrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - 
b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - 
 b)])/(a*d))/(a^2 - b^2) - (2*(A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sin[ 
c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]))/b)/ 
(3*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 3288

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

rule 3295

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f 
_.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr 
t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli 
pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] 
], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] 
&& PosQ[(a + b)/d]
 

rule 3472

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(d_.)*sin[(e_.) + (f_.)*( 
x_)]]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)), x_Symbol] :> Simp[2*(A 
*b - a*B)*(Cos[e + f*x]/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[d*Sin[ 
e + f*x]])), x] + Simp[d/(a^2 - b^2)   Int[(A*b - a*B + (a*A - b*B)*Sin[e + 
 f*x])/(Sqrt[a + b*Sin[e + f*x]]*(d*Sin[e + f*x])^(3/2)), x], x] /; FreeQ[{ 
a, b, d, e, f, A, B}, x] && NeQ[a^2 - b^2, 0]
 

rule 3473

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) 
^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* 
(c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] 
)/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + 
d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], 
x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && 
PosQ[(c + d)/b]
 

rule 3477

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S 
imp[(A - B)/(a - b)   Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* 
x]]), x], x] - Simp[(A*b - a*B)/(a - b)   Int[(1 + Sin[e + f*x])/((a + b*Si 
n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e 
, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 
0] && NeQ[A, B]
 

rule 3527

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^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + 
 f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 
2))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* 
d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b 
*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( 
A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], 
x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - 
b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
 

rule 3530

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_ 
)])^(3/2)), x_Symbol] :> Simp[C/(b*d)   Int[Sqrt[d*Sin[e + f*x]]/Sqrt[a + b 
*Sin[e + f*x]], x], x] + Simp[1/b   Int[(A*b + (b*B - a*C)*Sin[e + f*x])/(( 
a + b*Sin[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, d, 
e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
 

rule 4709

Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a 
+ b*x])^m*(c*Cos[a + b*x])^m   Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], 
x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2974\) vs. \(2(557)=1114\).

Time = 23.39 (sec) , antiderivative size = 2975, normalized size of antiderivative = 4.81

method	result	size
default	\(\text {Expression too large to display}\)	\(2975\)
parts	\(\text {Expression too large to display}\)	\(3007\)

Input:

int((A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2)/sec(d*x+c)^(1/2),x,method=_R 
ETURNVERBOSE)
 

Output:

2/3/d/(a-b)^2/(a+b)^2/a/b^2*(a+b*cos(d*x+c))^(1/2)/((1+cos(d*x+c))*a^2+cos 
(d*x+c)*(2*cos(d*x+c)+2)*a*b+cos(d*x+c)^2*(1+cos(d*x+c))*b^2)/sec(d*x+c)^( 
1/2)*(3*A*a^4*b^2*sin(d*x+c)-A*b^6*cos(d*x+c)*sin(d*x+c)+(-cos(d*x+c)^2-2* 
cos(d*x+c)-1)*A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c) 
)/(1+cos(d*x+c)))^(1/2)*b^6*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b) 
)^(1/2))+(-7*cos(d*x+c)^2-14*cos(d*x+c)-7)*C*(cos(d*x+c)/(1+cos(d*x+c)))^( 
1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b^4*EllipticE(-cs 
c(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))+(-6*cos(d*x+c)^2-12*cos(d*x+c)-6 
)*C*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x 
+c)))^(1/2)*a^5*b*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,(-(a-b)/(a+b))^(1/2 
))+(12*cos(d*x+c)^2+24*cos(d*x+c)+12)*C*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)* 
(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*b^3*EllipticPi(-csc(d* 
x+c)+cot(d*x+c),-1,(-(a-b)/(a+b))^(1/2))+C*(cos(d*x+c)/(1+cos(d*x+c)))^(1/ 
2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*b^3*EllipticF(-csc( 
d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(cos(d*x+c)^2+8*cos(d*x+c)+13+6*se 
c(d*x+c))+C*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1 
+cos(d*x+c)))^(1/2)*a^2*b^4*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b) 
)^(1/2))*(6*cos(d*x+c)^2+15*cos(d*x+c)+12+3*sec(d*x+c))+C*(cos(d*x+c)/(1+c 
os(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^4*b^2* 
EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,(-(a-b)/(a+b))^(1/2))*(12*cos(d*x+...
 

Fricas [F]

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

integrate((A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2)/sec(d*x+c)^(1/2),x, al 
gorithm="fricas")
 

Output:

integral((C*cos(d*x + c)^2 + A)*sqrt(b*cos(d*x + c) + a)/((b^3*cos(d*x + c 
)^3 + 3*a*b^2*cos(d*x + c)^2 + 3*a^2*b*cos(d*x + c) + a^3)*sqrt(sec(d*x + 
c))), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \] Input:

integrate((A+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**(5/2)/sec(d*x+c)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2)/sec(d*x+c)^(1/2),x, al 
gorithm="maxima")
 

Output:

integrate((C*cos(d*x + c)^2 + A)/((b*cos(d*x + c) + a)^(5/2)*sqrt(sec(d*x 
+ c))), x)
 

Giac [F]

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

integrate((A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2)/sec(d*x+c)^(1/2),x, al 
gorithm="giac")
 

Output:

integrate((C*cos(d*x + c)^2 + A)/((b*cos(d*x + c) + a)^(5/2)*sqrt(sec(d*x 
+ c))), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:

int((A + C*cos(c + d*x)^2)/((1/cos(c + d*x))^(1/2)*(a + b*cos(c + d*x))^(5 
/2)),x)
 

Output:

int((A + C*cos(c + d*x)^2)/((1/cos(c + d*x))^(1/2)*(a + b*cos(c + d*x))^(5 
/2)), x)
 

Reduce [F]

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

int((A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2)/sec(d*x+c)^(1/2),x)
 

Output:

int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2)/(cos(c + 
 d*x)**3*sec(c + d*x)*b**3 + 3*cos(c + d*x)**2*sec(c + d*x)*a*b**2 + 3*cos 
(c + d*x)*sec(c + d*x)*a**2*b + sec(c + d*x)*a**3),x)*c + int((sqrt(sec(c 
+ d*x))*sqrt(cos(c + d*x)*b + a))/(cos(c + d*x)**3*sec(c + d*x)*b**3 + 3*c 
os(c + d*x)**2*sec(c + d*x)*a*b**2 + 3*cos(c + d*x)*sec(c + d*x)*a**2*b + 
sec(c + d*x)*a**3),x)*a