\(\int (a+b \tan ^2(e+f x))^p \, dx\) [370]

Optimal result

Integrand size = 14, antiderivative size = 78 \[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,-p,\frac {3}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{f} \]

[Out]

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/((1+b*tan(f*x+e)^2/a)
^p)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3742, 441, 440} \[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {\tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},1,-p,\frac {3}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )}{f} \]

[In]

Int[(a + b*Tan[e + f*x]^2)^p,x]

[Out]

(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)

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 3742

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[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])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\left (\left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \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 (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{f} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(192\) vs. \(2(78)=156\).

Time = 0.46 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.46 \[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {3 a \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sin (2 (e+f x)) \left (a+b \tan ^2(e+f x)\right )^p}{6 a f \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+4 f \left (b p \operatorname {AppellF1}\left (\frac {3}{2},1-p,1,\frac {5}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )-a \operatorname {AppellF1}\left (\frac {3}{2},-p,2,\frac {5}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)} \]

[In]

Integrate[(a + b*Tan[e + f*x]^2)^p,x]

[Out]

(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sin[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), -T
an[e + f*x]^2])*Tan[e + f*x]^2)

Maple [F]

\[\int \left (a +b \tan \left (f x +e \right )^{2}\right )^{p}d x\]

[In]

int((a+b*tan(f*x+e)^2)^p,x)

[Out]

int((a+b*tan(f*x+e)^2)^p,x)

Fricas [F]

\[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \,d x } \]

[In]

integrate((a+b*tan(f*x+e)^2)^p,x, algorithm="fricas")

[Out]

integral((b*tan(f*x + e)^2 + a)^p, x)

Sympy [F]

\[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{p}\, dx \]

[In]

integrate((a+b*tan(f*x+e)**2)**p,x)

[Out]

Integral((a + b*tan(e + f*x)**2)**p, x)

Maxima [F]

\[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \,d x } \]

[In]

integrate((a+b*tan(f*x+e)^2)^p,x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e)^2 + a)^p, x)

Giac [F]

\[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \,d x } \]

[In]

integrate((a+b*tan(f*x+e)^2)^p,x, algorithm="giac")

[Out]

integrate((b*tan(f*x + e)^2 + a)^p, x)

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \]

[In]

int((a + b*tan(e + f*x)^2)^p,x)

[Out]

int((a + b*tan(e + f*x)^2)^p, x)