\(\int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx\) [754]

Optimal result

Integrand size = 27, antiderivative size = 626 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx=-\frac {2 \left (41 a^2 A b^4-15 A b^6-3 a^6 C-29 a^4 b^2 (2 A+C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{15 a^3 b^2 \sqrt {a+b} \left (a^2-b^2\right )^2 d}+\frac {2 \left (36 a^2 A b^3-5 a A b^4-15 A b^5+3 a^5 C+a^3 b^2 (13 A+5 C)-3 a^4 b (15 A+8 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{15 a^3 b \sqrt {a+b} \left (a^2-b^2\right )^2 d}-\frac {2 A \sqrt {a+b} \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a^4 d}+\frac {2 \left (A b^2+a^2 C\right ) \tan (c+d x)}{5 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{5/2}}-\frac {2 \left (5 A b^4-3 a^4 C-a^2 b^2 (13 A+5 C)\right ) \tan (c+d x)}{15 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (41 a^2 A b^4-15 A b^6-3 a^6 C-29 a^4 b^2 (2 A+C)\right ) \tan (c+d x)}{15 a^3 \left (a^2-b^2\right )^3 d \sqrt {a+b \sec (c+d x)}} \] Output:

-2/15*(41*a^2*A*b^4-15*A*b^6-3*a^6*C-29*a^4*b^2*(2*A+C))*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/b^2/(a+b)^(1/2)/(a^2- 
b^2)^2/d+2/15*(36*a^2*A*b^3-5*a*A*b^4-15*A*b^5+3*a^5*C+a^3*b^2*(13*A+5*C)- 
3*a^4*b*(15*A+8*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)^2/d-2*A*(a+b)^(1/2)*cot(d*x+c)*Ell 
ipticPi((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+2/5*(A* 
b^2+C*a^2)*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^(5/2)-2/15*(5*A*b^4-3 
*a^4*C-a^2*b^2*(13*A+5*C))*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^( 
3/2)-2/15*(41*a^2*A*b^4-15*A*b^6-3*a^6*C-29*a^4*b^2*(2*A+C))*tan(d*x+c)/a^ 
3/(a^2-b^2)^3/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. \(13317\) vs. \(2(626)=1252\).

Time = 24.55 (sec) , antiderivative size = 13317, normalized size of antiderivative = 21.27 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.81 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.07, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {3042, 4549, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4549

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4548

\(\displaystyle \frac {-\frac {2 \int -\frac {15 A \left (a^2-b^2\right )^2-\left (-3 C a^4-b^2 (13 A+5 C) a^2+5 A b^4\right ) \sec ^2(c+d x)+6 a b \left (A b^2-a^2 (5 A+4 C)\right ) \sec (c+d x)}{2 (a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {15 A \left (a^2-b^2\right )^2-\left (-3 C a^4-b^2 (13 A+5 C) a^2+5 A b^4\right ) \sec ^2(c+d x)+6 a b \left (A b^2-a^2 (5 A+4 C)\right ) \sec (c+d x)}{(a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {15 A \left (a^2-b^2\right )^2+\left (3 C a^4+b^2 (13 A+5 C) a^2-5 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+6 a b \left (A b^2-a^2 (5 A+4 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\)

\(\Big \downarrow \) 4548

\(\displaystyle \frac {\frac {-\frac {2 \int -\frac {15 A \left (a^2-b^2\right )^3+\left (-3 C a^6-29 b^2 (2 A+C) a^4+41 A b^4 a^2-15 A b^6\right ) \sec ^2(c+d x)-a b \left (9 (5 A+3 C) a^4-b^2 (23 A-5 C) a^2+10 A b^4\right ) \sec (c+d x)}{2 \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {15 A \left (a^2-b^2\right )^3+\left (-3 C a^6-29 b^2 (2 A+C) a^4+41 A b^4 a^2-15 A b^6\right ) \sec ^2(c+d x)-a b \left (9 (5 A+3 C) a^4-b^2 (23 A-5 C) a^2+10 A b^4\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {15 A \left (a^2-b^2\right )^3+\left (-3 C a^6-29 b^2 (2 A+C) a^4+41 A b^4 a^2-15 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-a b \left (9 (5 A+3 C) a^4-b^2 (23 A-5 C) a^2+10 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\)

\(\Big \downarrow \) 4546

\(\displaystyle \frac {\frac {\frac {\left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {15 A \left (a^2-b^2\right )^3+\left (3 C a^6+29 b^2 (2 A+C) a^4-41 A b^4 a^2-b \left (9 (5 A+3 C) a^4-b^2 (23 A-5 C) a^2+10 A b^4\right ) a+15 A b^6\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {15 A \left (a^2-b^2\right )^3+\left (3 C a^6+29 b^2 (2 A+C) a^4-41 A b^4 a^2-b \left (9 (5 A+3 C) a^4-b^2 (23 A-5 C) a^2+10 A b^4\right ) a+15 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\)

\(\Big \downarrow \) 4409

\(\displaystyle \frac {\frac {\frac {15 A \left (a^2-b^2\right )^3 \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx+\left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+(a-b) \left (3 a^5 C-3 a^4 b (15 A+8 C)+a^3 b^2 (13 A+5 C)+36 a^2 A b^3-5 a A b^4-15 A b^5\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {15 A \left (a^2-b^2\right )^3 \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+(a-b) \left (3 a^5 C-3 a^4 b (15 A+8 C)+a^3 b^2 (13 A+5 C)+36 a^2 A b^3-5 a A b^4-15 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\)

\(\Big \downarrow \) 4271

\(\displaystyle \frac {\frac {\frac {\left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+(a-b) \left (3 a^5 C-3 a^4 b (15 A+8 C)+a^3 b^2 (13 A+5 C)+36 a^2 A b^3-5 a A b^4-15 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {30 A \sqrt {a+b} \left (a^2-b^2\right )^3 \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\)

\(\Big \downarrow \) 4319

\(\displaystyle \frac {\frac {\frac {\left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {30 A \sqrt {a+b} \left (a^2-b^2\right )^3 \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}+\frac {2 (a-b) \sqrt {a+b} \left (3 a^5 C-3 a^4 b (15 A+8 C)+a^3 b^2 (13 A+5 C)+36 a^2 A b^3-5 a A b^4-15 A b^5\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}+\frac {\frac {\frac {-\frac {30 A \sqrt {a+b} \left (a^2-b^2\right )^3 \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}-\frac {2 (a-b) \sqrt {a+b} \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 d}+\frac {2 (a-b) \sqrt {a+b} \left (3 a^5 C-3 a^4 b (15 A+8 C)+a^3 b^2 (13 A+5 C)+36 a^2 A b^3-5 a A b^4-15 A b^5\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}\)

Input:

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

Output:

(2*(A*b^2 + a^2*C)*Tan[c + d*x])/(5*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^( 
5/2)) + ((-2*(5*A*b^4 - 3*a^4*C - a^2*b^2*(13*A + 5*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]*(41 
*a^2*A*b^4 - 15*A*b^6 - 3*a^6*C - 29*a^4*b^2*(2*A + C))*Cot[c + d*x]*Ellip 
ticE[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))])/(b 
^2*d) + (2*(a - b)*Sqrt[a + b]*(36*a^2*A*b^3 - 5*a*A*b^4 - 15*A*b^5 + 3*a^ 
5*C + a^3*b^2*(13*A + 5*C) - 3*a^4*b*(15*A + 8*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)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - 
 (30*A*Sqrt[a + b]*(a^2 - b^2)^3*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*(41*a^2*A*b^4 - 15*A*b^6 - 3*a^6*C - 29*a^4*b^2*(2*A + C))* 
Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]))/(3*a*(a^2 - b^2) 
))/(5*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 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 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] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4918\) vs. \(2(583)=1166\).

Time = 39.41 (sec) , antiderivative size = 4919, normalized size of antiderivative = 7.86

method	result	size
default	\(\text {Expression too large to display}\)	\(4919\)
parts	\(\text {Expression too large to display}\)	\(4957\)

Input:

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

Output:

2/15/d/(a-b)^3/(a+b)^3/b/a^3*(sin(d*x+c)*cos(d*x+c)*(35*cos(d*x+c)-10)*A*a 
*b^8+sin(d*x+c)*cos(d*x+c)^2*(-29*cos(d*x+c)+9)*C*a^7*b^2+sin(d*x+c)*cos(d 
*x+c)*(25*cos(d*x+c)^2-62*cos(d*x+c)+3)*C*a^6*b^3+sin(d*x+c)*cos(d*x+c)*(6 
0*cos(d*x+c)-27)*C*a^5*b^4+sin(d*x+c)*cos(d*x+c)*(-2*cos(d*x+c)^2-2*cos(d* 
x+c)+29)*C*a^4*b^5+sin(d*x+c)*cos(d*x+c)*(-5*cos(d*x+c)-5)*C*a^3*b^6+(-15* 
cos(d*x+c)^2-30*cos(d*x+c)-15)*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1)) 
^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*b^9*EllipticE(-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))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*b^9* 
EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+sin(d*x+c)*cos(d 
*x+c)^2*(74*cos(d*x+c)-100)*A*a^6*b^3+sin(d*x+c)*cos(d*x+c)*(41*cos(d*x+c) 
^2+129*cos(d*x+c)-45)*A*a^5*b^4+sin(d*x+c)*cos(d*x+c)*(-65*cos(d*x+c)^2+58 
*cos(d*x+c)+58)*A*a^4*b^5+C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2 
)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^9*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)+sin(d* 
x+c)*cos(d*x+c)*(-15*cos(d*x+c)^2-100*cos(d*x+c)+23)*A*a^3*b^6-58*A*a^7*b^ 
2*cos(d*x+c)^3*sin(d*x+c)+9*C*a^8*b*cos(d*x+c)^3*sin(d*x+c)-3*C*a^9*cos(d* 
x+c)^3*sin(d*x+c)+15*A*b^9*cos(d*x+c)*sin(d*x+c)+(90*cos(d*x+c)^2+180*cos( 
d*x+c)+90)*sin(d*x+c)^2*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)* 
(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^4*b^5*EllipticPi(-csc(d*x+c)+cot(d*...
 

Fricas [F]

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

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

Output:

integral((C*sec(d*x + c)^2 + A)*sqrt(b*sec(d*x + c) + a)/(b^4*sec(d*x + c) 
^4 + 4*a*b^3*sec(d*x + c)^3 + 6*a^2*b^2*sec(d*x + c)^2 + 4*a^3*b*sec(d*x + 
 c) + a^4), x)
 

Sympy [F]

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

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

Output:

Integral((A + C*sec(c + d*x)**2)/(a + b*sec(c + d*x))**(7/2), x)
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(sec(c + d*x)*b + a)/(sec(c + d*x)**4*b**4 + 4*sec(c + d*x)**3*a*b 
**3 + 6*sec(c + d*x)**2*a**2*b**2 + 4*sec(c + d*x)*a**3*b + a**4),x)*a + i 
nt((sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**2)/(sec(c + d*x)**4*b**4 + 4*se 
c(c + d*x)**3*a*b**3 + 6*sec(c + d*x)**2*a**2*b**2 + 4*sec(c + d*x)*a**3*b 
 + a**4),x)*c