Integrand size = 45, antiderivative size = 235 \[ \int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {A \sec ^{-n}(c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1+n)}+\frac {2^{1+n} (A n+B (1+n)-C (1+n)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+n,\frac {3}{2},\frac {1-\sec (c+d x)}{1+\sec (c+d x)}\right ) \sec ^{-n}(c+d x) \left (\frac {\sec (c+d x)}{1+\sec (c+d x)}\right )^{1+n} (a+a \sec (c+d x))^n \sin (c+d x)}{d (1+n)}+\frac {2^{\frac {3}{2}+n} C \operatorname {AppellF1}\left (\frac {1}{2},1+n,-\frac {1}{2}-n,\frac {3}{2},1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right ) (1+\sec (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \tan (c+d x)}{d} \] Output:
A*(a+a*sec(d*x+c))^n*sin(d*x+c)/d/(1+n)/(sec(d*x+c)^n)+2^(1+n)*(A*n+B*(1+n )-C*(1+n))*hypergeom([1/2, 1+n],[3/2],(1-sec(d*x+c))/(1+sec(d*x+c)))*(sec( d*x+c)/(1+sec(d*x+c)))^(1+n)*(a+a*sec(d*x+c))^n*sin(d*x+c)/d/(1+n)/(sec(d* x+c)^n)+2^(3/2+n)*C*AppellF1(1/2,1+n,-1/2-n,3/2,1-sec(d*x+c),1/2-1/2*sec(d *x+c))*(1+sec(d*x+c))^(-1/2-n)*(a+a*sec(d*x+c))^n*tan(d*x+c)/d
\[ \int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx \] Input:
Integrate[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]
Output:
Integrate[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 = 1.05 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3042, 4574, 3042, 4511, 3042, 4315, 3042, 4312, 142, 150}
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.
| \(\displaystyle \int \sec ^{-n-1}(c+d x) (a \sec (c+d x)+a)^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^{-n-1} \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^n \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4574 |
| \(\displaystyle \frac {\int \sec ^{-n}(c+d x) (\sec (c+d x) a+a)^n (a (n B+B+A n)+a C (n+1) \sec (c+d x))dx}{a (n+1)}+\frac {A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^{-n} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^n \left (a (n B+B+A n)+a C (n+1) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a (n+1)}+\frac {A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}\) |
| \(\Big \downarrow \) 4511 |
| \(\displaystyle \frac {a (A n+B n+B-C (n+1)) \int \sec ^{-n}(c+d x) (\sec (c+d x) a+a)^ndx+C (n+1) \int \sec ^{-n}(c+d x) (\sec (c+d x) a+a)^{n+1}dx}{a (n+1)}+\frac {A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A n+B n+B-C (n+1)) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{-n} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^ndx+C (n+1) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{-n} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{n+1}dx}{a (n+1)}+\frac {A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}\) |
| \(\Big \downarrow \) 4315 |
| \(\displaystyle \frac {a (A n+B n+B-C (n+1)) (\sec (c+d x)+1)^{-n} (a \sec (c+d x)+a)^n \int \sec ^{-n}(c+d x) (\sec (c+d x)+1)^ndx+a C (n+1) (\sec (c+d x)+1)^{-n} (a \sec (c+d x)+a)^n \int \sec ^{-n}(c+d x) (\sec (c+d x)+1)^{n+1}dx}{a (n+1)}+\frac {A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A n+B n+B-C (n+1)) (\sec (c+d x)+1)^{-n} (a \sec (c+d x)+a)^n \int \csc \left (c+d x+\frac {\pi }{2}\right )^{-n} \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )^ndx+a C (n+1) (\sec (c+d x)+1)^{-n} (a \sec (c+d x)+a)^n \int \csc \left (c+d x+\frac {\pi }{2}\right )^{-n} \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )^{n+1}dx}{a (n+1)}+\frac {A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}\) |
| \(\Big \downarrow \) 4312 |
| \(\displaystyle \frac {\frac {a (A n+B n+B-C (n+1)) \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \int \frac {\sec ^{-n-1}(c+d x) (\sec (c+d x)+1)^{n-\frac {1}{2}}}{\sqrt {1-\sec (c+d x)}}d(1-\sec (c+d x))}{d \sqrt {1-\sec (c+d x)}}+\frac {a C (n+1) \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \int \frac {\sec ^{-n-1}(c+d x) (\sec (c+d x)+1)^{n+\frac {1}{2}}}{\sqrt {1-\sec (c+d x)}}d(1-\sec (c+d x))}{d \sqrt {1-\sec (c+d x)}}}{a (n+1)}+\frac {A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}\) |
| \(\Big \downarrow \) 142 |
| \(\displaystyle \frac {\frac {a C (n+1) \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \int \frac {\sec ^{-n-1}(c+d x) (\sec (c+d x)+1)^{n+\frac {1}{2}}}{\sqrt {1-\sec (c+d x)}}d(1-\sec (c+d x))}{d \sqrt {1-\sec (c+d x)}}+\frac {a (A n+B n+B-C (n+1)) \sin (c+d x) \sec ^{1-n}(c+d x) \left (\frac {\sec (c+d x)+1}{1-\sec (c+d x)}\right )^{\frac {1}{2}-n} (a \sec (c+d x)+a)^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-n,-n,1-n,-\frac {2 \sec (c+d x)}{1-\sec (c+d x)}\right )}{d n (\sec (c+d x)+1)}}{a (n+1)}+\frac {A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle \frac {\frac {a (A n+B n+B-C (n+1)) \sin (c+d x) \sec ^{1-n}(c+d x) \left (\frac {\sec (c+d x)+1}{1-\sec (c+d x)}\right )^{\frac {1}{2}-n} (a \sec (c+d x)+a)^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-n,-n,1-n,-\frac {2 \sec (c+d x)}{1-\sec (c+d x)}\right )}{d n (\sec (c+d x)+1)}+\frac {a C 2^{n+\frac {3}{2}} (n+1) \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \operatorname {AppellF1}\left (\frac {1}{2},n+1,-n-\frac {1}{2},\frac {3}{2},1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right )}{d}}{a (n+1)}+\frac {A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}\) |
Input:
Int[Sec[c + d*x]^(-1 - n)*(a + a*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*S ec[c + d*x]^2),x]
Output:
(A*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + n)*Sec[c + d*x]^n) + ((a*( B + A*n + B*n - C*(1 + n))*Hypergeometric2F1[1/2 - n, -n, 1 - n, (-2*Sec[c + d*x])/(1 - Sec[c + d*x])]*Sec[c + d*x]^(1 - n)*((1 + Sec[c + d*x])/(1 - Sec[c + d*x]))^(1/2 - n)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*(1 + S ec[c + d*x])) + (2^(3/2 + n)*a*C*(1 + n)*AppellF1[1/2, 1 + n, -1/2 - n, 3/ 2, 1 - Sec[c + d*x], (1 - Sec[c + d*x])/2]*(1 + Sec[c + d*x])^(-1/2 - n)*( a + a*Sec[c + d*x])^n*Tan[c + d*x])/d)/(a*(1 + n))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f *x))))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] && !IntegerQ[n]
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[(-(a*(d/b))^n)*(Cot[e + f*x]/(a^(n - 2)*f*Sqrt [a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])) Subst[Int[(a - x)^(n - 1) *((2*a - x)^(m - 1/2)/Sqrt[x]), x], x, a - b*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && GtQ[a, 0] & & !IntegerQ[n] && GtQ[a*(d/b), 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a^IntPart[m]*((a + b*Csc[e + f*x])^FracPart[m ]/(1 + (b/a)*Csc[e + f*x])^FracPart[m]) Int[(1 + (b/a)*Csc[e + f*x])^m*(d *Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^ 2, 0] && !IntegerQ[m] && !GtQ[a, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(A*b - a*B)/b Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n, x], x] + Simp[B/b Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b , d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0]
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[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d*n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[ e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x] , x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
\[\int \sec \left (d x +c \right )^{-1-n} \left (a +a \sec \left (d x +c \right )\right )^{n} \left (A +B \sec \left (d x +c \right )+C \sec \left (d x +c \right )^{2}\right )d x\]
Input:
int(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 )
Output:
int(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 )
\[ \int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )^{-n - 1} \,d x } \] Input:
integrate(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")
Output:
integral((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^n*se c(d*x + c)^(-n - 1), x)
\[ \int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{- n - 1}{\left (c + d x \right )}\, dx \] Input:
integrate(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)
Output:
Integral((a*(sec(c + d*x) + 1))**n*(A + B*sec(c + d*x) + C*sec(c + d*x)**2 )*sec(c + d*x)**(-n - 1), x)
\[ \int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )^{-n - 1} \,d x } \] Input:
integrate(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")
Output:
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), x)
\[ \int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )^{-n - 1} \,d x } \] Input:
integrate(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")
Output:
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), x)
Timed out. \[ \int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{n+1}} \,d x \] Input:
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),x)
Output:
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), x)
\[ \int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \frac {\left (\sec \left (d x +c \right ) a +a \right )^{n}}{\sec \left (d x +c \right )^{n}}d x \right ) b +\left (\int \frac {\left (\sec \left (d x +c \right ) a +a \right )^{n}}{\sec \left (d x +c \right )^{n} \sec \left (d x +c \right )}d x \right ) a +\left (\int \frac {\left (\sec \left (d x +c \right ) a +a \right )^{n} \sec \left (d x +c \right )}{\sec \left (d x +c \right )^{n}}d x \right ) c \] Input:
int(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 )
Output:
int((sec(c + d*x)*a + a)**n/sec(c + d*x)**n,x)*b + int((sec(c + d*x)*a + a )**n/(sec(c + d*x)**n*sec(c + d*x)),x)*a + int(((sec(c + d*x)*a + a)**n*se c(c + d*x))/sec(c + d*x)**n,x)*c