Integrand size = 23, antiderivative size = 85 \[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,-p,\frac {3}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a+b}\right ) \tan (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^p \left (\frac {a+b+b \tan ^2(e+f x)}{a+b}\right )^{-p}}{f} \] Output:
AppellF1(1/2,2,-p,3/2,-tan(f*x+e)^2,-b*tan(f*x+e)^2/(a+b))*tan(f*x+e)*(a+b +b*tan(f*x+e)^2)^p/f/(((a+b+b*tan(f*x+e)^2)/(a+b))^p)
Leaf count is larger than twice the leaf count of optimal. \(1343\) vs. \(2(85)=170\).
Time = 14.16 (sec) , antiderivative size = 1343, normalized size of antiderivative = 15.80 \[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx =\text {Too large to display} \] Input:
Integrate[Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p,x]
Output:
(AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] *Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]*(3*(a + b)*AppellF1[1/ 2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*Ap pellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^ 2)/(a + b))])*Tan[e + f*x]^2))/(f*(AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x] ^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*(3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*Appe llF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2) /(a + b))])*Tan[e + f*x]^2) - (2*a*p*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f* x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sin[2*(e + f*x)]*Tan[e + f*x]*(3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b ))] + 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f* x]^2)/(a + b))] - 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -(( b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2))/(a + 2*b + a*Cos[2*(e + f*x) ]) + 2*(-2 + p)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f* x]^2)/(a + b))]*Tan[e + f*x]^2*(3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 2*(a + b)*Appel...
Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4634, 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.
\(\displaystyle \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \sec (e+f x)^2\right )^p}{\sec (e+f x)^2}dx\) |
\(\Big \downarrow \) 4634 |
\(\displaystyle \frac {\int \frac {\left (b \tan ^2(e+f x)+a+b\right )^p}{\left (\tan ^2(e+f x)+1\right )^2}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 334 |
\(\displaystyle \frac {\left (a+b \tan ^2(e+f x)+b\right )^p \left (\frac {b \tan ^2(e+f x)}{a+b}+1\right )^{-p} \int \frac {\left (\frac {b \tan ^2(e+f x)}{a+b}+1\right )^p}{\left (\tan ^2(e+f x)+1\right )^2}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 333 |
\(\displaystyle \frac {\tan (e+f x) \left (a+b \tan ^2(e+f x)+b\right )^p \left (\frac {b \tan ^2(e+f x)}{a+b}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},2,-p,\frac {3}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a+b}\right )}{f}\) |
Input:
Int[Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p,x]
Output:
(AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] *Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)
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[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_) )^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ [m/2] && IntegerQ[n/2]
\[\int \cos \left (f x +e \right )^{2} \left (a +b \sec \left (f x +e \right )^{2}\right )^{p}d x\]
Input:
int(cos(f*x+e)^2*(a+b*sec(f*x+e)^2)^p,x)
Output:
int(cos(f*x+e)^2*(a+b*sec(f*x+e)^2)^p,x)
\[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{2} \,d x } \] Input:
integrate(cos(f*x+e)^2*(a+b*sec(f*x+e)^2)^p,x, algorithm="fricas")
Output:
integral((b*sec(f*x + e)^2 + a)^p*cos(f*x + e)^2, x)
Timed out. \[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\text {Timed out} \] Input:
integrate(cos(f*x+e)**2*(a+b*sec(f*x+e)**2)**p,x)
Output:
Timed out
\[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{2} \,d x } \] Input:
integrate(cos(f*x+e)^2*(a+b*sec(f*x+e)^2)^p,x, algorithm="maxima")
Output:
integrate((b*sec(f*x + e)^2 + a)^p*cos(f*x + e)^2, x)
\[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{2} \,d x } \] Input:
integrate(cos(f*x+e)^2*(a+b*sec(f*x+e)^2)^p,x, algorithm="giac")
Output:
integrate((b*sec(f*x + e)^2 + a)^p*cos(f*x + e)^2, x)
Timed out. \[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int {\cos \left (e+f\,x\right )}^2\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^p \,d x \] Input:
int(cos(e + f*x)^2*(a + b/cos(e + f*x)^2)^p,x)
Output:
int(cos(e + f*x)^2*(a + b/cos(e + f*x)^2)^p, x)
\[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int \left (\sec \left (f x +e \right )^{2} b +a \right )^{p} \cos \left (f x +e \right )^{2}d x \] Input:
int(cos(f*x+e)^2*(a+b*sec(f*x+e)^2)^p,x)
Output:
int((sec(e + f*x)**2*b + a)**p*cos(e + f*x)**2,x)