Integrand size = 21, antiderivative size = 103 \[ \int \sec (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (\frac {1}{2},1+p,-p,\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^2(e+f x)^p \sin (e+f x) \left (\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )\right )^p \left (1-\frac {a \sin ^2(e+f x)}{a+b}\right )^{-p}}{f} \]
AppellF1(1/2,p+1,-p,3/2,sin(f*x+e)^2,a*sin(f*x+e)^2/(a+b))*(cos(f*x+e)^2)^ p*sin(f*x+e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^p/f/((1-a*sin(f*x+e)^2/(a +b))^p)
Leaf count is larger than twice the leaf count of optimal. \(1995\) vs. \(2(103)=206\).
Time = 17.42 (sec) , antiderivative size = 1995, normalized size of antiderivative = 19.37 \[ \int \sec (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx =\text {Too large to display} \]
(3*(a + b)*AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^ 2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*Sec[e + f*x]*(Sec[e + f*x]^2 )^(-1/2 + p)*(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x])/(f*(3*(a + b)*AppellF1 [1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b *p*AppellF1[3/2, 1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - (a + b)*AppellF1[3/2, 3/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)*((3*(a + b)*AppellF1[1/2, 1/2, -p, 3/ 2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(1/2 + p))/(3*(a + b)*AppellF1[1/2, 1/2, -p, 3/ 2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - (a + b) *AppellF1[3/2, 3/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b) )])*Tan[e + f*x]^2) - (6*a*(a + b)*p*AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^(-1/2 + p)*Sin[2*(e + f*x)]*Tan[e + f*x])/(3*(a + b) *AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b) )] + (2*b*p*AppellF1[3/2, 1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f *x]^2)/(a + b))] - (a + b)*AppellF1[3/2, 3/2, -p, 5/2, -Tan[e + f*x]^2, -( (b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (6*(a + b)*(-1/2 + p)*Appe llF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]...
Time = 0.31 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4636, 2057, 2058, 334, 333}
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.
\(\displaystyle \int \sec (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sec (e+f x) \left (a+b \sec (e+f x)^2\right )^pdx\) |
\(\Big \downarrow \) 4636 |
\(\displaystyle \frac {\int \frac {\left (a+\frac {b}{1-\sin ^2(e+f x)}\right )^p}{1-\sin ^2(e+f x)}d\sin (e+f x)}{f}\) |
\(\Big \downarrow \) 2057 |
\(\displaystyle \frac {\int \frac {\left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^p}{1-\sin ^2(e+f x)}d\sin (e+f x)}{f}\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \frac {\left (1-\sin ^2(e+f x)\right )^p \left (-a \sin ^2(e+f x)+a+b\right )^{-p} \left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^p \int \left (1-\sin ^2(e+f x)\right )^{-p-1} \left (-a \sin ^2(e+f x)+a+b\right )^pd\sin (e+f x)}{f}\) |
\(\Big \downarrow \) 334 |
\(\displaystyle \frac {\left (1-\sin ^2(e+f x)\right )^p \left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^p \left (1-\frac {a \sin ^2(e+f x)}{a+b}\right )^{-p} \int \left (1-\sin ^2(e+f x)\right )^{-p-1} \left (1-\frac {a \sin ^2(e+f x)}{a+b}\right )^pd\sin (e+f x)}{f}\) |
\(\Big \downarrow \) 333 |
\(\displaystyle \frac {\sin (e+f x) \left (1-\sin ^2(e+f x)\right )^p \left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^p \left (1-\frac {a \sin ^2(e+f x)}{a+b}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},p+1,-p,\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )}{f}\) |
(AppellF1[1/2, 1 + p, -p, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] *Sin[e + f*x]*(1 - Sin[e + f*x]^2)^p*((a + b - a*Sin[e + f*x]^2)/(1 - Sin[ e + f*x]^2))^p)/(f*(1 - (a*Sin[e + f*x]^2)/(a + b))^p)
3.3.100.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[ (1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && !(IntegerQ[p] || GtQ[a, 0])
Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ ))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[(a + b/(1 - ff^2*x^2)^(n/2))^p/(1 - ff^2*x^2)^((m + 1)/2), x], x , Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && !IntegerQ[p]
\[\int \sec \left (f x +e \right ) \left (a +b \sec \left (f x +e \right )^{2}\right )^{p}d x\]
\[ \int \sec (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \sec \left (f x + e\right ) \,d x } \]
\[ \int \sec (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{p} \sec {\left (e + f x \right )}\, dx \]
\[ \int \sec (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \sec \left (f x + e\right ) \,d x } \]
\[ \int \sec (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \sec \left (f x + e\right ) \,d x } \]
Timed out. \[ \int \sec (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int \frac {{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^p}{\cos \left (e+f\,x\right )} \,d x \]