Integrand size = 14, antiderivative size = 79 \[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,-p,\frac {3}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {a+b \tan ^2(e+f x)}{a}\right )^{-p}}{f} \] Output:
AppellF1(1/2,1,-p,3/2,-tan(f*x+e)^2,-b*tan(f*x+e)^2/a)*tan(f*x+e)*(a+b*tan (f*x+e)^2)^p/f/(((a+b*tan(f*x+e)^2)/a)^p)
Leaf count is larger than twice the leaf count of optimal. \(192\) vs. \(2(79)=158\).
Time = 0.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.43 \[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {3 a \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sin (2 (e+f x)) \left (a+b \tan ^2(e+f x)\right )^p}{6 a f \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+4 f \left (b p \operatorname {AppellF1}\left (\frac {3}{2},1-p,1,\frac {5}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )-a \operatorname {AppellF1}\left (\frac {3}{2},-p,2,\frac {5}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)} \] Input:
Integrate[(a + b*Tan[e + f*x]^2)^p,x]
Output:
(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*S in[2*(e + f*x)]*(a + b*Tan[e + f*x]^2)^p)/(6*a*f*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 4*f*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2 , 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)
Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4144, 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.
\(\displaystyle \int \left (a+b \tan ^2(e+f x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a+b \tan (e+f x)^2\right )^pdx\) |
\(\Big \downarrow \) 4144 |
\(\displaystyle \frac {\int \frac {\left (b \tan ^2(e+f x)+a\right )^p}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 334 |
\(\displaystyle \frac {\left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \int \frac {\left (\frac {b \tan ^2(e+f x)}{a}+1\right )^p}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 333 |
\(\displaystyle \frac {\tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},1,-p,\frac {3}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )}{f}\) |
Input:
Int[(a + b*Tan[e + f*x]^2)^p,x]
Output:
(AppellF1[1/2, 1, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^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[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(a + b* (ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
\[\int \left (a +b \tan \left (f x +e \right )^{2}\right )^{p}d x\]
Input:
int((a+b*tan(f*x+e)^2)^p,x)
Output:
int((a+b*tan(f*x+e)^2)^p,x)
\[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \,d x } \] Input:
integrate((a+b*tan(f*x+e)^2)^p,x, algorithm="fricas")
Output:
integral((b*tan(f*x + e)^2 + a)^p, x)
\[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{p}\, dx \] Input:
integrate((a+b*tan(f*x+e)**2)**p,x)
Output:
Integral((a + b*tan(e + f*x)**2)**p, x)
\[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \,d x } \] Input:
integrate((a+b*tan(f*x+e)^2)^p,x, algorithm="maxima")
Output:
integrate((b*tan(f*x + e)^2 + a)^p, x)
\[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \,d x } \] Input:
integrate((a+b*tan(f*x+e)^2)^p,x, algorithm="giac")
Output:
integrate((b*tan(f*x + e)^2 + a)^p, x)
Timed out. \[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \] Input:
int((a + b*tan(e + f*x)^2)^p,x)
Output:
int((a + b*tan(e + f*x)^2)^p, x)
\[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int \left (\tan \left (f x +e \right )^{2} b +a \right )^{p}d x \] Input:
int((a+b*tan(f*x+e)^2)^p,x)
Output:
int((tan(e + f*x)**2*b + a)**p,x)