Integrand size = 102, antiderivative size = 38 \[ \int \left (\frac {\sec ^{-n}(c+d x) (a+a \sec (c+d x))^n (-a (B+A n+B n)-a C (1+n) \sec (c+d x))}{a (1+n)}+\sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )\right ) \, dx=\frac {A \sec ^{-n}(c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1+n)} \]
Time = 0.13 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \left (\frac {\sec ^{-n}(c+d x) (a+a \sec (c+d x))^n (-a (B+A n+B n)-a C (1+n) \sec (c+d x))}{a (1+n)}+\sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )\right ) \, dx=\frac {A \sec ^{-n}(c+d x) (a (1+\sec (c+d x)))^n \sin (c+d x)}{d (1+n)} \]
Integrate[((a + a*Sec[c + d*x])^n*(-(a*(B + A*n + B*n)) - a*C*(1 + n)*Sec[ c + d*x]))/(a*(1 + n)*Sec[c + d*x]^n) + Sec[c + d*x]^(-1 - n)*(a + a*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Time = 0.96 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {2009}
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.
| \(\displaystyle \int \left (\frac {\sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n (-a (A n+B n+B)-a C (n+1) \sec (c+d x))}{a (n+1)}+\sec ^{-n-1}(c+d x) (a \sec (c+d x)+a)^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}\) |
Int[((a + a*Sec[c + d*x])^n*(-(a*(B + A*n + B*n)) - a*C*(1 + n)*Sec[c + d* x]))/(a*(1 + n)*Sec[c + d*x]^n) + Sec[c + d*x]^(-1 - n)*(a + a*Sec[c + d*x ])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
3.7.35.3.1 Defintions of rubi rules used
\[\int \left (\frac {\left (a +a \sec \left (d x +c \right )\right )^{n} \left (-a \left (A n +B n +B \right )-a C \left (1+n \right ) \sec \left (d x +c \right )\right ) \sec \left (d x +c \right )^{-n}}{a \left (1+n \right )}+\sec \left (d x +c \right )^{-1-n} \left (a +a \sec \left (d x +c \right )\right )^{n} \left (A +B \sec \left (d x +c \right )+C \sec \left (d x +c \right )^{2}\right )\right )d x\]
int((a+a*sec(d*x+c))^n*(-a*(A*n+B*n+B)-a*C*(1+n)*sec(d*x+c))/a/(1+n)/(sec( d*x+c)^n)+sec(d*x+c)^(-1-n)*(a+a*sec(d*x+c))^n*(A+B*sec(d*x+c)+C*sec(d*x+c )^2),x)
int((a+a*sec(d*x+c))^n*(-a*(A*n+B*n+B)-a*C*(1+n)*sec(d*x+c))/a/(1+n)/(sec( d*x+c)^n)+sec(d*x+c)^(-1-n)*(a+a*sec(d*x+c))^n*(A+B*sec(d*x+c)+C*sec(d*x+c )^2),x)
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.53 \[ \int \left (\frac {\sec ^{-n}(c+d x) (a+a \sec (c+d x))^n (-a (B+A n+B n)-a C (1+n) \sec (c+d x))}{a (1+n)}+\sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )\right ) \, dx=\frac {A \left (\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}\right )^{n} \frac {1}{\cos \left (d x + c\right )}^{-n - 1} \sin \left (d x + c\right )}{{\left (d n + d\right )} \cos \left (d x + c\right )} \]
integrate((a+a*sec(d*x+c))^n*(-a*(A*n+B*n+B)-a*C*(1+n)*sec(d*x+c))/a/(1+n) /(sec(d*x+c)^n)+sec(d*x+c)^(-1-n)*(a+a*sec(d*x+c))^n*(A+B*sec(d*x+c)+C*sec (d*x+c)^2),x, algorithm="fricas")
A*((a*cos(d*x + c) + a)/cos(d*x + c))^n*(1/cos(d*x + c))^(-n - 1)*sin(d*x + c)/((d*n + d)*cos(d*x + c))
\[ \int \left (\frac {\sec ^{-n}(c+d x) (a+a \sec (c+d x))^n (-a (B+A n+B n)-a C (1+n) \sec (c+d x))}{a (1+n)}+\sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )\right ) \, dx=\frac {\int A \left (a \sec {\left (c + d x \right )} + a\right )^{n} \sec ^{- n - 1}{\left (c + d x \right )}\, dx + \int \left (- B \left (a \sec {\left (c + d x \right )} + a\right )^{n} \sec ^{- n}{\left (c + d x \right )}\right )\, dx + \int \left (- A n \left (a \sec {\left (c + d x \right )} + a\right )^{n} \sec ^{- n}{\left (c + d x \right )}\right )\, dx + \int A n \left (a \sec {\left (c + d x \right )} + a\right )^{n} \sec ^{- n - 1}{\left (c + d x \right )}\, dx + \int \left (- B n \left (a \sec {\left (c + d x \right )} + a\right )^{n} \sec ^{- n}{\left (c + d x \right )}\right )\, dx + \int B \left (a \sec {\left (c + d x \right )} + a\right )^{n} \sec {\left (c + d x \right )} \sec ^{- n - 1}{\left (c + d x \right )}\, dx + \int \left (- C \left (a \sec {\left (c + d x \right )} + a\right )^{n} \sec {\left (c + d x \right )} \sec ^{- n}{\left (c + d x \right )}\right )\, dx + \int C \left (a \sec {\left (c + d x \right )} + a\right )^{n} \sec ^{2}{\left (c + d x \right )} \sec ^{- n - 1}{\left (c + d x \right )}\, dx + \int B n \left (a \sec {\left (c + d x \right )} + a\right )^{n} \sec {\left (c + d x \right )} \sec ^{- n - 1}{\left (c + d x \right )}\, dx + \int \left (- C n \left (a \sec {\left (c + d x \right )} + a\right )^{n} \sec {\left (c + d x \right )} \sec ^{- n}{\left (c + d x \right )}\right )\, dx + \int C n \left (a \sec {\left (c + d x \right )} + a\right )^{n} \sec ^{2}{\left (c + d x \right )} \sec ^{- n - 1}{\left (c + d x \right )}\, dx}{n + 1} \]
integrate((a+a*sec(d*x+c))**n*(-a*(A*n+B*n+B)-a*C*(1+n)*sec(d*x+c))/a/(1+n )/(sec(d*x+c)**n)+sec(d*x+c)**(-1-n)*(a+a*sec(d*x+c))**n*(A+B*sec(d*x+c)+C *sec(d*x+c)**2),x)
(Integral(A*(a*sec(c + d*x) + a)**n*sec(c + d*x)**(-n - 1), x) + Integral( -B*(a*sec(c + d*x) + a)**n/sec(c + d*x)**n, x) + Integral(-A*n*(a*sec(c + d*x) + a)**n/sec(c + d*x)**n, x) + Integral(A*n*(a*sec(c + d*x) + a)**n*se c(c + d*x)**(-n - 1), x) + Integral(-B*n*(a*sec(c + d*x) + a)**n/sec(c + d *x)**n, x) + Integral(B*(a*sec(c + d*x) + a)**n*sec(c + d*x)*sec(c + d*x)* *(-n - 1), x) + Integral(-C*(a*sec(c + d*x) + a)**n*sec(c + d*x)/sec(c + d *x)**n, x) + Integral(C*(a*sec(c + d*x) + a)**n*sec(c + d*x)**2*sec(c + d* x)**(-n - 1), x) + Integral(B*n*(a*sec(c + d*x) + a)**n*sec(c + d*x)*sec(c + d*x)**(-n - 1), x) + Integral(-C*n*(a*sec(c + d*x) + a)**n*sec(c + d*x) /sec(c + d*x)**n, x) + Integral(C*n*(a*sec(c + d*x) + a)**n*sec(c + d*x)** 2*sec(c + d*x)**(-n - 1), x))/(n + 1)
Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (38) = 76\).
Time = 4.78 (sec) , antiderivative size = 310, normalized size of antiderivative = 8.16 \[ \int \left (\frac {\sec ^{-n}(c+d x) (a+a \sec (c+d x))^n (-a (B+A n+B n)-a C (1+n) \sec (c+d x))}{a (1+n)}+\sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )\right ) \, dx=\frac {{\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )}^{n} A a^{n} \cos \left (-{\left (d n + d\right )} x + 2 \, n \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right ) + 1\right ) - c\right ) \sin \left (c n\right ) - {\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )}^{n} A a^{n} \cos \left (-{\left (d n - d\right )} x + 2 \, n \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right ) + 1\right ) + c\right ) \sin \left (c n\right ) - {\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )}^{n} A a^{n} \cos \left (c n\right ) \sin \left (-{\left (d n + d\right )} x + 2 \, n \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right ) + 1\right ) - c\right ) + {\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )}^{n} A a^{n} \cos \left (c n\right ) \sin \left (-{\left (d n - d\right )} x + 2 \, n \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right ) + 1\right ) + c\right )}{2 \, {\left ({\left (d n + d\right )} 2^{n} \cos \left (c n\right )^{2} + {\left (d n + d\right )} 2^{n} \sin \left (c n\right )^{2}\right )}} \]
integrate((a+a*sec(d*x+c))^n*(-a*(A*n+B*n+B)-a*C*(1+n)*sec(d*x+c))/a/(1+n) /(sec(d*x+c)^n)+sec(d*x+c)^(-1-n)*(a+a*sec(d*x+c))^n*(A+B*sec(d*x+c)+C*sec (d*x+c)^2),x, algorithm="maxima")
1/2*((cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)^n*A*a^n*cos(-( d*n + d)*x + 2*n*arctan2(sin(d*x + c), cos(d*x + c) + 1) - c)*sin(c*n) - ( cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)^n*A*a^n*cos(-(d*n - d)*x + 2*n*arctan2(sin(d*x + c), cos(d*x + c) + 1) + c)*sin(c*n) - (cos(d* x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)^n*A*a^n*cos(c*n)*sin(-(d*n + d)*x + 2*n*arctan2(sin(d*x + c), cos(d*x + c) + 1) - c) + (cos(d*x + c) ^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)^n*A*a^n*cos(c*n)*sin(-(d*n - d)* x + 2*n*arctan2(sin(d*x + c), cos(d*x + c) + 1) + c))/((d*n + d)*2^n*cos(c *n)^2 + (d*n + d)*2^n*sin(c*n)^2)
\[ \int \left (\frac {\sec ^{-n}(c+d x) (a+a \sec (c+d x))^n (-a (B+A n+B n)-a C (1+n) \sec (c+d x))}{a (1+n)}+\sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )^{-n - 1} - \frac {{\left (C a {\left (n + 1\right )} \sec \left (d x + c\right ) + {\left (A n + B n + B\right )} a\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{a {\left (n + 1\right )} \sec \left (d x + c\right )^{n}} \,d x } \]
integrate((a+a*sec(d*x+c))^n*(-a*(A*n+B*n+B)-a*C*(1+n)*sec(d*x+c))/a/(1+n) /(sec(d*x+c)^n)+sec(d*x+c)^(-1-n)*(a+a*sec(d*x+c))^n*(A+B*sec(d*x+c)+C*sec (d*x+c)^2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^n*s ec(d*x + c)^(-n - 1) - (C*a*(n + 1)*sec(d*x + c) + (A*n + B*n + B)*a)*(a*s ec(d*x + c) + a)^n/(a*(n + 1)*sec(d*x + c)^n), x)
Timed out. \[ \int \left (\frac {\sec ^{-n}(c+d x) (a+a \sec (c+d x))^n (-a (B+A n+B n)-a C (1+n) \sec (c+d x))}{a (1+n)}+\sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )\right ) \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{n+1}}-\frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n\,\left (a\,\left (B+A\,n+B\,n\right )+\frac {C\,a\,\left (n+1\right )}{\cos \left (c+d\,x\right )}\right )}{a\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^n\,\left (n+1\right )} \,d x \]
int(((a + a/cos(c + d*x))^n*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/co s(c + d*x))^(n + 1) - ((a + a/cos(c + d*x))^n*(a*(B + A*n + B*n) + (C*a*(n + 1))/cos(c + d*x)))/(a*(1/cos(c + d*x))^n*(n + 1)),x)