Integrand size = 36, antiderivative size = 169 \[ \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-3-m} \, dx=-\frac {(a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-3-m} \tan (e+f x)}{f (1+2 m)}+\frac {2 (a+a \sec (e+f x))^{1+m} (c-c \sec (e+f x))^{-3-m} \tan (e+f x)}{a f \left (3+8 m+4 m^2\right )}-\frac {2 (a+a \sec (e+f x))^{2+m} (c-c \sec (e+f x))^{-3-m} \tan (e+f x)}{a^2 f (1+2 m) \left (15+16 m+4 m^2\right )} \]
-(a+a*sec(f*x+e))^m*(c-c*sec(f*x+e))^(-3-m)*tan(f*x+e)/f/(1+2*m)+2*(a+a*se c(f*x+e))^(1+m)*(c-c*sec(f*x+e))^(-3-m)*tan(f*x+e)/a/f/(4*m^2+8*m+3)-2*(a+ a*sec(f*x+e))^(2+m)*(c-c*sec(f*x+e))^(-3-m)*tan(f*x+e)/a^2/f/(1+2*m)/(4*m^ 2+16*m+15)
Time = 2.47 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.62 \[ \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-3-m} \, dx=\frac {(a (1+\sec (e+f x)))^m (c-c \sec (e+f x))^{-m} \left (7+12 m+4 m^2-2 (3+2 m) \sec (e+f x)+2 \sec ^2(e+f x)\right ) \tan (e+f x)}{c^3 f (1+2 m) (3+2 m) (5+2 m) (-1+\sec (e+f x))^3} \]
((a*(1 + Sec[e + f*x]))^m*(7 + 12*m + 4*m^2 - 2*(3 + 2*m)*Sec[e + f*x] + 2 *Sec[e + f*x]^2)*Tan[e + f*x])/(c^3*f*(1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(-1 + Sec[e + f*x])^3*(c - c*Sec[e + f*x])^m)
Time = 0.82 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 4439, 3042, 4439, 3042, 4438}
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.
| \(\displaystyle \int \sec (e+f x) (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^m \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{-m-3}dx\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle -\frac {2 \int \sec (e+f x) (\sec (e+f x) a+a)^{m+1} (c-c \sec (e+f x))^{-m-3}dx}{a (2 m+1)}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-3}}{f (2 m+1)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^{m+1} \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{-m-3}dx}{a (2 m+1)}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-3}}{f (2 m+1)}\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle -\frac {2 \left (-\frac {\int \sec (e+f x) (\sec (e+f x) a+a)^{m+2} (c-c \sec (e+f x))^{-m-3}dx}{a (2 m+3)}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{m+1} (c-c \sec (e+f x))^{-m-3}}{f (2 m+3)}\right )}{a (2 m+1)}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-3}}{f (2 m+1)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (-\frac {\int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^{m+2} \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{-m-3}dx}{a (2 m+3)}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{m+1} (c-c \sec (e+f x))^{-m-3}}{f (2 m+3)}\right )}{a (2 m+1)}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-3}}{f (2 m+1)}\) |
| \(\Big \downarrow \) 4438 |
| \(\displaystyle -\frac {\tan (e+f x) (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-3}}{f (2 m+1)}-\frac {2 \left (\frac {\tan (e+f x) (a \sec (e+f x)+a)^{m+2} (c-c \sec (e+f x))^{-m-3}}{a f (2 m+3) (2 m+5)}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{m+1} (c-c \sec (e+f x))^{-m-3}}{f (2 m+3)}\right )}{a (2 m+1)}\) |
-(((a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-3 - m)*Tan[e + f*x])/(f*( 1 + 2*m))) - (2*(-(((a + a*Sec[e + f*x])^(1 + m)*(c - c*Sec[e + f*x])^(-3 - m)*Tan[e + f*x])/(f*(3 + 2*m))) + ((a + a*Sec[e + f*x])^(2 + m)*(c - c*S ec[e + f*x])^(-3 - m)*Tan[e + f*x])/(a*f*(3 + 2*m)*(5 + 2*m))))/(a*(1 + 2* m))
3.2.62.3.1 Defintions of rubi rules used
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[b*Cot[e + f*x] *(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] /; Fre eQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] & & EqQ[m + n + 1, 0] && NeQ[2*m + 1, 0]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[b*Cot[e + f*x] *(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] + Simp [(m + n + 1)/(a*(2*m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* (c + d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ [b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[m + n + 1, 0] && NeQ[2*m + 1, 0 ] && !LtQ[n, 0] && !(IGtQ[n + 1/2, 0] && LtQ[n + 1/2, -(m + n)])
\[\int \sec \left (f x +e \right ) \left (a +a \sec \left (f x +e \right )\right )^{m} \left (c -c \sec \left (f x +e \right )\right )^{-3-m}d x\]
Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.71 \[ \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-3-m} \, dx=-\frac {{\left ({\left (4 \, m^{2} + 12 \, m + 7\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (2 \, m + 3\right )} \cos \left (f x + e\right ) + 2\right )} \left (\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}\right )^{m} \left (\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}\right )^{-m - 3} \sin \left (f x + e\right )}{{\left (8 \, f m^{3} + 36 \, f m^{2} + 46 \, f m + 15 \, f\right )} \cos \left (f x + e\right )^{3}} \]
-((4*m^2 + 12*m + 7)*cos(f*x + e)^2 - 2*(2*m + 3)*cos(f*x + e) + 2)*((a*co s(f*x + e) + a)/cos(f*x + e))^m*((c*cos(f*x + e) - c)/cos(f*x + e))^(-m - 3)*sin(f*x + e)/((8*f*m^3 + 36*f*m^2 + 46*f*m + 15*f)*cos(f*x + e)^3)
\[ \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-3-m} \, dx=\int \left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{m} \left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{- m - 3} \sec {\left (e + f x \right )}\, dx \]
Time = 0.31 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.92 \[ \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-3-m} \, dx=\frac {{\left ({\left (4 \, m^{2} + 8 \, m + 3\right )} \left (-a\right )^{m} - \frac {2 \, {\left (4 \, m^{2} + 12 \, m + 5\right )} \left (-a\right )^{m} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {{\left (4 \, m^{2} + 16 \, m + 15\right )} \left (-a\right )^{m} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )} c^{-m - 3} {\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{4 \, {\left (8 \, m^{3} + 36 \, m^{2} + 46 \, m + 15\right )} f \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )^{2 \, m} \sin \left (f x + e\right )^{5}} \]
1/4*((4*m^2 + 8*m + 3)*(-a)^m - 2*(4*m^2 + 12*m + 5)*(-a)^m*sin(f*x + e)^2 /(cos(f*x + e) + 1)^2 + (4*m^2 + 16*m + 15)*(-a)^m*sin(f*x + e)^4/(cos(f*x + e) + 1)^4)*c^(-m - 3)*(cos(f*x + e) + 1)^5/((8*m^3 + 36*m^2 + 46*m + 15 )*f*(sin(f*x + e)/(cos(f*x + e) + 1))^(2*m)*sin(f*x + e)^5)
\[ \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-3-m} \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{m} {\left (-c \sec \left (f x + e\right ) + c\right )}^{-m - 3} \sec \left (f x + e\right ) \,d x } \]
Time = 22.28 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.72 \[ \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-3-m} \, dx=-\frac {\left (\cos \left (3\,e+3\,f\,x\right )-\sin \left (3\,e+3\,f\,x\right )\,1{}\mathrm {i}\right )\,\left (\frac {\sin \left (e+f\,x\right )\,{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^m\,\left (\cos \left (3\,e+3\,f\,x\right )+\sin \left (3\,e+3\,f\,x\right )\,1{}\mathrm {i}\right )\,\left (4\,m^2+12\,m+15\right )\,2{}\mathrm {i}}{f\,\left (m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}\right )}-\frac {\sin \left (2\,e+2\,f\,x\right )\,\left (8\,m+12\right )\,{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^m\,\left (\cos \left (3\,e+3\,f\,x\right )+\sin \left (3\,e+3\,f\,x\right )\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{f\,\left (m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}\right )}+\frac {\sin \left (3\,e+3\,f\,x\right )\,{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^m\,\left (\cos \left (3\,e+3\,f\,x\right )+\sin \left (3\,e+3\,f\,x\right )\,1{}\mathrm {i}\right )\,\left (4\,m^2+12\,m+7\right )\,2{}\mathrm {i}}{f\,\left (m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}\right )}\right )}{8\,{\cos \left (e+f\,x\right )}^3\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{m+3}} \]
-((cos(3*e + 3*f*x) - sin(3*e + 3*f*x)*1i)*((sin(e + f*x)*(a + a/cos(e + f *x))^m*(cos(3*e + 3*f*x) + sin(3*e + 3*f*x)*1i)*(12*m + 4*m^2 + 15)*2i)/(f *(m*46i + m^2*36i + m^3*8i + 15i)) - (sin(2*e + 2*f*x)*(8*m + 12)*(a + a/c os(e + f*x))^m*(cos(3*e + 3*f*x) + sin(3*e + 3*f*x)*1i)*2i)/(f*(m*46i + m^ 2*36i + m^3*8i + 15i)) + (sin(3*e + 3*f*x)*(a + a/cos(e + f*x))^m*(cos(3*e + 3*f*x) + sin(3*e + 3*f*x)*1i)*(12*m + 4*m^2 + 7)*2i)/(f*(m*46i + m^2*36 i + m^3*8i + 15i))))/(8*cos(e + f*x)^3*(c - c/cos(e + f*x))^(m + 3))