\(\int \frac {(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^{\frac {7}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx\) [1533]

Optimal result

Integrand size = 45, antiderivative size = 585 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx=-\frac {2 \left (48 A b^4+25 a^3 b B-40 a b^3 B-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{15 a^5 \sqrt {a+b} d \sqrt {\sec (c+d x)}}-\frac {2 \left (48 A b^3+4 a b^2 (9 A-10 B)+6 a^2 b (2 A-5 B+5 C)+a^3 (9 A-5 B+15 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{15 a^4 \sqrt {a+b} d \sqrt {\sec (c+d x)}}+\frac {2 \left (24 A b^3+5 a^3 B-20 a b^2 B-a^2 (9 A b-15 b C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d}+\frac {2 \left (A b^2-a (b B-a C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (6 A b^2-5 a b B-a^2 (A-5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d} \] Output:

-2/15*(48*A*b^4+25*B*a^3*b-40*B*a*b^3-6*a^2*b^2*(4*A-5*C)-3*a^4*(3*A+5*C)) 
*cos(d*x+c)^(1/2)*csc(d*x+c)*EllipticE((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^5/(a+b)^(1/2)/d/sec(d*x+c)^(1/2)-2/15*(48*A*b 
^3+4*a*b^2*(9*A-10*B)+6*a^2*b*(2*A-5*B+5*C)+a^3*(9*A-5*B+15*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^4/(a+b)^(1/2)/d/sec(d*x+c)^(1/2)+2/15*(24*A*b^3+5*B*a^3- 
20*B*a*b^2-a^2*(9*A*b-15*C*b))*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(3/2)*sin 
(d*x+c)/a^3/(a^2-b^2)/d+2*(A*b^2-a*(B*b-C*a))*sec(d*x+c)^(5/2)*sin(d*x+c)/ 
a/(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)-2/5*(6*A*b^2-5*B*a*b-a^2*(A-5*C))*(a+ 
b*cos(d*x+c))^(1/2)*sec(d*x+c)^(5/2)*sin(d*x+c)/a^2/(a^2-b^2)/d
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(4900\) vs. \(2(585)=1170\).

Time = 27.60 (sec) , antiderivative size = 4900, normalized size of antiderivative = 8.38 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a 
+ b*Cos[c + d*x])^(3/2),x]
 

Output:

(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(9*a^4*A + 24*a^2*A*b^2 - 
 48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 15*a^4*C - 30*a^2*b^2*C)*Sin[c + d*x 
])/(15*a^4*(a^2 - b^2)) + (2*Sec[c + d*x]*(-9*A*b*Sin[c + d*x] + 5*a*B*Sin 
[c + d*x]))/(15*a^3) + (2*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2 
*b^2*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Sec[c 
+ d*x]*Tan[c + d*x])/(5*a^2)))/d + (2*((-3*a*A)/(5*(a^2 - b^2)*Sqrt[a + b* 
Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*A*b^2)/(5*a*(a^2 - b^2)*Sqrt[a + b* 
Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^4)/(5*a^3*(a^2 - b^2)*Sqrt[a + 
 b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*b*B)/(3*(a^2 - b^2)*Sqrt[a + b*C 
os[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*b^3*B)/(3*a^2*(a^2 - b^2)*Sqrt[a + b 
*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a*C)/((a^2 - b^2)*Sqrt[a + b*Cos[c + 
 d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^2*C)/(a*(a^2 - b^2)*Sqrt[a + b*Cos[c + d 
*x]]*Sqrt[Sec[c + d*x]]) - (4*A*b*Sqrt[Sec[c + d*x]])/(5*(a^2 - b^2)*Sqrt[ 
a + b*Cos[c + d*x]]) - (12*A*b^3*Sqrt[Sec[c + d*x]])/(5*a^2*(a^2 - b^2)*Sq 
rt[a + b*Cos[c + d*x]]) + (16*A*b^5*Sqrt[Sec[c + d*x]])/(5*a^4*(a^2 - b^2) 
*Sqrt[a + b*Cos[c + d*x]]) + (a*B*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)*Sqrt[ 
a + b*Cos[c + d*x]]) + (7*b^2*B*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)*Sqrt[ 
a + b*Cos[c + d*x]]) - (8*b^4*B*Sqrt[Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*Sqr 
t[a + b*Cos[c + d*x]]) - (2*b*C*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*Sqrt[a + 
b*Cos[c + d*x]]) + (2*b^3*C*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*Sqrt[a...
 

Rubi [A] (verified)

Time = 3.01 (sec) , antiderivative size = 582, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.356, Rules used = {3042, 4709, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 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.

\(\displaystyle \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sec (c+d x)^{7/2} \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )}{(a+b \cos (c+d x))^{3/2}}dx\)

\(\Big \downarrow \) 4709

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {C \cos ^2(c+d x)+B \cos (c+d x)+A}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 3534

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \int -\frac {-\left ((A-5 C) a^2\right )-5 b B a+(A b+C b-a B) \cos (c+d x) a+6 A b^2-4 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)}{2 \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\int \frac {-\left ((A-5 C) a^2\right )-5 b B a+(A b+C b-a B) \cos (c+d x) a+6 A b^2-4 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a \left (a^2-b^2\right )}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\int \frac {-\left ((A-5 C) a^2\right )-5 b B a+(A b+C b-a B) \sin \left (c+d x+\frac {\pi }{2}\right ) a+6 A b^2-4 \left (A b^2-a (b B-a C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}\right )\)

\(\Big \downarrow \) 3534

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \int -\frac {5 B a^3-3 b (3 A-5 C) a^2-20 b^2 B a+\left ((3 A+5 C) a^2-5 b B a+2 A b^2\right ) \cos (c+d x) a+24 A b^3-2 b \left (-\left ((A-5 C) a^2\right )-5 b B a+6 A b^2\right ) \cos ^2(c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}}{a \left (a^2-b^2\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {5 B a^3-(9 A b-15 b C) a^2-20 b^2 B a+\left ((3 A+5 C) a^2-5 b B a+2 A b^2\right ) \cos (c+d x) a+24 A b^3-2 b \left (-\left ((A-5 C) a^2\right )-5 b B a+6 A b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}}{a \left (a^2-b^2\right )}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {5 B a^3-(9 A b-15 b C) a^2-20 b^2 B a+\left ((3 A+5 C) a^2-5 b B a+2 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+24 A b^3-2 b \left (-\left ((A-5 C) a^2\right )-5 b B a+6 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}}{a \left (a^2-b^2\right )}\right )\)

\(\Big \downarrow \) 3534

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {-3 (3 A+5 C) a^4+25 b B a^3-6 b^2 (4 A-5 C) a^2-40 b^3 B a+\left (-5 B a^3+3 b (A+5 C) a^2-10 b^2 B a+12 A b^3\right ) \cos (c+d x) a+48 A b^4}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}}{a \left (a^2-b^2\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-3 (3 A+5 C) a^4+25 b B a^3-6 b^2 (4 A-5 C) a^2-40 b^3 B a+\left (-5 B a^3+3 b (A+5 C) a^2-10 b^2 B a+12 A b^3\right ) \cos (c+d x) a+48 A b^4}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-3 (3 A+5 C) a^4+25 b B a^3-6 b^2 (4 A-5 C) a^2-40 b^3 B a+\left (-5 B a^3+3 b (A+5 C) a^2-10 b^2 B a+12 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+48 A b^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\right )\)

\(\Big \downarrow \) 3477

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {(a-b) \left (a^3 (9 A-5 B+15 C)+6 a^2 b (2 A-5 B+5 C)+4 a b^2 (9 A-10 B)+48 A b^3\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {(a-b) \left (a^3 (9 A-5 B+15 C)+6 a^2 b (2 A-5 B+5 C)+4 a b^2 (9 A-10 B)+48 A b^3\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\right )\)

\(\Big \downarrow \) 3295

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (a^3 (9 A-5 B+15 C)+6 a^2 b (2 A-5 B+5 C)+4 a b^2 (9 A-10 B)+48 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\right )\)

\(\Big \downarrow \) 3473

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (a^3 (9 A-5 B+15 C)+6 a^2 b (2 A-5 B+5 C)+4 a b^2 (9 A-10 B)+48 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\right )\)

Input:

Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + b*Co 
s[c + d*x])^(3/2),x]
 

Output:

Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(A*b^2 - a*(b*B - a*C))*Sin[c + 
d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]) - ((2* 
(6*A*b^2 - 5*a*b*B - a^2*(A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x]) 
/(5*a*d*Cos[c + d*x]^(5/2)) - (-1/3*((2*(a - b)*Sqrt[a + b]*(48*A*b^4 + 25 
*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*Cot[c + 
 d*x]*EllipticE[ArcSin[Sqrt[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^2*d) + (2*(a - b)*Sqrt[a + b]*(48*A*b^3 + 
 4*a*b^2*(9*A - 10*B) + 6*a^2*b*(2*A - 5*B + 5*C) + a^3*(9*A - 5*B + 15*C) 
)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[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*(24*A*b^3 + 5*a^3*B - 
 20*a*b^2*B - 3*a^2*b*(3*A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/ 
(3*a*d*Cos[c + d*x]^(3/2)))/(5*a))/(a*(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 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 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 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 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. \(3479\) vs. \(2(535)=1070\).

Time = 64.16 (sec) , antiderivative size = 3480, normalized size of antiderivative = 5.95

method	result	size
parts	\(\text {Expression too large to display}\)	\(3480\)
default	\(\text {Expression too large to display}\)	\(3507\)

Input:

int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(7/2)/(a+b*cos(d*x+c))^(3/2 
),x,method=_RETURNVERBOSE)
 

Output:

2/5*A/d/(a+b)/(a-b)/a^4*((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*c 
os(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^5*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a 
-b)/(a+b))^(1/2))*(3*cos(d*x+c)^4+6*cos(d*x+c)^3+3*cos(d*x+c)^2)+(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 
^4*b*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(3*cos(d*x+c)^ 
4+6*cos(d*x+c)^3+3*cos(d*x+c)^2)+(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^2*EllipticE(-csc(d*x+c)+cot 
(d*x+c),(-(a-b)/(a+b))^(1/2))*(8*cos(d*x+c)^4+16*cos(d*x+c)^3+8*cos(d*x+c) 
^2)+(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^3*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2)) 
*(8*cos(d*x+c)^4+16*cos(d*x+c)^3+8*cos(d*x+c)^2)+(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*b^4*EllipticE(- 
csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(-16*cos(d*x+c)^4-32*cos(d*x+c 
)^3-16*cos(d*x+c)^2)+(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^5*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b)/ 
(a+b))^(1/2))*(-16*cos(d*x+c)^4-32*cos(d*x+c)^3-16*cos(d*x+c)^2)+(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*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(-3*cos(d*x+c)^4 
-6*cos(d*x+c)^3-3*cos(d*x+c)^2)+(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^4*b*EllipticF(-csc(d*x+c)+cot...
 

Fricas [F]

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

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(7/2)/(a+b*cos(d*x+c) 
)^(3/2),x, algorithm="fricas")
 

Output:

integral((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a)* 
sec(d*x + c)^(7/2)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**(7/2)/(a+b*cos(d*x+ 
c))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(7/2)/(a+b*cos(d*x+c) 
)^(3/2),x, algorithm="maxima")
 

Output:

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sec(d*x + c)^(7/2)/(b*co 
s(d*x + c) + a)^(3/2), x)
 

Giac [F]

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

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(7/2)/(a+b*cos(d*x+c) 
)^(3/2),x, algorithm="giac")
 

Output:

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sec(d*x + c)^(7/2)/(b*co 
s(d*x + c) + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:

int(((1/cos(c + d*x))^(7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + 
b*cos(c + d*x))^(3/2),x)
 

Output:

int(((1/cos(c + d*x))^(7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + 
b*cos(c + d*x))^(3/2), x)
 

Reduce [F]

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

int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(7/2)/(a+b*cos(d*x+c))^(3/2 
),x)
 

Output:

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