\(\int \frac {\sec ^4(c+d x) (A+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\) [691]

Optimal result

Integrand size = 33, antiderivative size = 381 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {\left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) \text {arctanh}(\sin (c+d x))}{2 b^5 d}-\frac {a \left (6 A b^6+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}-\frac {a \left (a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)+12 a^4 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2 b^2 (A-10 C)-b^4 (4 A-C)+6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4-4 a^4 C+7 a^2 b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \] Output:

1/2*(2*A*b^2+(12*a^2+b^2)*C)*arctanh(sin(d*x+c))/b^5/d-a*(6*A*b^6+a^4*b^2* 
(2*A-29*C)-5*a^2*b^4*(A-4*C)+12*a^6*C)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2 
*c)/(a+b)^(1/2))/(a-b)^(5/2)/b^5/(a+b)^(5/2)/d-1/2*a*(a^2*b^2*(2*A-21*C)-b 
^4*(5*A-6*C)+12*a^4*C)*tan(d*x+c)/b^4/(a^2-b^2)^2/d+1/2*(a^2*b^2*(A-10*C)- 
b^4*(4*A-C)+6*a^4*C)*sec(d*x+c)*tan(d*x+c)/b^3/(a^2-b^2)^2/d-1/2*(A*b^2+C* 
a^2)*sec(d*x+c)^3*tan(d*x+c)/b/(a^2-b^2)/d/(a+b*sec(d*x+c))^2+1/2*(3*A*b^4 
-4*C*a^4+7*C*a^2*b^2)*sec(d*x+c)^2*tan(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*sec(d 
*x+c))
 

Mathematica [A] (warning: unable to verify)

Time = 4.62 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.47 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {(b+a \cos (c+d x)) \sec (c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (\frac {4 a \left (6 A b^6+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^2}{\left (a^2-b^2\right )^{5/2}}-2 \left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) (b+a \cos (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) (b+a \cos (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b^2 C (b+a \cos (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {12 a b C (b+a \cos (c+d x))^2 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {b^2 C (b+a \cos (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {12 a b C (b+a \cos (c+d x))^2 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {2 a^2 b^2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{(-a+b) (a+b)}+\frac {2 a^2 b \left (5 A b^4-6 a^4 C+a^2 b^2 (-2 A+9 C)\right ) (b+a \cos (c+d x)) \sin (c+d x)}{(a-b)^2 (a+b)^2}\right )}{2 b^5 d (A+2 C+A \cos (2 (c+d x))) (a+b \sec (c+d x))^3} \] Input:

Integrate[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x 
]
 

Output:

((b + a*Cos[c + d*x])*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((4*a*(6*A*b^6 + 
 a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTanh[((-a + b)* 
Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2)/(a^2 - b^2)^(5/ 
2) - 2*(2*A*b^2 + (12*a^2 + b^2)*C)*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d* 
x)/2] - Sin[(c + d*x)/2]] + 2*(2*A*b^2 + (12*a^2 + b^2)*C)*(b + a*Cos[c + 
d*x])^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^2*C*(b + a*Cos[c + d 
*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - (12*a*b*C*(b + a*Cos[c + 
 d*x])^2*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (b^2*C* 
(b + a*Cos[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (12*a*b* 
C*(b + a*Cos[c + d*x])^2*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d* 
x)/2]) + (2*a^2*b^2*(A*b^2 + a^2*C)*Sin[c + d*x])/((-a + b)*(a + b)) + (2* 
a^2*b*(5*A*b^4 - 6*a^4*C + a^2*b^2*(-2*A + 9*C))*(b + a*Cos[c + d*x])*Sin[ 
c + d*x])/((a - b)^2*(a + b)^2)))/(2*b^5*d*(A + 2*C + A*Cos[2*(c + d*x)])* 
(a + b*Sec[c + d*x])^3)
 

Rubi [A] (verified)

Time = 2.87 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.07, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 4587, 3042, 4586, 3042, 4580, 27, 3042, 4570, 3042, 4486, 3042, 4257, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4587

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4586

\(\displaystyle -\frac {-\frac {\int \frac {\sec ^2(c+d x) \left (2 \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right ) \sec ^2(c+d x)-a b \left (-C a^2+3 A b^2+4 b^2 C\right ) \sec (c+d x)+2 \left (-4 C a^4+7 b^2 C a^2+3 A b^4\right )\right )}{a+b \sec (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (2 \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-a b \left (-C a^2+3 A b^2+4 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 \left (-4 C a^4+7 b^2 C a^2+3 A b^4\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4580

\(\displaystyle -\frac {-\frac {\frac {\int \frac {2 \sec (c+d x) \left (-a \left (12 C a^4+b^2 (2 A-21 C) a^2-b^4 (5 A-6 C)\right ) \sec ^2(c+d x)-b \left (2 C a^4-b^2 (A+4 C) a^2-b^4 (2 A+C)\right ) \sec (c+d x)+a \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right )\right )}{a+b \sec (c+d x)}dx}{2 b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\int \frac {\sec (c+d x) \left (-a \left (12 C a^4+b^2 (2 A-21 C) a^2-b^4 (5 A-6 C)\right ) \sec ^2(c+d x)-b \left (2 C a^4-b^2 (A+4 C) a^2-b^4 (2 A+C)\right ) \sec (c+d x)+a \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right )\right )}{a+b \sec (c+d x)}dx}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (-a \left (12 C a^4+b^2 (2 A-21 C) a^2-b^4 (5 A-6 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-b \left (2 C a^4-b^2 (A+4 C) a^2-b^4 (2 A+C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4570

\(\displaystyle -\frac {-\frac {\frac {\frac {\int \frac {\sec (c+d x) \left (\left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) \sec (c+d x) \left (a^2-b^2\right )^2+a b \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right )\right )}{a+b \sec (c+d x)}dx}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \tan (c+d x)}{b d}}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )^2+a b \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \tan (c+d x)}{b d}}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4486

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (C \left (12 a^2+b^2\right )+2 A b^2\right ) \int \sec (c+d x)dx}{b}-\frac {a \left (12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{b}}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \tan (c+d x)}{b d}}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (C \left (12 a^2+b^2\right )+2 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b}-\frac {a \left (12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \tan (c+d x)}{b d}}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4257

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (C \left (12 a^2+b^2\right )+2 A b^2\right ) \text {arctanh}(\sin (c+d x))}{b d}-\frac {a \left (12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \tan (c+d x)}{b d}}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4318

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (C \left (12 a^2+b^2\right )+2 A b^2\right ) \text {arctanh}(\sin (c+d x))}{b d}-\frac {a \left (12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{b^2}}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \tan (c+d x)}{b d}}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (C \left (12 a^2+b^2\right )+2 A b^2\right ) \text {arctanh}(\sin (c+d x))}{b d}-\frac {a \left (12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{b^2}}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \tan (c+d x)}{b d}}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3138

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (C \left (12 a^2+b^2\right )+2 A b^2\right ) \text {arctanh}(\sin (c+d x))}{b d}-\frac {2 a \left (12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{b^2 d}}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \tan (c+d x)}{b d}}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \tan (c+d x) \sec (c+d x)}{b d}+\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (C \left (12 a^2+b^2\right )+2 A b^2\right ) \text {arctanh}(\sin (c+d x))}{b d}-\frac {2 a \left (12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b d \sqrt {a-b} \sqrt {a+b}}}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \tan (c+d x)}{b d}}{b}}{b \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}\)

Input:

Int[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]
 

Output:

-1/2*((A*b^2 + a^2*C)*Sec[c + d*x]^3*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b 
*Sec[c + d*x])^2) - (-(((3*A*b^4 - 4*a^4*C + 7*a^2*b^2*C)*Sec[c + d*x]^2*T 
an[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))) - (((a^2*b^2*(A - 10* 
C) - b^4*(4*A - C) + 6*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(b*d) + ((((a^2 - 
 b^2)^2*(2*A*b^2 + (12*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(b*d) - (2*a*( 
6*A*b^6 + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTanh[( 
Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)) 
/b - (a*(a^2*b^2*(2*A - 21*C) - b^4*(5*A - 6*C) + 12*a^4*C)*Tan[c + d*x])/ 
(b*d))/b)/(b*(a^2 - b^2)))/(2*b*(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 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 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 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

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 4486

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

rule 4570

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

rule 4580

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[ 
(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x 
_Symbol] :> Simp[(-C)*Csc[e + f*x]*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 
1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3))   Int[Csc[e + f*x]*(a + b*Csc[e 
+ f*x])^m*Simp[a*C + b*(C*(m + 2) + A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B* 
(m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] & 
& NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]
 

rule 4586

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[(-d)*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + 
b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 1)/(b*f*(a^2 - b^2)*(m + 1)) 
), x] + Simp[d/(b*(a^2 - b^2)*(m + 1))   Int[(a + b*Csc[e + f*x])^(m + 1)*( 
d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1) + b*(a*A 
 - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n 
+ b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C 
}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]
 

rule 4587

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

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {-\frac {C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (2 A \,b^{2}+12 C \,a^{2}+C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{5}}+\frac {C \left (6 a +b \right )}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 a \left (\frac {\frac {\left (2 A \,a^{2} b^{2}-a A \,b^{3}-6 A \,b^{4}+6 a^{4} C -a^{3} b C -10 C \,a^{2} b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (2 A \,a^{2} b^{2}+a A \,b^{3}-6 A \,b^{4}+6 a^{4} C +a^{3} b C -10 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (2 A \,a^{4} b^{2}-5 a^{2} A \,b^{4}+6 A \,b^{6}+12 a^{6} C -29 a^{4} b^{2} C +20 C \,a^{2} b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}+\frac {C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-2 A \,b^{2}-12 C \,a^{2}-C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{5}}+\frac {C \left (6 a +b \right )}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(462\)
default	\(\frac {-\frac {C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (2 A \,b^{2}+12 C \,a^{2}+C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{5}}+\frac {C \left (6 a +b \right )}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 a \left (\frac {\frac {\left (2 A \,a^{2} b^{2}-a A \,b^{3}-6 A \,b^{4}+6 a^{4} C -a^{3} b C -10 C \,a^{2} b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (2 A \,a^{2} b^{2}+a A \,b^{3}-6 A \,b^{4}+6 a^{4} C +a^{3} b C -10 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (2 A \,a^{4} b^{2}-5 a^{2} A \,b^{4}+6 A \,b^{6}+12 a^{6} C -29 a^{4} b^{2} C +20 C \,a^{2} b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}+\frac {C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-2 A \,b^{2}-12 C \,a^{2}-C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{5}}+\frac {C \left (6 a +b \right )}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(462\)
risch	\(\text {Expression too large to display}\)	\(2026\)

Input:

int(sec(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x,method=_RETURNVER 
BOSE)
 

Output:

1/d*(-1/2*C/b^3/(tan(1/2*d*x+1/2*c)+1)^2+1/2*(2*A*b^2+12*C*a^2+C*b^2)/b^5* 
ln(tan(1/2*d*x+1/2*c)+1)+1/2*C*(6*a+b)/b^4/(tan(1/2*d*x+1/2*c)+1)+2*a/b^5* 
((1/2*(2*A*a^2*b^2-A*a*b^3-6*A*b^4+6*C*a^4-C*a^3*b-10*C*a^2*b^2)*a*b/(a-b) 
/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*b*a*(2*A*a^2*b^2+A*a*b^3-6*A*b^4 
+6*C*a^4+C*a^3*b-10*C*a^2*b^2)/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c))/(tan(1/2* 
d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2-1/2*(2*A*a^4*b^2-5*A*a^2*b^4+ 
6*A*b^6+12*C*a^6-29*C*a^4*b^2+20*C*a^2*b^4)/(a^4-2*a^2*b^2+b^4)/((a+b)*(a- 
b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))+1/2*C/b^3 
/(tan(1/2*d*x+1/2*c)-1)^2+1/2/b^5*(-2*A*b^2-12*C*a^2-C*b^2)*ln(tan(1/2*d*x 
+1/2*c)-1)+1/2*C*(6*a+b)/b^4/(tan(1/2*d*x+1/2*c)-1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1014 vs. \(2 (362) = 724\).

Time = 17.86 (sec) , antiderivative size = 2084, normalized size of antiderivative = 5.47 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm= 
"fricas")
 

Output:

[1/4*(((12*C*a^9 + (2*A - 29*C)*a^7*b^2 - 5*(A - 4*C)*a^5*b^4 + 6*A*a^3*b^ 
6)*cos(d*x + c)^4 + 2*(12*C*a^8*b + (2*A - 29*C)*a^6*b^3 - 5*(A - 4*C)*a^4 
*b^5 + 6*A*a^2*b^7)*cos(d*x + c)^3 + (12*C*a^7*b^2 + (2*A - 29*C)*a^5*b^4 
- 5*(A - 4*C)*a^3*b^6 + 6*A*a*b^8)*cos(d*x + c)^2)*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*cos(d*x + c)^2 + 2*a*b*cos 
(d*x + c) + b^2)) + ((12*C*a^10 + (2*A - 35*C)*a^8*b^2 - 3*(2*A - 11*C)*a^ 
6*b^4 + 3*(2*A - 3*C)*a^4*b^6 - (2*A + C)*a^2*b^8)*cos(d*x + c)^4 + 2*(12* 
C*a^9*b + (2*A - 35*C)*a^7*b^3 - 3*(2*A - 11*C)*a^5*b^5 + 3*(2*A - 3*C)*a^ 
3*b^7 - (2*A + C)*a*b^9)*cos(d*x + c)^3 + (12*C*a^8*b^2 + (2*A - 35*C)*a^6 
*b^4 - 3*(2*A - 11*C)*a^4*b^6 + 3*(2*A - 3*C)*a^2*b^8 - (2*A + C)*b^10)*co 
s(d*x + c)^2)*log(sin(d*x + c) + 1) - ((12*C*a^10 + (2*A - 35*C)*a^8*b^2 - 
 3*(2*A - 11*C)*a^6*b^4 + 3*(2*A - 3*C)*a^4*b^6 - (2*A + C)*a^2*b^8)*cos(d 
*x + c)^4 + 2*(12*C*a^9*b + (2*A - 35*C)*a^7*b^3 - 3*(2*A - 11*C)*a^5*b^5 
+ 3*(2*A - 3*C)*a^3*b^7 - (2*A + C)*a*b^9)*cos(d*x + c)^3 + (12*C*a^8*b^2 
+ (2*A - 35*C)*a^6*b^4 - 3*(2*A - 11*C)*a^4*b^6 + 3*(2*A - 3*C)*a^2*b^8 - 
(2*A + C)*b^10)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) + 2*(C*a^6*b^4 - 3* 
C*a^4*b^6 + 3*C*a^2*b^8 - C*b^10 - (12*C*a^9*b + (2*A - 33*C)*a^7*b^3 - (7 
*A - 27*C)*a^5*b^5 + (5*A - 6*C)*a^3*b^7)*cos(d*x + c)^3 - (18*C*a^8*b^2 + 
 (3*A - 50*C)*a^6*b^4 - (9*A - 43*C)*a^4*b^6 + (6*A - 11*C)*a^2*b^8)*co...
 

Sympy [F]

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

integrate(sec(d*x+c)**4*(A+C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**3,x)
 

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(sec(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,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. 1187 vs. \(2 (362) = 724\).

Time = 0.40 (sec) , antiderivative size = 1187, normalized size of antiderivative = 3.12 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm= 
"giac")
                                                                                    
                                                                                    
 

Output:

-1/2*(2*(12*C*a^7 + 2*A*a^5*b^2 - 29*C*a^5*b^2 - 5*A*a^3*b^4 + 20*C*a^3*b^ 
4 + 6*A*a*b^6)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-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^ 
4*b^5 - 2*a^2*b^7 + b^9)*sqrt(-a^2 + b^2)) - 2*(12*C*a^7*tan(1/2*d*x + 1/2 
*c)^7 - 18*C*a^6*b*tan(1/2*d*x + 1/2*c)^7 + 2*A*a^5*b^2*tan(1/2*d*x + 1/2* 
c)^7 - 17*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 - 3*A*a^4*b^3*tan(1/2*d*x + 1/2 
*c)^7 + 33*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 5*A*a^3*b^4*tan(1/2*d*x + 1/ 
2*c)^7 - 2*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 + 6*A*a^2*b^5*tan(1/2*d*x + 1/ 
2*c)^7 - 13*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 4*C*a*b^6*tan(1/2*d*x + 1/2 
*c)^7 + C*b^7*tan(1/2*d*x + 1/2*c)^7 - 36*C*a^7*tan(1/2*d*x + 1/2*c)^5 + 1 
8*C*a^6*b*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 67 
*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 3*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 2 
9*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 + 15*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 - 
 26*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 
+ 5*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 4*C*a*b^6*tan(1/2*d*x + 1/2*c)^5 + 
3*C*b^7*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^7*tan(1/2*d*x + 1/2*c)^3 + 18*C*a^ 
6*b*tan(1/2*d*x + 1/2*c)^3 + 6*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 67*C*a^5 
*b^2*tan(1/2*d*x + 1/2*c)^3 + 3*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 29*C*a^ 
4*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 + 26*C* 
a^3*b^4*tan(1/2*d*x + 1/2*c)^3 - 6*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 5...
 

Mupad [B] (verification not implemented)

Time = 26.18 (sec) , antiderivative size = 10408, normalized size of antiderivative = 27.32 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \] Input:

int((A + C/cos(c + d*x)^2)/(cos(c + d*x)^4*(a + b/cos(c + d*x))^3),x)
 

Output:

- ((tan(c/2 + (d*x)/2)*(12*C*a^6 - C*b^6 - 6*A*a^2*b^4 + A*a^3*b^3 + 2*A*a 
^4*b^2 + 8*C*a^2*b^4 - 10*C*a^3*b^3 - 23*C*a^4*b^2 + 5*C*a*b^5 + 6*C*a^5*b 
))/((a + b)*(b^6 - 2*a*b^5 + a^2*b^4)) - (tan(c/2 + (d*x)/2)^3*(36*C*a^7 + 
 3*C*b^7 - 6*A*a^2*b^5 - 15*A*a^3*b^4 + 3*A*a^4*b^3 + 6*A*a^5*b^2 + 5*C*a^ 
2*b^5 + 26*C*a^3*b^4 - 29*C*a^4*b^3 - 67*C*a^5*b^2 - 4*C*a*b^6 + 18*C*a^6* 
b))/((a + b)^2*(b^6 - 2*a*b^5 + a^2*b^4)) - (tan(c/2 + (d*x)/2)^5*(3*C*b^7 
 - 36*C*a^7 - 6*A*a^2*b^5 + 15*A*a^3*b^4 + 3*A*a^4*b^3 - 6*A*a^5*b^2 + 5*C 
*a^2*b^5 - 26*C*a^3*b^4 - 29*C*a^4*b^3 + 67*C*a^5*b^2 + 4*C*a*b^6 + 18*C*a 
^6*b))/((a + b)^2*(b^6 - 2*a*b^5 + a^2*b^4)) + (tan(c/2 + (d*x)/2)^7*(C*b^ 
6 - 12*C*a^6 + 6*A*a^2*b^4 + A*a^3*b^3 - 2*A*a^4*b^2 - 8*C*a^2*b^4 - 10*C* 
a^3*b^3 + 23*C*a^4*b^2 + 5*C*a*b^5 + 6*C*a^5*b))/((a*b^4 - b^5)*(a + b)^2) 
)/(d*(2*a*b + tan(c/2 + (d*x)/2)^4*(6*a^2 - 2*b^2) - tan(c/2 + (d*x)/2)^2* 
(4*a*b + 4*a^2) + tan(c/2 + (d*x)/2)^6*(4*a*b - 4*a^2) + tan(c/2 + (d*x)/2 
)^8*(a^2 - 2*a*b + b^2) + a^2 + b^2)) - (atan(((((8*tan(c/2 + (d*x)/2)*(4* 
A^2*b^14 + 288*C^2*a^14 + C^2*b^14 - 8*A^2*a*b^13 - 2*C^2*a*b^13 - 288*C^2 
*a^13*b + 24*A^2*a^2*b^12 + 32*A^2*a^3*b^11 - 52*A^2*a^4*b^10 - 48*A^2*a^5 
*b^9 + 57*A^2*a^6*b^8 + 32*A^2*a^7*b^7 - 32*A^2*a^8*b^6 - 8*A^2*a^9*b^5 + 
8*A^2*a^10*b^4 + 21*C^2*a^2*b^12 - 40*C^2*a^3*b^11 + 74*C^2*a^4*b^10 - 108 
*C^2*a^5*b^9 + 18*C^2*a^6*b^8 + 872*C^2*a^7*b^7 - 827*C^2*a^8*b^6 - 1538*C 
^2*a^9*b^5 + 1538*C^2*a^10*b^4 + 1104*C^2*a^11*b^3 - 1104*C^2*a^12*b^2 ...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 5588, normalized size of antiderivative = 14.67 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 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**8*b*c - 8*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**3 + 116*sqrt( - a**2 + b**2)*a 
tan((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**6*b**3*c + 20*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**5 - 80*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** 
4*b**5*c - 24*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**7 + 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**8*b*c + 8*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** 
7*b**3 - 116*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**6*b**3*c - 20*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**5*b**5 + 80*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**4*b**5*c + 24 
*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sq...