\(\int \frac {\cos (c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx\) [974]

Optimal result

Integrand size = 41, antiderivative size = 618 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=-\frac {\left (26 a^2 A b^2-15 A b^4-14 a^3 b B+6 a b^3 B-a^4 (3 A-8 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3 a^3 (a-b) b (a+b)^{3/2} d}-\frac {\left (15 A b^4+a b^3 (5 A-6 B)-a^2 b^2 (21 A+2 B)-6 a^4 C-a^3 b (3 A-2 (6 B+C))\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3 a^3 b \sqrt {a+b} \left (a^2-b^2\right ) d}+\frac {\sqrt {a+b} (5 A b-2 a B) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a^4 d}+\frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {b \left (5 A b^2-2 a b B-a^2 (3 A-2 C)\right ) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {b \left (26 a^2 A b^2-15 A b^4-14 a^3 b B+6 a b^3 B-a^4 (3 A-8 C)\right ) \tan (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}} \] Output:

-1/3*(26*A*a^2*b^2-15*A*b^4-14*B*a^3*b+6*B*a*b^3-a^4*(3*A-8*C))*cot(d*x+c) 
*EllipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-s 
ec(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^4+a*b^3*(5*A-6*B)-a^2*b^2*(21*A+2*B)-6*a^4*C-a^3*b*(3* 
A-12*B-2*C))*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-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^( 
1/2)/a^3/b/(a+b)^(1/2)/(a^2-b^2)/d+(a+b)^(1/2)*(5*A*b-2*B*a)*cot(d*x+c)*El 
lipticPi((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*(5*A*b^2-2*B*a*b-a^2*(3*A-2*C))*ta 
n(d*x+c)/a^2/(a^2-b^2)/d/(a+b*sec(d*x+c))^(3/2)-1/3*b*(26*A*a^2*b^2-15*A*b 
^4-14*B*a^3*b+6*B*a*b^3-a^4*(3*A-8*C))*tan(d*x+c)/a^3/(a^2-b^2)^2/d/(a+b*s 
ec(d*x+c))^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2607\) vs. \(2(618)=1236\).

Time = 22.21 (sec) , antiderivative size = 2607, normalized size of antiderivative = 4.22 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^ 
2)*((-4*(-10*a^2*A*b^2 + 6*A*b^4 + 7*a^3*b*B - 3*a*b^3*B - 4*a^4*C)*Sin[c 
+ d*x])/(3*a^3*(-a^2 + b^2)^2) + (4*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + 
d*x] + a^2*b^2*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) 
 + (4*(-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] - 5*a^4*b*C*Sin[c + d*x] + a^2*b^3*C*Si 
n[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2* 
B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) - (2*(b + 
 a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + 
d*x]^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*Tan[(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] - 8*a^5*C*Tan[(c 
 + d*x)/2] - 8*a^4*b*C*Tan[(c + d*x)/2] - 6*a^5*A*Tan[(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 + 16*a^5*C*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...
 

Rubi [A] (verified)

Time = 2.89 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.06, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.415, Rules used = {3042, 4592, 27, 3042, 4548, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4592

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4548

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4548

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {2 b \tan (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {2 \int -\frac {3 (5 A b-2 a B) \left (a^2-b^2\right )^2-b \left (-\left ((3 A-8 C) a^4\right )-14 b B a^3+26 A b^2 a^2+6 b^3 B a-15 A b^4\right ) \sec ^2(c+d x)+2 a \left (-3 C a^4+6 b B a^3-b^2 (9 A+C) a^2-2 b^3 B a+5 A b^4\right ) \sec (c+d x)}{2 \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {3 (5 A b-2 a B) \left (a^2-b^2\right )^2-b \left (-\left ((3 A-8 C) a^4\right )-14 b B a^3+26 A b^2 a^2+6 b^3 B a-15 A b^4\right ) \sec ^2(c+d x)+2 a \left (-3 C a^4+6 b B a^3-b^2 (9 A+C) a^2-2 b^3 B a+5 A b^4\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {3 (5 A b-2 a B) \left (a^2-b^2\right )^2-b \left (-\left ((3 A-8 C) a^4\right )-14 b B a^3+26 A b^2 a^2+6 b^3 B a-15 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (-3 C a^4+6 b B a^3-b^2 (9 A+C) a^2-2 b^3 B a+5 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 4546

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {3 (5 A b-2 a B) \left (a^2-b^2\right )^2+\left (b \left (-\left ((3 A-8 C) a^4\right )-14 b B a^3+26 A b^2 a^2+6 b^3 B a-15 A b^4\right )+2 a \left (-3 C a^4+6 b B a^3-b^2 (9 A+C) a^2-2 b^3 B a+5 A b^4\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-b \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {3 (5 A b-2 a B) \left (a^2-b^2\right )^2+\left (b \left (-\left ((3 A-8 C) a^4\right )-14 b B a^3+26 A b^2 a^2+6 b^3 B a-15 A b^4\right )+2 a \left (-3 C a^4+6 b B a^3-b^2 (9 A+C) a^2-2 b^3 B a+5 A b^4\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 4409

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {3 \left (a^2-b^2\right )^2 (5 A b-2 a B) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx-b \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+(a-b) \left (-6 a^4 C-a^3 b (3 A-2 (6 B+C))-a^2 b^2 (21 A+2 B)+a b^3 (5 A-6 B)+15 A b^4\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {3 \left (a^2-b^2\right )^2 (5 A b-2 a B) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+(a-b) \left (-6 a^4 C-a^3 b (3 A-2 (6 B+C))-a^2 b^2 (21 A+2 B)+a b^3 (5 A-6 B)+15 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 4271

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {(a-b) \left (-6 a^4 C-a^3 b (3 A-2 (6 B+C))-a^2 b^2 (21 A+2 B)+a b^3 (5 A-6 B)+15 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 (5 A b-2 a B) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 4319

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {-b \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 (5 A b-2 a B) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-6 a^4 C-a^3 b (3 A-2 (6 B+C))-a^2 b^2 (21 A+2 B)+a b^3 (5 A-6 B)+15 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}}{a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 b \tan (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}+\frac {\frac {-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 (5 A b-2 a B) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-6 a^4 C-a^3 b (3 A-2 (6 B+C))-a^2 b^2 (21 A+2 B)+a b^3 (5 A-6 B)+15 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}}{a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}}{2 a}\)

Input:

Int[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(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*(5*A*b^2 - 2*a*b 
*B - a^2*(3*A - 2*C))*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]*(26*a^2*A*b^2 - 15*A*b^4 - 14*a^3*b*B 
+ 6*a*b^3*B - a^4*(3*A - 8*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Se 
c[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))])/(b*d) + (2*(a - b)*Sqrt[a + 
 b]*(15*A*b^4 + a*b^3*(5*A - 6*B) - a^2*b^2*(21*A + 2*B) - 6*a^4*C - a^3*b 
*(3*A - 2*(6*B + C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x 
]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqr 
t[-((b*(1 + Sec[c + d*x]))/(a - b))])/(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 
*(26*a^2*A*b^2 - 15*A*b^4 - 14*a^3*b*B + 6*a*b^3*B - a^4*(3*A - 8*C))*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 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 4592

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))^(m_), 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] + Simp[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 + A*n + C*n)* 
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, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5012\) vs. \(2(576)=1152\).

Time = 16.65 (sec) , antiderivative size = 5013, normalized size of antiderivative = 8.11

method	result	size
default	\(\text {Expression too large to display}\)	\(5013\)

Input:

int(cos(d*x+c)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(5/2),x,me 
thod=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F]

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

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

Output:

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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