\(\int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx\) [390]

Optimal result

Integrand size = 31, antiderivative size = 582 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx=\frac {\left (3 a^4 A-26 a^2 A b^2+15 A b^4+14 a^3 b B-6 a b^3 B\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3 a^3 (a-b) b (a+b)^{3/2} d}-\frac {\left (15 A b^3+a b^2 (5 A-6 B)-3 a^3 (A-4 B)-a^2 b (21 A+2 B)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3 a^3 \sqrt {a+b} \left (a^2-b^2\right ) d}+\frac {\sqrt {a+b} (5 A b-2 a B) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a^4 d}+\frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}+\frac {b \left (3 a^2 A-5 A b^2+2 a b B\right ) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac {b \left (3 a^4 A-26 a^2 A b^2+15 A b^4+14 a^3 b B-6 a b^3 B\right ) \tan (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}} \] Output:

1/3*(3*A*a^4-26*A*a^2*b^2+15*A*b^4+14*B*a^3*b-6*B*a*b^3)*cot(d*x+c)*Ellipt 
icE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+ 
c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^3/(a-b)/b/(a+b)^(3/2)/d 
-1/3*(15*A*b^3+a*b^2*(5*A-6*B)-3*a^3*(A-4*B)-a^2*b*(21*A+2*B))*cot(d*x+c)* 
EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-se 
c(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^3/(a+b)^(1/2)/(a^ 
2-b^2)/d+(a+b)^(1/2)*(5*A*b-2*B*a)*cot(d*x+c)*EllipticPi((a+b*sec(d*x+c))^ 
(1/2)/(a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1 
/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^4/d+A*sin(d*x+c)/a/d/(a+b*sec(d*x+c) 
)^(3/2)+1/3*b*(3*A*a^2-5*A*b^2+2*B*a*b)*tan(d*x+c)/a^2/(a^2-b^2)/d/(a+b*se 
c(d*x+c))^(3/2)+1/3*b*(3*A*a^4-26*A*a^2*b^2+15*A*b^4+14*B*a^3*b-6*B*a*b^3) 
*tan(d*x+c)/a^3/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2366\) vs. \(2(582)=1164\).

Time = 19.98 (sec) , antiderivative size = 2366, normalized size of antiderivative = 4.07 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((-2*b*(-10*a^2*A*b + 6*A*b^3 + 7*a 
^3*B - 3*a*b^2*B)*Sin[c + d*x])/(3*a^3*(-a^2 + b^2)^2) + (2*(A*b^4*Sin[c + 
 d*x] - a*b^3*B*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) 
+ (2*(-11*a^2*A*b^3*Sin[c + d*x] + 7*A*b^5*Sin[c + d*x] + 8*a^3*b^2*B*Sin[ 
c + d*x] - 4*a*b^4*B*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d* 
x]))))/(d*(a + b*Sec[c + d*x])^(5/2)) - ((b + a*Cos[c + d*x])^(5/2)*Sec[c 
+ d*x]^(5/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + 
d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(3*a^5*A*Tan[( 
c + d*x)/2] + 3*a^4*A*b*Tan[(c + d*x)/2] - 26*a^3*A*b^2*Tan[(c + d*x)/2] - 
 26*a^2*A*b^3*Tan[(c + d*x)/2] + 15*a*A*b^4*Tan[(c + d*x)/2] + 15*A*b^5*Ta 
n[(c + d*x)/2] + 14*a^4*b*B*Tan[(c + d*x)/2] + 14*a^3*b^2*B*Tan[(c + d*x)/ 
2] - 6*a^2*b^3*B*Tan[(c + d*x)/2] - 6*a*b^4*B*Tan[(c + d*x)/2] - 6*a^5*A*T 
an[(c + d*x)/2]^3 + 52*a^3*A*b^2*Tan[(c + d*x)/2]^3 - 30*a*A*b^4*Tan[(c + 
d*x)/2]^3 - 28*a^4*b*B*Tan[(c + d*x)/2]^3 + 12*a^2*b^3*B*Tan[(c + d*x)/2]^ 
3 + 3*a^5*A*Tan[(c + d*x)/2]^5 - 3*a^4*A*b*Tan[(c + d*x)/2]^5 - 26*a^3*A*b 
^2*Tan[(c + d*x)/2]^5 + 26*a^2*A*b^3*Tan[(c + d*x)/2]^5 + 15*a*A*b^4*Tan[( 
c + d*x)/2]^5 - 15*A*b^5*Tan[(c + d*x)/2]^5 + 14*a^4*b*B*Tan[(c + d*x)/2]^ 
5 - 14*a^3*b^2*B*Tan[(c + d*x)/2]^5 - 6*a^2*b^3*B*Tan[(c + d*x)/2]^5 + 6*a 
*b^4*B*Tan[(c + d*x)/2]^5 - 30*a^4*A*b*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...
 

Rubi [A] (verified)

Time = 2.67 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.06, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {3042, 4522, 27, 3042, 4549, 27, 3042, 4548, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\)

\(\Big \downarrow \) 4522

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\int \frac {-3 A b \sec ^2(c+d x)+5 A b-2 a B}{2 (a+b \sec (c+d x))^{5/2}}dx}{a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\int \frac {-3 A b \sec ^2(c+d x)+5 A b-2 a B}{(a+b \sec (c+d x))^{5/2}}dx}{2 a}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4549

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {-\frac {2 \int -\frac {-b \left (3 A a^2+2 b B a-5 A b^2\right ) \sec ^2(c+d x)-6 a b (A b-a B) \sec (c+d x)+3 \left (a^2-b^2\right ) (5 A b-2 a B)}{2 (a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^2 A+2 a b B-5 A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\int \frac {-b \left (3 A a^2+2 b B a-5 A b^2\right ) \sec ^2(c+d x)-6 a b (A b-a B) \sec (c+d x)+3 \left (a^2-b^2\right ) (5 A b-2 a B)}{(a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^2 A+2 a b B-5 A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\int \frac {-b \left (3 A a^2+2 b B a-5 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-6 a b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (a^2-b^2\right ) (5 A b-2 a B)}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^2 A+2 a b B-5 A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 4548

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {-\frac {2 \int -\frac {3 (5 A b-2 a B) \left (a^2-b^2\right )^2+b \left (3 A a^4+14 b B a^3-26 A b^2 a^2-6 b^3 B a+15 A b^4\right ) \sec ^2(c+d x)-2 a b \left (-6 B a^3+9 A b a^2+2 b^2 B a-5 A b^3\right ) \sec (c+d x)}{2 \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^2 A+2 a b B-5 A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {3 (5 A b-2 a B) \left (a^2-b^2\right )^2+b \left (3 A a^4+14 b B a^3-26 A b^2 a^2-6 b^3 B a+15 A b^4\right ) \sec ^2(c+d x)-2 a b \left (-6 B a^3+9 A b a^2+2 b^2 B a-5 A b^3\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^2 A+2 a b B-5 A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {3 (5 A b-2 a B) \left (a^2-b^2\right )^2+b \left (3 A a^4+14 b B a^3-26 A b^2 a^2-6 b^3 B a+15 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left (-6 B a^3+9 A b a^2+2 b^2 B a-5 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^2 A+2 a b B-5 A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 4546

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {3 (5 A b-2 a B) \left (a^2-b^2\right )^2+\left (-2 a b \left (-6 B a^3+9 A b a^2+2 b^2 B a-5 A b^3\right )-b \left (3 A a^4+14 b B a^3-26 A b^2 a^2-6 b^3 B a+15 A b^4\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^2 A+2 a b B-5 A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {3 (5 A b-2 a B) \left (a^2-b^2\right )^2+\left (-2 a b \left (-6 B a^3+9 A b a^2+2 b^2 B a-5 A b^3\right )-b \left (3 A a^4+14 b B a^3-26 A b^2 a^2-6 b^3 B a+15 A b^4\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^2 A+2 a b B-5 A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 4409

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {3 \left (a^2-b^2\right )^2 (5 A b-2 a B) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx+b (a-b) \left (-3 a^3 (A-4 B)-a^2 b (21 A+2 B)+a b^2 (5 A-6 B)+15 A b^3\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx+b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^2 A+2 a b B-5 A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {3 \left (a^2-b^2\right )^2 (5 A b-2 a B) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b (a-b) \left (-3 a^3 (A-4 B)-a^2 b (21 A+2 B)+a b^2 (5 A-6 B)+15 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^2 A+2 a b B-5 A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 4271

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {b (a-b) \left (-3 a^3 (A-4 B)-a^2 b (21 A+2 B)+a b^2 (5 A-6 B)+15 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 (5 A b-2 a B) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^2 A+2 a b B-5 A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 4319

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 (5 A b-2 a B) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}+\frac {2 (a-b) \sqrt {a+b} \left (-3 a^3 (A-4 B)-a^2 b (21 A+2 B)+a b^2 (5 A-6 B)+15 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^2 A+2 a b B-5 A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 (5 A b-2 a B) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}+\frac {2 (a-b) \sqrt {a+b} \left (-3 a^3 (A-4 B)-a^2 b (21 A+2 B)+a b^2 (5 A-6 B)+15 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {2 (a-b) \sqrt {a+b} \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}}{a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 b \left (3 a^2 A+2 a b B-5 A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

Input:

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

Output:

(A*Sin[c + d*x])/(a*d*(a + b*Sec[c + d*x])^(3/2)) - ((-2*b*(3*a^2*A - 5*A* 
b^2 + 2*a*b*B)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2) 
) + (((-2*(a - b)*Sqrt[a + b]*(3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14*a^3* 
b*B - 6*a*b^3*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sq 
rt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-(( 
b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (2*(a - b)*Sqrt[a + b]*(15*A*b^3 
+ a*b^2*(5*A - 6*B) - 3*a^3*(A - 4*B) - a^2*b*(21*A + 2*B))*Cot[c + d*x]*E 
llipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sq 
rt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))] 
)/d - (6*Sqrt[a + b]*(a^2 - b^2)^2*(5*A*b - 2*a*B)*Cot[c + d*x]*EllipticPi 
[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)] 
*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b 
))])/(a*d))/(a*(a^2 - b^2)) - (2*b*(3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14 
*a^3*b*B - 6*a*b^3*B)*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d* 
x]]))/(3*a*(a^2 - b^2)))/(2*a)
 

Defintions of rubi rules used

rule 27

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

rule 3042

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

rule 4271

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a 
 + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) 
*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ 
c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && 
NeQ[a^2 - b^2, 0]
 

rule 4319

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* 
x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt 
[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, 
 f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4409

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ 
.) + (a_)], x_Symbol] :> Simp[c   Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + 
Simp[d   Int[Csc[e + f*x]/Sqrt[a + b*Csc[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 4492

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c 
sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a 
 + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e 
+ f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + 
 f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, 
 f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
 

rule 4522

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( 
a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*Cot[e 
 + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*n)), x] + Sim 
p[1/(a*d*n)   Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*B* 
n - A*b*(m + n + 1) + A*a*(n + 1)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f* 
x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] 
 && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
 

rule 4546

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

rule 4548

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

rule 4549

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4147\) vs. \(2(541)=1082\).

Time = 13.97 (sec) , antiderivative size = 4148, normalized size of antiderivative = 7.13

method	result	size
default	\(\text {Expression too large to display}\)	\(4148\)

Input:

int(cos(d*x+c)*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(5/2),x,method=_RETURNVER 
BOSE)
                                                                                    
                                                                                    
 

Output:

1/3/d/(a-b)^2/(a+b)^2/a^3*(6*A*a^5*b*cos(d*x+c)^2*sin(d*x+c)-6*B*a*b^5*cos 
(d*x+c)*sin(d*x+c)+B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos( 
d*x+c)/(1+cos(d*x+c)))^(1/2)*a^6*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/( 
a+b))^(1/2))*(6*cos(d*x+c)^3+12*cos(d*x+c)^2+6*cos(d*x+c))+3*A*a^6*cos(d*x 
+c)^3*sin(d*x+c)+15*A*b^6*cos(d*x+c)*sin(d*x+c)-14*B*a^5*b*cos(d*x+c)^2*si 
n(d*x+c)+(10*cos(d*x+c)^2+20*cos(d*x+c)+10)*A*(1/(a+b)*(b+a*cos(d*x+c))/(1 
+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a*b^5*EllipticF(-csc 
(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+(12*cos(d*x+c)^3+30*cos(d*x+c)^2+2 
4*cos(d*x+c)+6)*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x 
+c)/(1+cos(d*x+c)))^(1/2)*a^5*b*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/(a 
+b))^(1/2))+(2*cos(d*x+c)^3+16*cos(d*x+c)^2+26*cos(d*x+c)+12)*B*(1/(a+b)*( 
b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^ 
4*b^2*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+(-4*cos(d*x+c) 
^3-6*cos(d*x+c)^2+2)*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(co 
s(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))+(-4*cos(d*x+c)^2-8*cos(d*x+c)-4)*B*(1/(a+b)*(b+a*cos(d* 
x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^2*b^4*Elli 
pticF(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+(30*cos(d*x+c)^2+60*cos( 
d*x+c)+30)*A*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(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...
 

Fricas [F]

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

integrate(cos(d*x+c)*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(5/2),x, algorithm= 
"fricas")
 

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(cos(d*x+c)*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(5/2),x, algorithm= 
"maxima")
 

Output:

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

Giac [F]

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

integrate(cos(d*x+c)*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(5/2),x, algorithm= 
"giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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