\(\int \tan ^6(e+f x) (a+b \tan ^2(e+f x))^p \, dx\) [367]

Optimal result

Integrand size = 23, antiderivative size = 83 \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (\frac {7}{2},1,-p,\frac {9}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \tan ^7(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{7 f} \]

[Out]

1/7*AppellF1(7/2,1,-p,9/2,-tan(f*x+e)^2,-b*tan(f*x+e)^2/a)*tan(f*x+e)^7*(a+b*tan(f*x+e)^2)^p/f/((1+b*tan(f*x+e
)^2/a)^p)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 525, 524} \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {\tan ^7(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {7}{2},1,-p,\frac {9}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )}{7 f} \]

[In]

Int[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p,x]

[Out]

(AppellF1[7/2, 1, -p, 9/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^7*(a + b*Tan[e + f*x]^2)^p)/
(7*f*(1 + (b*Tan[e + f*x]^2)/a)^p)

Rule 524

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

Rule 525

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

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((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[(d*ff*(x/c))^m*((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, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^6 \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 {x^6 \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 {7}{2},1,-p,\frac {9}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \tan ^7(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}}{7 f} \\ \end{align*}

Mathematica [F]

\[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx \]

[In]

Integrate[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p,x]

[Out]

Integrate[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p, x]

Maple [F]

\[\int \tan \left (f x +e \right )^{6} \left (a +b \tan \left (f x +e \right )^{2}\right )^{p}d x\]

[In]

int(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^p,x)

[Out]

int(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^p,x)

Fricas [F]

\[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )^{6} \,d x } \]

[In]

integrate(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^p,x, algorithm="fricas")

[Out]

integral((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^6, x)

Sympy [F(-1)]

Timed out. \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\text {Timed out} \]

[In]

integrate(tan(f*x+e)**6*(a+b*tan(f*x+e)**2)**p,x)

[Out]

Timed out

Maxima [F]

\[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )^{6} \,d x } \]

[In]

integrate(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^p,x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^6, x)

Giac [F]

\[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )^{6} \,d x } \]

[In]

integrate(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^p,x, algorithm="giac")

[Out]

integrate((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^6, x)

Mupad [F(-1)]

Timed out. \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^6\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \]

[In]

int(tan(e + f*x)^6*(a + b*tan(e + f*x)^2)^p,x)

[Out]

int(tan(e + f*x)^6*(a + b*tan(e + f*x)^2)^p, x)