\(\int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx\) [934]

Optimal result

Integrand size = 48, antiderivative size = 336 \[ \int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx=\frac {(b B-a C) x}{a^4}-\frac {b \left (8 a^6 b B-8 a^4 b^3 B+7 a^2 b^5 B-2 b^7 B-10 a^7 C+5 a^5 b^2 C-7 a^3 b^4 C+2 a b^6 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4 b B-17 a^2 b^3 B+6 b^5 B-37 a^5 C+13 a^3 b^2 C-6 a b^4 C\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \] Output:

(B*b-C*a)*x/a^4-b*(8*B*a^6*b-8*B*a^4*b^3+7*B*a^2*b^5-2*B*b^7-10*C*a^7+5*C* 
a^5*b^2-7*C*a^3*b^4+2*C*a*b^6)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b 
)^(1/2))/a^4/(a-b)^(7/2)/(a+b)^(7/2)/d+1/3*b^2*(B*b-2*C*a)*tan(d*x+c)/a/(a 
^2-b^2)/d/(a+b*sec(d*x+c))^3+1/6*b^2*(8*B*a^2*b-3*B*b^3-13*C*a^3+3*C*a*b^2 
)*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2+1/6*b^2*(26*B*a^4*b-17*B 
*a^2*b^3+6*B*b^5-37*C*a^5+13*C*a^3*b^2-6*C*a*b^4)*tan(d*x+c)/a^3/(a^2-b^2) 
^3/d/(a+b*sec(d*x+c))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1097\) vs. \(2(336)=672\).

Time = 5.80 (sec) , antiderivative size = 1097, normalized size of antiderivative = 3.26 \[ \int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx=\frac {(b+a \cos (c+d x)) \sec ^3(c+d x) (b B-a C+b C \sec (c+d x)) \left (\frac {24 b \left (8 a^6 b B-8 a^4 b^3 B+7 a^2 b^5 B-2 b^7 B-10 a^7 C+5 a^5 b^2 C-7 a^3 b^4 C+2 a b^6 C\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^3}{\left (a^2-b^2\right )^{7/2}}+\frac {36 a^8 b^2 B c-84 a^6 b^4 B c+36 a^4 b^6 B c+36 a^2 b^8 B c-24 b^{10} B c-36 a^9 b c C+84 a^7 b^3 c C-36 a^5 b^5 c C-36 a^3 b^7 c C+24 a b^9 c C+36 a^8 b^2 B d x-84 a^6 b^4 B d x+36 a^4 b^6 B d x+36 a^2 b^8 B d x-24 b^{10} B d x-36 a^9 b C d x+84 a^7 b^3 C d x-36 a^5 b^5 C d x-36 a^3 b^7 C d x+24 a b^9 C d x-18 a \left (a^2-b^2\right )^3 \left (a^2+4 b^2\right ) (-b B+a C) (c+d x) \cos (c+d x)-36 a^2 b \left (a^2-b^2\right )^3 (-b B+a C) (c+d x) \cos (2 (c+d x))+6 a^9 b B c \cos (3 (c+d x))-18 a^7 b^3 B c \cos (3 (c+d x))+18 a^5 b^5 B c \cos (3 (c+d x))-6 a^3 b^7 B c \cos (3 (c+d x))-6 a^{10} c C \cos (3 (c+d x))+18 a^8 b^2 c C \cos (3 (c+d x))-18 a^6 b^4 c C \cos (3 (c+d x))+6 a^4 b^6 c C \cos (3 (c+d x))+6 a^9 b B d x \cos (3 (c+d x))-18 a^7 b^3 B d x \cos (3 (c+d x))+18 a^5 b^5 B d x \cos (3 (c+d x))-6 a^3 b^7 B d x \cos (3 (c+d x))-6 a^{10} C d x \cos (3 (c+d x))+18 a^8 b^2 C d x \cos (3 (c+d x))-18 a^6 b^4 C d x \cos (3 (c+d x))+6 a^4 b^6 C d x \cos (3 (c+d x))+36 a^7 b^3 B \sin (c+d x)+72 a^5 b^5 B \sin (c+d x)-57 a^3 b^7 B \sin (c+d x)+24 a b^9 B \sin (c+d x)-54 a^8 b^2 C \sin (c+d x)-111 a^6 b^4 C \sin (c+d x)+39 a^4 b^6 C \sin (c+d x)-24 a^2 b^8 C \sin (c+d x)+120 a^6 b^4 B \sin (2 (c+d x))-90 a^4 b^6 B \sin (2 (c+d x))+30 a^2 b^8 B \sin (2 (c+d x))-174 a^7 b^3 C \sin (2 (c+d x))+84 a^5 b^5 C \sin (2 (c+d x))-30 a^3 b^7 C \sin (2 (c+d x))+36 a^7 b^3 B \sin (3 (c+d x))-32 a^5 b^5 B \sin (3 (c+d x))+11 a^3 b^7 B \sin (3 (c+d x))-54 a^8 b^2 C \sin (3 (c+d x))+37 a^6 b^4 C \sin (3 (c+d x))-13 a^4 b^6 C \sin (3 (c+d x))}{\left (a^2-b^2\right )^3}\right )}{24 a^4 d (b C+(b B-a C) \cos (c+d x)) (a+b \sec (c+d x))^4} \] Input:

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

Output:

((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(b*B - a*C + b*C*Sec[c + d*x])*((24*b 
*(8*a^6*b*B - 8*a^4*b^3*B + 7*a^2*b^5*B - 2*b^7*B - 10*a^7*C + 5*a^5*b^2*C 
 - 7*a^3*b^4*C + 2*a*b^6*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - 
 b^2]]*(b + a*Cos[c + d*x])^3)/(a^2 - b^2)^(7/2) + (36*a^8*b^2*B*c - 84*a^ 
6*b^4*B*c + 36*a^4*b^6*B*c + 36*a^2*b^8*B*c - 24*b^10*B*c - 36*a^9*b*c*C + 
 84*a^7*b^3*c*C - 36*a^5*b^5*c*C - 36*a^3*b^7*c*C + 24*a*b^9*c*C + 36*a^8* 
b^2*B*d*x - 84*a^6*b^4*B*d*x + 36*a^4*b^6*B*d*x + 36*a^2*b^8*B*d*x - 24*b^ 
10*B*d*x - 36*a^9*b*C*d*x + 84*a^7*b^3*C*d*x - 36*a^5*b^5*C*d*x - 36*a^3*b 
^7*C*d*x + 24*a*b^9*C*d*x - 18*a*(a^2 - b^2)^3*(a^2 + 4*b^2)*(-(b*B) + a*C 
)*(c + d*x)*Cos[c + d*x] - 36*a^2*b*(a^2 - b^2)^3*(-(b*B) + a*C)*(c + d*x) 
*Cos[2*(c + d*x)] + 6*a^9*b*B*c*Cos[3*(c + d*x)] - 18*a^7*b^3*B*c*Cos[3*(c 
 + d*x)] + 18*a^5*b^5*B*c*Cos[3*(c + d*x)] - 6*a^3*b^7*B*c*Cos[3*(c + d*x) 
] - 6*a^10*c*C*Cos[3*(c + d*x)] + 18*a^8*b^2*c*C*Cos[3*(c + d*x)] - 18*a^6 
*b^4*c*C*Cos[3*(c + d*x)] + 6*a^4*b^6*c*C*Cos[3*(c + d*x)] + 6*a^9*b*B*d*x 
*Cos[3*(c + d*x)] - 18*a^7*b^3*B*d*x*Cos[3*(c + d*x)] + 18*a^5*b^5*B*d*x*C 
os[3*(c + d*x)] - 6*a^3*b^7*B*d*x*Cos[3*(c + d*x)] - 6*a^10*C*d*x*Cos[3*(c 
 + d*x)] + 18*a^8*b^2*C*d*x*Cos[3*(c + d*x)] - 18*a^6*b^4*C*d*x*Cos[3*(c + 
 d*x)] + 6*a^4*b^6*C*d*x*Cos[3*(c + d*x)] + 36*a^7*b^3*B*Sin[c + d*x] + 72 
*a^5*b^5*B*Sin[c + d*x] - 57*a^3*b^7*B*Sin[c + d*x] + 24*a*b^9*B*Sin[c + d 
*x] - 54*a^8*b^2*C*Sin[c + d*x] - 111*a^6*b^4*C*Sin[c + d*x] + 39*a^4*b...
 

Rubi [A] (verified)

Time = 1.95 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.22, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.354, Rules used = {2014, 3042, 4411, 25, 3042, 4548, 25, 3042, 4548, 27, 3042, 4407, 3042, 4318, 3042, 3138, 221}

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.

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

\(\Big \downarrow \) 2014

\(\displaystyle \frac {\int \frac {C \sec (c+d x) b^3+(b B-a C) b^2}{(a+b \sec (c+d x))^4}dx}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {C \csc \left (c+d x+\frac {\pi }{2}\right ) b^3+(b B-a C) b^2}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx}{b^2}\)

\(\Big \downarrow \) 4411

\(\displaystyle \frac {\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\int -\frac {2 (b B-2 a C) \sec ^2(c+d x) b^4-3 a (b B-2 a C) \sec (c+d x) b^3+3 \left (a^2-b^2\right ) (b B-a C) b^2}{(a+b \sec (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}}{b^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {2 (b B-2 a C) \sec ^2(c+d x) b^4-3 a (b B-2 a C) \sec (c+d x) b^3+3 \left (a^2-b^2\right ) (b B-a C) b^2}{(a+b \sec (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}+\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {2 (b B-2 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^4-3 a (b B-2 a C) \csc \left (c+d x+\frac {\pi }{2}\right ) b^3+3 \left (a^2-b^2\right ) (b B-a C) b^2}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 a \left (a^2-b^2\right )}+\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}}{b^2}\)

\(\Big \downarrow \) 4548

\(\displaystyle \frac {\frac {\frac {b^4 \left (-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int -\frac {\left (-13 C a^3+8 b B a^2+3 b^2 C a-3 b^3 B\right ) \sec ^2(c+d x) b^4-2 a \left (-9 C a^3+6 b B a^2-b^2 C a-b^3 B\right ) \sec (c+d x) b^3+6 \left (a^2-b^2\right )^2 (b B-a C) b^2}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}+\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}}{b^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (-13 C a^3+8 b B a^2+3 b^2 C a-3 b^3 B\right ) \sec ^2(c+d x) b^4-2 a \left (-9 C a^3+6 b B a^2-b^2 C a-b^3 B\right ) \sec (c+d x) b^3+6 \left (a^2-b^2\right )^2 (b B-a C) b^2}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b^4 \left (-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (-13 C a^3+8 b B a^2+3 b^2 C a-3 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^4-2 a \left (-9 C a^3+6 b B a^2-b^2 C a-b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) b^3+6 \left (a^2-b^2\right )^2 (b B-a C) b^2}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b^4 \left (-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}}{b^2}\)

\(\Big \downarrow \) 4548

\(\displaystyle \frac {\frac {\frac {\frac {b^4 \left (-37 a^5 C+26 a^4 b B+13 a^3 b^2 C-17 a^2 b^3 B-6 a b^4 C+6 b^5 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int -\frac {3 \left (2 b^2 \left (a^2-b^2\right )^3 (b B-a C)-a b^3 \left (-8 C a^5+6 b B a^4-b^2 C a^3-2 b^3 B a^2-b^4 C a+b^5 B\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {b^4 \left (-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}}{b^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {3 \int \frac {2 b^2 \left (a^2-b^2\right )^3 (b B-a C)-a b^3 \left (-8 C a^5+6 b B a^4-b^2 C a^3-2 b^3 B a^2-b^4 C a+b^5 B\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {b^4 \left (-37 a^5 C+26 a^4 b B+13 a^3 b^2 C-17 a^2 b^3 B-6 a b^4 C+6 b^5 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^4 \left (-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {3 \int \frac {2 b^2 \left (a^2-b^2\right )^3 (b B-a C)-a b^3 \left (-8 C a^5+6 b B a^4-b^2 C a^3-2 b^3 B a^2-b^4 C a+b^5 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a \left (a^2-b^2\right )}+\frac {b^4 \left (-37 a^5 C+26 a^4 b B+13 a^3 b^2 C-17 a^2 b^3 B-6 a b^4 C+6 b^5 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^4 \left (-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}}{b^2}\)

\(\Big \downarrow \) 4407

\(\displaystyle \frac {\frac {\frac {\frac {3 \left (\frac {2 b^2 x \left (a^2-b^2\right )^3 (b B-a C)}{a}-\frac {b^3 \left (-10 a^7 C+8 a^6 b B+5 a^5 b^2 C-8 a^4 b^3 B-7 a^3 b^4 C+7 a^2 b^5 B+2 a b^6 C-2 b^7 B\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}\right )}{a \left (a^2-b^2\right )}+\frac {b^4 \left (-37 a^5 C+26 a^4 b B+13 a^3 b^2 C-17 a^2 b^3 B-6 a b^4 C+6 b^5 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^4 \left (-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {3 \left (\frac {2 b^2 x \left (a^2-b^2\right )^3 (b B-a C)}{a}-\frac {b^3 \left (-10 a^7 C+8 a^6 b B+5 a^5 b^2 C-8 a^4 b^3 B-7 a^3 b^4 C+7 a^2 b^5 B+2 a b^6 C-2 b^7 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a \left (a^2-b^2\right )}+\frac {b^4 \left (-37 a^5 C+26 a^4 b B+13 a^3 b^2 C-17 a^2 b^3 B-6 a b^4 C+6 b^5 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^4 \left (-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}}{b^2}\)

\(\Big \downarrow \) 4318

\(\displaystyle \frac {\frac {\frac {\frac {3 \left (\frac {2 b^2 x \left (a^2-b^2\right )^3 (b B-a C)}{a}-\frac {b^2 \left (-10 a^7 C+8 a^6 b B+5 a^5 b^2 C-8 a^4 b^3 B-7 a^3 b^4 C+7 a^2 b^5 B+2 a b^6 C-2 b^7 B\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a}\right )}{a \left (a^2-b^2\right )}+\frac {b^4 \left (-37 a^5 C+26 a^4 b B+13 a^3 b^2 C-17 a^2 b^3 B-6 a b^4 C+6 b^5 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^4 \left (-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {3 \left (\frac {2 b^2 x \left (a^2-b^2\right )^3 (b B-a C)}{a}-\frac {b^2 \left (-10 a^7 C+8 a^6 b B+5 a^5 b^2 C-8 a^4 b^3 B-7 a^3 b^4 C+7 a^2 b^5 B+2 a b^6 C-2 b^7 B\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a}\right )}{a \left (a^2-b^2\right )}+\frac {b^4 \left (-37 a^5 C+26 a^4 b B+13 a^3 b^2 C-17 a^2 b^3 B-6 a b^4 C+6 b^5 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^4 \left (-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}}{b^2}\)

\(\Big \downarrow \) 3138

\(\displaystyle \frac {\frac {\frac {\frac {3 \left (\frac {2 b^2 x \left (a^2-b^2\right )^3 (b B-a C)}{a}-\frac {2 b^2 \left (-10 a^7 C+8 a^6 b B+5 a^5 b^2 C-8 a^4 b^3 B-7 a^3 b^4 C+7 a^2 b^5 B+2 a b^6 C-2 b^7 B\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a \left (a^2-b^2\right )}+\frac {b^4 \left (-37 a^5 C+26 a^4 b B+13 a^3 b^2 C-17 a^2 b^3 B-6 a b^4 C+6 b^5 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^4 \left (-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}}{b^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {\frac {b^4 \left (-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\frac {b^4 \left (-37 a^5 C+26 a^4 b B+13 a^3 b^2 C-17 a^2 b^3 B-6 a b^4 C+6 b^5 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {3 \left (\frac {2 b^2 x \left (a^2-b^2\right )^3 (b B-a C)}{a}-\frac {2 b^3 \left (-10 a^7 C+8 a^6 b B+5 a^5 b^2 C-8 a^4 b^3 B-7 a^3 b^4 C+7 a^2 b^5 B+2 a b^6 C-2 b^7 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}}{b^2}\)

Input:

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

Output:

((b^4*(b*B - 2*a*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^ 
3) + ((b^4*(8*a^2*b*B - 3*b^3*B - 13*a^3*C + 3*a*b^2*C)*Tan[c + d*x])/(2*a 
*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((3*((2*b^2*(a^2 - b^2)^3*(b*B - 
a*C)*x)/a - (2*b^3*(8*a^6*b*B - 8*a^4*b^3*B + 7*a^2*b^5*B - 2*b^7*B - 10*a 
^7*C + 5*a^5*b^2*C - 7*a^3*b^4*C + 2*a*b^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(c 
+ d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)))/(a*(a^2 - b^2)) + 
 (b^4*(26*a^4*b*B - 17*a^2*b^3*B + 6*b^5*B - 37*a^5*C + 13*a^3*b^2*C - 6*a 
*b^4*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])))/(2*a*(a^2 - 
b^2)))/(3*a*(a^2 - b^2)))/b^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 2014

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_S 
ymbol] :> Simp[1/b^2   Int[u*(a + b*v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], 
x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] && LeQ 
[m, -1]
 

rule 3042

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

rule 3138

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ 
e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + b + 
(a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] 
 && NeQ[a^2 - b^2, 0]
 

rule 4318

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo 
l] :> Simp[1/b   Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, 
f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4407

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
 (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a   Int[Csc[e + f* 
x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c 
- a*d, 0]
 

rule 4411

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d 
_.) + (c_)), x_Symbol] :> Simp[b*(b*c - a*d)*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[c*(a^2 - b^2)*(m + 1) - (a*(b*c - 
 a*d)*(m + 1))*Csc[e + f*x] + b*(b*c - a*d)*(m + 2)*Csc[e + f*x]^2, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && N 
eQ[a^2 - b^2, 0] && IntegerQ[2*m]
 

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]
 

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.57

method	result	size
derivativedivides	\(\frac {\frac {2 \left (B b -C a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {2 b \left (\frac {-\frac {\left (12 B \,a^{4} b +4 a^{3} b^{2} B -6 B \,a^{2} b^{3}-a \,b^{4} B +2 B \,b^{5}-18 a^{5} C -7 a^{4} b C +4 C \,a^{3} b^{2}+C \,a^{2} b^{3}-2 C a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (18 B \,a^{4} b -11 B \,a^{2} b^{3}+3 B \,b^{5}-27 a^{5} C +10 C \,a^{3} b^{2}-3 C a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (12 B \,a^{4} b -4 a^{3} b^{2} B -6 B \,a^{2} b^{3}+a \,b^{4} B +2 B \,b^{5}-18 a^{5} C +7 a^{4} b C +4 C \,a^{3} b^{2}-C \,a^{2} b^{3}-2 C a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (8 B \,a^{6} b -8 B \,a^{4} b^{3}+7 a^{2} b^{5} B -2 b^{7} B -10 a^{7} C +5 a^{5} b^{2} C -7 C \,a^{3} b^{4}+2 C a \,b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{4}}}{d}\)	\(526\)
default	\(\frac {\frac {2 \left (B b -C a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {2 b \left (\frac {-\frac {\left (12 B \,a^{4} b +4 a^{3} b^{2} B -6 B \,a^{2} b^{3}-a \,b^{4} B +2 B \,b^{5}-18 a^{5} C -7 a^{4} b C +4 C \,a^{3} b^{2}+C \,a^{2} b^{3}-2 C a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (18 B \,a^{4} b -11 B \,a^{2} b^{3}+3 B \,b^{5}-27 a^{5} C +10 C \,a^{3} b^{2}-3 C a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (12 B \,a^{4} b -4 a^{3} b^{2} B -6 B \,a^{2} b^{3}+a \,b^{4} B +2 B \,b^{5}-18 a^{5} C +7 a^{4} b C +4 C \,a^{3} b^{2}-C \,a^{2} b^{3}-2 C a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (8 B \,a^{6} b -8 B \,a^{4} b^{3}+7 a^{2} b^{5} B -2 b^{7} B -10 a^{7} C +5 a^{5} b^{2} C -7 C \,a^{3} b^{4}+2 C a \,b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{4}}}{d}\)	\(526\)
risch	\(\text {Expression too large to display}\)	\(2131\)

Input:

int((B*a*b-C*a^2+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^5,x 
,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/d*(2*(B*b-C*a)/a^4*arctan(tan(1/2*d*x+1/2*c))+2*b/a^4*((-1/2*(12*B*a^4*b 
+4*B*a^3*b^2-6*B*a^2*b^3-B*a*b^4+2*B*b^5-18*C*a^5-7*C*a^4*b+4*C*a^3*b^2+C* 
a^2*b^3-2*C*a*b^4)*a*b/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^ 
5+2/3*(18*B*a^4*b-11*B*a^2*b^3+3*B*b^5-27*C*a^5+10*C*a^3*b^2-3*C*a*b^4)*a* 
b/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*(12*B*a^4*b-4*B 
*a^3*b^2-6*B*a^2*b^3+B*a*b^4+2*B*b^5-18*C*a^5+7*C*a^4*b+4*C*a^3*b^2-C*a^2* 
b^3-2*C*a*b^4)*a*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c))/(ta 
n(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3-1/2*(8*B*a^6*b-8*B*a^4* 
b^3+7*B*a^2*b^5-2*B*b^7-10*C*a^7+5*C*a^5*b^2-7*C*a^3*b^4+2*C*a*b^6)/(a^6-3 
*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2* 
c)/((a+b)*(a-b))^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1175 vs. \(2 (322) = 644\).

Time = 0.26 (sec) , antiderivative size = 2408, normalized size of antiderivative = 7.17 \[ \int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/12*(12*(C*a^12 - B*a^11*b - 4*C*a^10*b^2 + 4*B*a^9*b^3 + 6*C*a^8*b^4 - 
 6*B*a^7*b^5 - 4*C*a^6*b^6 + 4*B*a^5*b^7 + C*a^4*b^8 - B*a^3*b^9)*d*x*cos( 
d*x + c)^3 + 36*(C*a^11*b - B*a^10*b^2 - 4*C*a^9*b^3 + 4*B*a^8*b^4 + 6*C*a 
^7*b^5 - 6*B*a^6*b^6 - 4*C*a^5*b^7 + 4*B*a^4*b^8 + C*a^3*b^9 - B*a^2*b^10) 
*d*x*cos(d*x + c)^2 + 36*(C*a^10*b^2 - B*a^9*b^3 - 4*C*a^8*b^4 + 4*B*a^7*b 
^5 + 6*C*a^6*b^6 - 6*B*a^5*b^7 - 4*C*a^4*b^8 + 4*B*a^3*b^9 + C*a^2*b^10 - 
B*a*b^11)*d*x*cos(d*x + c) + 12*(C*a^9*b^3 - B*a^8*b^4 - 4*C*a^7*b^5 + 4*B 
*a^6*b^6 + 6*C*a^5*b^7 - 6*B*a^4*b^8 - 4*C*a^3*b^9 + 4*B*a^2*b^10 + C*a*b^ 
11 - B*b^12)*d*x + 3*(10*C*a^7*b^4 - 8*B*a^6*b^5 - 5*C*a^5*b^6 + 8*B*a^4*b 
^7 + 7*C*a^3*b^8 - 7*B*a^2*b^9 - 2*C*a*b^10 + 2*B*b^11 + (10*C*a^10*b - 8* 
B*a^9*b^2 - 5*C*a^8*b^3 + 8*B*a^7*b^4 + 7*C*a^6*b^5 - 7*B*a^5*b^6 - 2*C*a^ 
4*b^7 + 2*B*a^3*b^8)*cos(d*x + c)^3 + 3*(10*C*a^9*b^2 - 8*B*a^8*b^3 - 5*C* 
a^7*b^4 + 8*B*a^6*b^5 + 7*C*a^5*b^6 - 7*B*a^4*b^7 - 2*C*a^3*b^8 + 2*B*a^2* 
b^9)*cos(d*x + c)^2 + 3*(10*C*a^8*b^3 - 8*B*a^7*b^4 - 5*C*a^6*b^5 + 8*B*a^ 
5*b^6 + 7*C*a^4*b^7 - 7*B*a^3*b^8 - 2*C*a^2*b^9 + 2*B*a*b^10)*cos(d*x + c) 
)*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 
 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*c 
os(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + 2*(37*C*a^8*b^4 - 26*B*a^7*b^ 
5 - 50*C*a^6*b^6 + 43*B*a^5*b^7 + 19*C*a^4*b^8 - 23*B*a^3*b^9 - 6*C*a^2*b^ 
10 + 6*B*a*b^11 + (54*C*a^10*b^2 - 36*B*a^9*b^3 - 91*C*a^8*b^4 + 68*B*a...
 

Sympy [F]

\[ \int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx=- \int \left (- \frac {B b}{a^{4} + 4 a^{3} b \sec {\left (c + d x \right )} + 6 a^{2} b^{2} \sec ^{2}{\left (c + d x \right )} + 4 a b^{3} \sec ^{3}{\left (c + d x \right )} + b^{4} \sec ^{4}{\left (c + d x \right )}}\right )\, dx - \int \frac {C a}{a^{4} + 4 a^{3} b \sec {\left (c + d x \right )} + 6 a^{2} b^{2} \sec ^{2}{\left (c + d x \right )} + 4 a b^{3} \sec ^{3}{\left (c + d x \right )} + b^{4} \sec ^{4}{\left (c + d x \right )}}\, dx - \int \left (- \frac {C b \sec {\left (c + d x \right )}}{a^{4} + 4 a^{3} b \sec {\left (c + d x \right )} + 6 a^{2} b^{2} \sec ^{2}{\left (c + d x \right )} + 4 a b^{3} \sec ^{3}{\left (c + d x \right )} + b^{4} \sec ^{4}{\left (c + d x \right )}}\right )\, dx \] Input:

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

Output:

-Integral(-B*b/(a**4 + 4*a**3*b*sec(c + d*x) + 6*a**2*b**2*sec(c + d*x)**2 
 + 4*a*b**3*sec(c + d*x)**3 + b**4*sec(c + d*x)**4), x) - Integral(C*a/(a* 
*4 + 4*a**3*b*sec(c + d*x) + 6*a**2*b**2*sec(c + d*x)**2 + 4*a*b**3*sec(c 
+ d*x)**3 + b**4*sec(c + d*x)**4), x) - Integral(-C*b*sec(c + d*x)/(a**4 + 
 4*a**3*b*sec(c + d*x) + 6*a**2*b**2*sec(c + d*x)**2 + 4*a*b**3*sec(c + d* 
x)**3 + b**4*sec(c + d*x)**4), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 860 vs. \(2 (322) = 644\).

Time = 0.41 (sec) , antiderivative size = 860, normalized size of antiderivative = 2.56 \[ \int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*(3*(10*C*a^7*b - 8*B*a^6*b^2 - 5*C*a^5*b^3 + 8*B*a^4*b^4 + 7*C*a^3*b^5 
 - 7*B*a^2*b^6 - 2*C*a*b^7 + 2*B*b^8)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sg 
n(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/ 
sqrt(-a^2 + b^2)))/((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*sqrt(-a^2 + b 
^2)) - 3*(C*a - B*b)*(d*x + c)/a^4 + (54*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 
- 36*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 - 87*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^ 
5 + 60*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^4*b^5*tan(1/2*d*x + 1/2*c) 
^5 + 42*C*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 - 45*B*a^3*b^6*tan(1/2*d*x + 1/2* 
c)^5 + 6*B*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 - 15*C*a^2*b^7*tan(1/2*d*x + 1/2 
*c)^5 + 15*B*a*b^8*tan(1/2*d*x + 1/2*c)^5 + 6*C*a*b^8*tan(1/2*d*x + 1/2*c) 
^5 - 6*B*b^9*tan(1/2*d*x + 1/2*c)^5 - 108*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^3 
 + 72*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 + 148*C*a^5*b^4*tan(1/2*d*x + 1/2*c 
)^3 - 116*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 - 52*C*a^3*b^6*tan(1/2*d*x + 1/ 
2*c)^3 + 56*B*a^2*b^7*tan(1/2*d*x + 1/2*c)^3 + 12*C*a*b^8*tan(1/2*d*x + 1/ 
2*c)^3 - 12*B*b^9*tan(1/2*d*x + 1/2*c)^3 + 54*C*a^7*b^2*tan(1/2*d*x + 1/2* 
c) - 36*B*a^6*b^3*tan(1/2*d*x + 1/2*c) + 87*C*a^6*b^3*tan(1/2*d*x + 1/2*c) 
 - 60*B*a^5*b^4*tan(1/2*d*x + 1/2*c) + 6*B*a^4*b^5*tan(1/2*d*x + 1/2*c) - 
42*C*a^4*b^5*tan(1/2*d*x + 1/2*c) + 45*B*a^3*b^6*tan(1/2*d*x + 1/2*c) + 6* 
B*a^2*b^7*tan(1/2*d*x + 1/2*c) + 15*C*a^2*b^7*tan(1/2*d*x + 1/2*c) - 15*B* 
a*b^8*tan(1/2*d*x + 1/2*c) + 6*C*a*b^8*tan(1/2*d*x + 1/2*c) - 6*B*b^9*t...
 

Mupad [B] (verification not implemented)

Time = 25.45 (sec) , antiderivative size = 7573, normalized size of antiderivative = 22.54 \[ \int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx=\text {Too large to display} \] Input:

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

Output:

(log(tan(c/2 + (d*x)/2) + 1i)*(B*b - C*a)*1i)/(a^4*d) - ((4*tan(c/2 + (d*x 
)/2)^3*(3*B*b^7 - 11*B*a^2*b^5 + 18*B*a^4*b^3 + 10*C*a^3*b^4 - 27*C*a^5*b^ 
2 - 3*C*a*b^6))/(3*(a + b)^2*(a^5 - 2*a^4*b + a^3*b^2)) + (tan(c/2 + (d*x) 
/2)*(2*B*b^7 - 6*B*a^2*b^5 - 4*B*a^3*b^4 + 12*B*a^4*b^3 - C*a^2*b^5 + 4*C* 
a^3*b^4 + 7*C*a^4*b^3 - 18*C*a^5*b^2 + B*a*b^6 - 2*C*a*b^6))/((a + b)*(3*a 
^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)) + (tan(c/2 + (d*x)/2)^5*(2*B*b^7 - 6*B* 
a^2*b^5 + 4*B*a^3*b^4 + 12*B*a^4*b^3 + C*a^2*b^5 + 4*C*a^3*b^4 - 7*C*a^4*b 
^3 - 18*C*a^5*b^2 - B*a*b^6 - 2*C*a*b^6))/((a^3*b - a^4)*(a + b)^3))/(d*(t 
an(c/2 + (d*x)/2)^2*(3*a*b^2 - 3*a^2*b - 3*a^3 + 3*b^3) - tan(c/2 + (d*x)/ 
2)^4*(3*a*b^2 + 3*a^2*b - 3*a^3 - 3*b^3) + 3*a*b^2 + 3*a^2*b + a^3 + b^3 - 
 tan(c/2 + (d*x)/2)^6*(3*a*b^2 - 3*a^2*b + a^3 - b^3))) - (log(tan(c/2 + ( 
d*x)/2) - 1i)*(B*b*1i - C*a*1i))/(a^4*d) + (b*atan(((b*((a + b)^7*(a - b)^ 
7)^(1/2)*((8*tan(c/2 + (d*x)/2)*(8*B^2*b^16 + 4*C^2*a^16 - 8*B^2*a*b^15 - 
8*C^2*a^15*b - 48*B^2*a^2*b^14 + 48*B^2*a^3*b^13 + 117*B^2*a^4*b^12 - 120* 
B^2*a^5*b^11 - 164*B^2*a^6*b^10 + 160*B^2*a^7*b^9 + 156*B^2*a^8*b^8 - 120* 
B^2*a^9*b^7 - 92*B^2*a^10*b^6 + 48*B^2*a^11*b^5 + 44*B^2*a^12*b^4 - 8*B^2* 
a^13*b^3 + 4*B^2*a^14*b^2 + 8*C^2*a^2*b^14 - 8*C^2*a^3*b^13 - 48*C^2*a^4*b 
^12 + 48*C^2*a^5*b^11 + 105*C^2*a^6*b^10 - 120*C^2*a^7*b^9 - 130*C^2*a^8*b 
^8 + 160*C^2*a^9*b^7 + 145*C^2*a^10*b^6 - 120*C^2*a^11*b^5 - 64*C^2*a^12*b 
^4 + 48*C^2*a^13*b^3 + 80*C^2*a^14*b^2 - 16*B*C*a*b^15 - 8*B*C*a^15*b +...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 3721, normalized size of antiderivative = 11.07 \[ \int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(60*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sq 
rt( - a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*a**10*b*c - 48*sqrt( - a* 
*2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b 
**2))*cos(c + d*x)*sin(c + d*x)**2*a**9*b**3 - 30*sqrt( - a**2 + b**2)*ata 
n((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*cos(c + 
d*x)*sin(c + d*x)**2*a**8*b**3*c + 48*sqrt( - a**2 + b**2)*atan((tan((c + 
d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*cos(c + d*x)*sin(c + 
 d*x)**2*a**7*b**5 + 42*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - ta 
n((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*a**6* 
b**5*c - 42*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/ 
2)*b)/sqrt( - a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*a**5*b**7 - 12*sq 
rt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - 
a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*a**4*b**7*c + 12*sqrt( - a**2 + 
 b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2) 
)*cos(c + d*x)*sin(c + d*x)**2*a**3*b**9 - 60*sqrt( - a**2 + b**2)*atan((t 
an((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*cos(c + d*x) 
*a**10*b*c + 48*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d 
*x)/2)*b)/sqrt( - a**2 + b**2))*cos(c + d*x)*a**9*b**3 - 150*sqrt( - a**2 
+ b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2 
))*cos(c + d*x)*a**8*b**3*c + 96*sqrt( - a**2 + b**2)*atan((tan((c + d*...