\(\int \tan ^6(e+f x) (a+b \tan ^2(e+f x))^2 \, dx\) [204]

Optimal result

Integrand size = 23, antiderivative size = 113 \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=-(a-b)^2 x+\frac {(a-b)^2 \tan (e+f x)}{f}-\frac {(a-b)^2 \tan ^3(e+f x)}{3 f}+\frac {(a-b)^2 \tan ^5(e+f x)}{5 f}+\frac {(2 a-b) b \tan ^7(e+f x)}{7 f}+\frac {b^2 \tan ^9(e+f x)}{9 f} \]

[Out]

-(a-b)^2*x+(a-b)^2*tan(f*x+e)/f-1/3*(a-b)^2*tan(f*x+e)^3/f+1/5*(a-b)^2*tan(f*x+e)^5/f+1/7*(2*a-b)*b*tan(f*x+e)
^7/f+1/9*b^2*tan(f*x+e)^9/f

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 472, 209} \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=\frac {b (2 a-b) \tan ^7(e+f x)}{7 f}+\frac {(a-b)^2 \tan ^5(e+f x)}{5 f}-\frac {(a-b)^2 \tan ^3(e+f x)}{3 f}+\frac {(a-b)^2 \tan (e+f x)}{f}-x (a-b)^2+\frac {b^2 \tan ^9(e+f x)}{9 f} \]

[In]

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

[Out]

-((a - b)^2*x) + ((a - b)^2*Tan[e + f*x])/f - ((a - b)^2*Tan[e + f*x]^3)/(3*f) + ((a - b)^2*Tan[e + f*x]^5)/(5
*f) + ((2*a - b)*b*Tan[e + f*x]^7)/(7*f) + (b^2*Tan[e + f*x]^9)/(9*f)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

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 )^2}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left ((a-b)^2-(a-b)^2 x^2+(a-b)^2 x^4+(2 a-b) b x^6+b^2 x^8+\frac {-a^2+2 a b-b^2}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(a-b)^2 \tan (e+f x)}{f}-\frac {(a-b)^2 \tan ^3(e+f x)}{3 f}+\frac {(a-b)^2 \tan ^5(e+f x)}{5 f}+\frac {(2 a-b) b \tan ^7(e+f x)}{7 f}+\frac {b^2 \tan ^9(e+f x)}{9 f}-\frac {(a-b)^2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -(a-b)^2 x+\frac {(a-b)^2 \tan (e+f x)}{f}-\frac {(a-b)^2 \tan ^3(e+f x)}{3 f}+\frac {(a-b)^2 \tan ^5(e+f x)}{5 f}+\frac {(2 a-b) b \tan ^7(e+f x)}{7 f}+\frac {b^2 \tan ^9(e+f x)}{9 f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(243\) vs. \(2(113)=226\).

Time = 0.10 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.15 \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=-\frac {a^2 \arctan (\tan (e+f x))}{f}+\frac {2 a b \arctan (\tan (e+f x))}{f}-\frac {b^2 \arctan (\tan (e+f x))}{f}+\frac {a^2 \tan (e+f x)}{f}-\frac {2 a b \tan (e+f x)}{f}+\frac {b^2 \tan (e+f x)}{f}-\frac {a^2 \tan ^3(e+f x)}{3 f}+\frac {2 a b \tan ^3(e+f x)}{3 f}-\frac {b^2 \tan ^3(e+f x)}{3 f}+\frac {a^2 \tan ^5(e+f x)}{5 f}-\frac {2 a b \tan ^5(e+f x)}{5 f}+\frac {b^2 \tan ^5(e+f x)}{5 f}+\frac {2 a b \tan ^7(e+f x)}{7 f}-\frac {b^2 \tan ^7(e+f x)}{7 f}+\frac {b^2 \tan ^9(e+f x)}{9 f} \]

[In]

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

[Out]

-((a^2*ArcTan[Tan[e + f*x]])/f) + (2*a*b*ArcTan[Tan[e + f*x]])/f - (b^2*ArcTan[Tan[e + f*x]])/f + (a^2*Tan[e +
f*x])/f - (2*a*b*Tan[e + f*x])/f + (b^2*Tan[e + f*x])/f - (a^2*Tan[e + f*x]^3)/(3*f) + (2*a*b*Tan[e + f*x]^3)
/(3*f) - (b^2*Tan[e + f*x]^3)/(3*f) + (a^2*Tan[e + f*x]^5)/(5*f) - (2*a*b*Tan[e + f*x]^5)/(5*f) + (b^2*Tan[e +
f*x]^5)/(5*f) + (2*a*b*Tan[e + f*x]^7)/(7*f) - (b^2*Tan[e + f*x]^7)/(7*f) + (b^2*Tan[e + f*x]^9)/(9*f)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.11

method	result	size
norman	\(\left (-a^{2}+2 a b -b^{2}\right ) x +\frac {\left (a^{2}-2 a b +b^{2}\right ) \tan \left (f x +e \right )}{f}+\frac {b^{2} \tan \left (f x +e \right )^{9}}{9 f}-\frac {\left (a^{2}-2 a b +b^{2}\right ) \tan \left (f x +e \right )^{3}}{3 f}+\frac {\left (a^{2}-2 a b +b^{2}\right ) \tan \left (f x +e \right )^{5}}{5 f}+\frac {\left (2 a -b \right ) b \tan \left (f x +e \right )^{7}}{7 f}\)	\(125\)
parts	\(\frac {a^{2} \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {b^{2} \left (\frac {\tan \left (f x +e \right )^{9}}{9}-\frac {\tan \left (f x +e \right )^{7}}{7}+\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {2 a b \left (\frac {\tan \left (f x +e \right )^{7}}{7}-\frac {\tan \left (f x +e \right )^{5}}{5}+\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\)	\(161\)
derivativedivides	\(\frac {\frac {b^{2} \tan \left (f x +e \right )^{9}}{9}+\frac {2 a b \tan \left (f x +e \right )^{7}}{7}-\frac {b^{2} \tan \left (f x +e \right )^{7}}{7}+\frac {a^{2} \tan \left (f x +e \right )^{5}}{5}-\frac {2 a b \tan \left (f x +e \right )^{5}}{5}+\frac {b^{2} \tan \left (f x +e \right )^{5}}{5}-\frac {a^{2} \tan \left (f x +e \right )^{3}}{3}+\frac {2 a b \tan \left (f x +e \right )^{3}}{3}-\frac {b^{2} \tan \left (f x +e \right )^{3}}{3}+a^{2} \tan \left (f x +e \right )-2 a b \tan \left (f x +e \right )+b^{2} \tan \left (f x +e \right )+\left (-a^{2}+2 a b -b^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\)	\(173\)
default	\(\frac {\frac {b^{2} \tan \left (f x +e \right )^{9}}{9}+\frac {2 a b \tan \left (f x +e \right )^{7}}{7}-\frac {b^{2} \tan \left (f x +e \right )^{7}}{7}+\frac {a^{2} \tan \left (f x +e \right )^{5}}{5}-\frac {2 a b \tan \left (f x +e \right )^{5}}{5}+\frac {b^{2} \tan \left (f x +e \right )^{5}}{5}-\frac {a^{2} \tan \left (f x +e \right )^{3}}{3}+\frac {2 a b \tan \left (f x +e \right )^{3}}{3}-\frac {b^{2} \tan \left (f x +e \right )^{3}}{3}+a^{2} \tan \left (f x +e \right )-2 a b \tan \left (f x +e \right )+b^{2} \tan \left (f x +e \right )+\left (-a^{2}+2 a b -b^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\)	\(173\)
parallelrisch	\(-\frac {-35 b^{2} \tan \left (f x +e \right )^{9}-90 a b \tan \left (f x +e \right )^{7}+45 b^{2} \tan \left (f x +e \right )^{7}-63 a^{2} \tan \left (f x +e \right )^{5}+126 a b \tan \left (f x +e \right )^{5}-63 b^{2} \tan \left (f x +e \right )^{5}+105 a^{2} \tan \left (f x +e \right )^{3}-210 a b \tan \left (f x +e \right )^{3}+105 b^{2} \tan \left (f x +e \right )^{3}+315 a^{2} f x -630 a b f x +315 b^{2} f x -315 a^{2} \tan \left (f x +e \right )+630 a b \tan \left (f x +e \right )-315 b^{2} \tan \left (f x +e \right )}{315 f}\)	\(173\)
risch	\(-x \,a^{2}+2 x a b -x \,b^{2}+\frac {2 i \left (483 a^{2}+563 b^{2}+32508 a^{2} {\mathrm e}^{8 i \left (f x +e \right )}+24402 a^{2} {\mathrm e}^{6 i \left (f x +e \right )}+26292 b^{2} {\mathrm e}^{6 i \left (f x +e \right )}+11718 a^{2} {\mathrm e}^{4 i \left (f x +e \right )}+13968 b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+3402 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}+3492 b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+5670 a^{2} {\mathrm e}^{14 i \left (f x +e \right )}+6300 b^{2} {\mathrm e}^{14 i \left (f x +e \right )}-12600 a b \,{\mathrm e}^{14 i \left (f x +e \right )}-1056 a b -2520 a b \,{\mathrm e}^{16 i \left (f x +e \right )}-36120 a b \,{\mathrm e}^{12 i \left (f x +e \right )}-63000 a b \,{\mathrm e}^{10 i \left (f x +e \right )}-52584 a b \,{\mathrm e}^{6 i \left (f x +e \right )}-6984 a b \,{\mathrm e}^{2 i \left (f x +e \right )}-70056 a b \,{\mathrm e}^{8 i \left (f x +e \right )}-25416 a b \,{\mathrm e}^{4 i \left (f x +e \right )}+16170 a^{2} {\mathrm e}^{12 i \left (f x +e \right )}+21000 b^{2} {\mathrm e}^{12 i \left (f x +e \right )}+28350 a^{2} {\mathrm e}^{10 i \left (f x +e \right )}+31500 b^{2} {\mathrm e}^{10 i \left (f x +e \right )}+39438 b^{2} {\mathrm e}^{8 i \left (f x +e \right )}+1575 b^{2} {\mathrm e}^{16 i \left (f x +e \right )}+945 a^{2} {\mathrm e}^{16 i \left (f x +e \right )}\right )}{315 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{9}}\)	\(381\)

[In]

int(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^2,x,method=_RETURNVERBOSE)

[Out]

(-a^2+2*a*b-b^2)*x+(a^2-2*a*b+b^2)/f*tan(f*x+e)+1/9*b^2*tan(f*x+e)^9/f-1/3*(a^2-2*a*b+b^2)/f*tan(f*x+e)^3+1/5*
(a^2-2*a*b+b^2)/f*tan(f*x+e)^5+1/7*(2*a-b)*b*tan(f*x+e)^7/f

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.02 \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=\frac {35 \, b^{2} \tan \left (f x + e\right )^{9} + 45 \, {\left (2 \, a b - b^{2}\right )} \tan \left (f x + e\right )^{7} + 63 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{5} - 105 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{3} - 315 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} f x + 315 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )}{315 \, f} \]

[In]

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

[Out]

1/315*(35*b^2*tan(f*x + e)^9 + 45*(2*a*b - b^2)*tan(f*x + e)^7 + 63*(a^2 - 2*a*b + b^2)*tan(f*x + e)^5 - 105*(
a^2 - 2*a*b + b^2)*tan(f*x + e)^3 - 315*(a^2 - 2*a*b + b^2)*f*x + 315*(a^2 - 2*a*b + b^2)*tan(f*x + e))/f

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (88) = 176\).

Time = 0.30 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.88 \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=\begin {cases} - a^{2} x + \frac {a^{2} \tan ^{5}{\left (e + f x \right )}}{5 f} - \frac {a^{2} \tan ^{3}{\left (e + f x \right )}}{3 f} + \frac {a^{2} \tan {\left (e + f x \right )}}{f} + 2 a b x + \frac {2 a b \tan ^{7}{\left (e + f x \right )}}{7 f} - \frac {2 a b \tan ^{5}{\left (e + f x \right )}}{5 f} + \frac {2 a b \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 a b \tan {\left (e + f x \right )}}{f} - b^{2} x + \frac {b^{2} \tan ^{9}{\left (e + f x \right )}}{9 f} - \frac {b^{2} \tan ^{7}{\left (e + f x \right )}}{7 f} + \frac {b^{2} \tan ^{5}{\left (e + f x \right )}}{5 f} - \frac {b^{2} \tan ^{3}{\left (e + f x \right )}}{3 f} + \frac {b^{2} \tan {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a + b \tan ^{2}{\left (e \right )}\right )^{2} \tan ^{6}{\left (e \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-a**2*x + a**2*tan(e + f*x)**5/(5*f) - a**2*tan(e + f*x)**3/(3*f) + a**2*tan(e + f*x)/f + 2*a*b*x +
2*a*b*tan(e + f*x)**7/(7*f) - 2*a*b*tan(e + f*x)**5/(5*f) + 2*a*b*tan(e + f*x)**3/(3*f) - 2*a*b*tan(e + f*x)/
f - b**2*x + b**2*tan(e + f*x)**9/(9*f) - b**2*tan(e + f*x)**7/(7*f) + b**2*tan(e + f*x)**5/(5*f) - b**2*tan(e
+ f*x)**3/(3*f) + b**2*tan(e + f*x)/f, Ne(f, 0)), (x*(a + b*tan(e)**2)**2*tan(e)**6, True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04 \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=\frac {35 \, b^{2} \tan \left (f x + e\right )^{9} + 45 \, {\left (2 \, a b - b^{2}\right )} \tan \left (f x + e\right )^{7} + 63 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{5} - 105 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{3} - 315 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} {\left (f x + e\right )} + 315 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )}{315 \, f} \]

[In]

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

[Out]

1/315*(35*b^2*tan(f*x + e)^9 + 45*(2*a*b - b^2)*tan(f*x + e)^7 + 63*(a^2 - 2*a*b + b^2)*tan(f*x + e)^5 - 105*(
a^2 - 2*a*b + b^2)*tan(f*x + e)^3 - 315*(a^2 - 2*a*b + b^2)*(f*x + e) + 315*(a^2 - 2*a*b + b^2)*tan(f*x + e))/
f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2455 vs. \(2 (105) = 210\).

Time = 8.00 (sec) , antiderivative size = 2455, normalized size of antiderivative = 21.73 \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/315*(315*a^2*f*x*tan(f*x)^9*tan(e)^9 - 630*a*b*f*x*tan(f*x)^9*tan(e)^9 + 315*b^2*f*x*tan(f*x)^9*tan(e)^9 -
2835*a^2*f*x*tan(f*x)^8*tan(e)^8 + 5670*a*b*f*x*tan(f*x)^8*tan(e)^8 - 2835*b^2*f*x*tan(f*x)^8*tan(e)^8 + 315*a
^2*tan(f*x)^9*tan(e)^8 - 630*a*b*tan(f*x)^9*tan(e)^8 + 315*b^2*tan(f*x)^9*tan(e)^8 + 315*a^2*tan(f*x)^8*tan(e)
^9 - 630*a*b*tan(f*x)^8*tan(e)^9 + 315*b^2*tan(f*x)^8*tan(e)^9 + 11340*a^2*f*x*tan(f*x)^7*tan(e)^7 - 22680*a*b
*f*x*tan(f*x)^7*tan(e)^7 + 11340*b^2*f*x*tan(f*x)^7*tan(e)^7 - 105*a^2*tan(f*x)^9*tan(e)^6 + 210*a*b*tan(f*x)^
9*tan(e)^6 - 105*b^2*tan(f*x)^9*tan(e)^6 - 2835*a^2*tan(f*x)^8*tan(e)^7 + 5670*a*b*tan(f*x)^8*tan(e)^7 - 2835*
b^2*tan(f*x)^8*tan(e)^7 - 2835*a^2*tan(f*x)^7*tan(e)^8 + 5670*a*b*tan(f*x)^7*tan(e)^8 - 2835*b^2*tan(f*x)^7*ta
n(e)^8 - 105*a^2*tan(f*x)^6*tan(e)^9 + 210*a*b*tan(f*x)^6*tan(e)^9 - 105*b^2*tan(f*x)^6*tan(e)^9 - 26460*a^2*f
*x*tan(f*x)^6*tan(e)^6 + 52920*a*b*f*x*tan(f*x)^6*tan(e)^6 - 26460*b^2*f*x*tan(f*x)^6*tan(e)^6 + 63*a^2*tan(f*
x)^9*tan(e)^4 - 126*a*b*tan(f*x)^9*tan(e)^4 + 63*b^2*tan(f*x)^9*tan(e)^4 + 945*a^2*tan(f*x)^8*tan(e)^5 - 1890*
a*b*tan(f*x)^8*tan(e)^5 + 945*b^2*tan(f*x)^8*tan(e)^5 + 11340*a^2*tan(f*x)^7*tan(e)^6 - 22680*a*b*tan(f*x)^7*t
an(e)^6 + 11340*b^2*tan(f*x)^7*tan(e)^6 + 11340*a^2*tan(f*x)^6*tan(e)^7 - 22680*a*b*tan(f*x)^6*tan(e)^7 + 1134
0*b^2*tan(f*x)^6*tan(e)^7 + 945*a^2*tan(f*x)^5*tan(e)^8 - 1890*a*b*tan(f*x)^5*tan(e)^8 + 945*b^2*tan(f*x)^5*ta
n(e)^8 + 63*a^2*tan(f*x)^4*tan(e)^9 - 126*a*b*tan(f*x)^4*tan(e)^9 + 63*b^2*tan(f*x)^4*tan(e)^9 + 39690*a^2*f*x
*tan(f*x)^5*tan(e)^5 - 79380*a*b*f*x*tan(f*x)^5*tan(e)^5 + 39690*b^2*f*x*tan(f*x)^5*tan(e)^5 + 90*a*b*tan(f*x)
^9*tan(e)^2 - 45*b^2*tan(f*x)^9*tan(e)^2 - 252*a^2*tan(f*x)^8*tan(e)^3 + 1134*a*b*tan(f*x)^8*tan(e)^3 - 567*b^
2*tan(f*x)^8*tan(e)^3 - 2835*a^2*tan(f*x)^7*tan(e)^4 + 7560*a*b*tan(f*x)^7*tan(e)^4 - 3780*b^2*tan(f*x)^7*tan(
e)^4 - 24885*a^2*tan(f*x)^6*tan(e)^5 + 52920*a*b*tan(f*x)^6*tan(e)^5 - 26460*b^2*tan(f*x)^6*tan(e)^5 - 24885*a
^2*tan(f*x)^5*tan(e)^6 + 52920*a*b*tan(f*x)^5*tan(e)^6 - 26460*b^2*tan(f*x)^5*tan(e)^6 - 2835*a^2*tan(f*x)^4*t
an(e)^7 + 7560*a*b*tan(f*x)^4*tan(e)^7 - 3780*b^2*tan(f*x)^4*tan(e)^7 - 252*a^2*tan(f*x)^3*tan(e)^8 + 1134*a*b
*tan(f*x)^3*tan(e)^8 - 567*b^2*tan(f*x)^3*tan(e)^8 + 90*a*b*tan(f*x)^2*tan(e)^9 - 45*b^2*tan(f*x)^2*tan(e)^9 -
39690*a^2*f*x*tan(f*x)^4*tan(e)^4 + 79380*a*b*f*x*tan(f*x)^4*tan(e)^4 - 39690*b^2*f*x*tan(f*x)^4*tan(e)^4 + 3
5*b^2*tan(f*x)^9 - 180*a*b*tan(f*x)^8*tan(e) + 405*b^2*tan(f*x)^8*tan(e) + 378*a^2*tan(f*x)^7*tan(e)^2 - 2016*
a*b*tan(f*x)^7*tan(e)^2 + 2268*b^2*tan(f*x)^7*tan(e)^2 + 3990*a^2*tan(f*x)^6*tan(e)^3 - 11760*a*b*tan(f*x)^6*t
an(e)^3 + 8820*b^2*tan(f*x)^6*tan(e)^3 + 32130*a^2*tan(f*x)^5*tan(e)^4 - 70560*a*b*tan(f*x)^5*tan(e)^4 + 39690
*b^2*tan(f*x)^5*tan(e)^4 + 32130*a^2*tan(f*x)^4*tan(e)^5 - 70560*a*b*tan(f*x)^4*tan(e)^5 + 39690*b^2*tan(f*x)^
4*tan(e)^5 + 3990*a^2*tan(f*x)^3*tan(e)^6 - 11760*a*b*tan(f*x)^3*tan(e)^6 + 8820*b^2*tan(f*x)^3*tan(e)^6 + 378
*a^2*tan(f*x)^2*tan(e)^7 - 2016*a*b*tan(f*x)^2*tan(e)^7 + 2268*b^2*tan(f*x)^2*tan(e)^7 - 180*a*b*tan(f*x)*tan(
e)^8 + 405*b^2*tan(f*x)*tan(e)^8 + 35*b^2*tan(e)^9 + 26460*a^2*f*x*tan(f*x)^3*tan(e)^3 - 52920*a*b*f*x*tan(f*x
)^3*tan(e)^3 + 26460*b^2*f*x*tan(f*x)^3*tan(e)^3 + 90*a*b*tan(f*x)^7 - 45*b^2*tan(f*x)^7 - 252*a^2*tan(f*x)^6*
tan(e) + 1134*a*b*tan(f*x)^6*tan(e) - 567*b^2*tan(f*x)^6*tan(e) - 2835*a^2*tan(f*x)^5*tan(e)^2 + 7560*a*b*tan(
f*x)^5*tan(e)^2 - 3780*b^2*tan(f*x)^5*tan(e)^2 - 24885*a^2*tan(f*x)^4*tan(e)^3 + 52920*a*b*tan(f*x)^4*tan(e)^3
- 26460*b^2*tan(f*x)^4*tan(e)^3 - 24885*a^2*tan(f*x)^3*tan(e)^4 + 52920*a*b*tan(f*x)^3*tan(e)^4 - 26460*b^2*t
an(f*x)^3*tan(e)^4 - 2835*a^2*tan(f*x)^2*tan(e)^5 + 7560*a*b*tan(f*x)^2*tan(e)^5 - 3780*b^2*tan(f*x)^2*tan(e)^
5 - 252*a^2*tan(f*x)*tan(e)^6 + 1134*a*b*tan(f*x)*tan(e)^6 - 567*b^2*tan(f*x)*tan(e)^6 + 90*a*b*tan(e)^7 - 45*
b^2*tan(e)^7 - 11340*a^2*f*x*tan(f*x)^2*tan(e)^2 + 22680*a*b*f*x*tan(f*x)^2*tan(e)^2 - 11340*b^2*f*x*tan(f*x)^
2*tan(e)^2 + 63*a^2*tan(f*x)^5 - 126*a*b*tan(f*x)^5 + 63*b^2*tan(f*x)^5 + 945*a^2*tan(f*x)^4*tan(e) - 1890*a*b
*tan(f*x)^4*tan(e) + 945*b^2*tan(f*x)^4*tan(e) + 11340*a^2*tan(f*x)^3*tan(e)^2 - 22680*a*b*tan(f*x)^3*tan(e)^2
+ 11340*b^2*tan(f*x)^3*tan(e)^2 + 11340*a^2*tan(f*x)^2*tan(e)^3 - 22680*a*b*tan(f*x)^2*tan(e)^3 + 11340*b^2*t
an(f*x)^2*tan(e)^3 + 945*a^2*tan(f*x)*tan(e)^4 - 1890*a*b*tan(f*x)*tan(e)^4 + 945*b^2*tan(f*x)*tan(e)^4 + 63*a
^2*tan(e)^5 - 126*a*b*tan(e)^5 + 63*b^2*tan(e)^5 + 2835*a^2*f*x*tan(f*x)*tan(e) - 5670*a*b*f*x*tan(f*x)*tan(e)
+ 2835*b^2*f*x*tan(f*x)*tan(e) - 105*a^2*tan(f*x)^3 + 210*a*b*tan(f*x)^3 - 105*b^2*tan(f*x)^3 - 2835*a^2*tan(
f*x)^2*tan(e) + 5670*a*b*tan(f*x)^2*tan(e) - 2835*b^2*tan(f*x)^2*tan(e) - 2835*a^2*tan(f*x)*tan(e)^2 + 5670*a*
b*tan(f*x)*tan(e)^2 - 2835*b^2*tan(f*x)*tan(e)^2 - 105*a^2*tan(e)^3 + 210*a*b*tan(e)^3 - 105*b^2*tan(e)^3 - 31
5*a^2*f*x + 630*a*b*f*x - 315*b^2*f*x + 315*a^2*tan(f*x) - 630*a*b*tan(f*x) + 315*b^2*tan(f*x) + 315*a^2*tan(e
) - 630*a*b*tan(e) + 315*b^2*tan(e))/(f*tan(f*x)^9*tan(e)^9 - 9*f*tan(f*x)^8*tan(e)^8 + 36*f*tan(f*x)^7*tan(e)
^7 - 84*f*tan(f*x)^6*tan(e)^6 + 126*f*tan(f*x)^5*tan(e)^5 - 126*f*tan(f*x)^4*tan(e)^4 + 84*f*tan(f*x)^3*tan(e)
^3 - 36*f*tan(f*x)^2*tan(e)^2 + 9*f*tan(f*x)*tan(e) - f)

Mupad [B] (verification not implemented)

Time = 11.91 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.37 \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=\frac {{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (\frac {2\,a\,b}{7}-\frac {b^2}{7}\right )}{f}-\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (e+f\,x\right )\,{\left (a-b\right )}^2}{a^2-2\,a\,b+b^2}\right )\,{\left (a-b\right )}^2}{f}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2-2\,a\,b+b^2\right )}{f}+\frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^9}{9\,f}-\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {a^2}{3}-\frac {2\,a\,b}{3}+\frac {b^2}{3}\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (\frac {a^2}{5}-\frac {2\,a\,b}{5}+\frac {b^2}{5}\right )}{f} \]

[In]

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

[Out]

(tan(e + f*x)^7*((2*a*b)/7 - b^2/7))/f - (atan((tan(e + f*x)*(a - b)^2)/(a^2 - 2*a*b + b^2))*(a - b)^2)/f + (t
an(e + f*x)*(a^2 - 2*a*b + b^2))/f + (b^2*tan(e + f*x)^9)/(9*f) - (tan(e + f*x)^3*(a^2/3 - (2*a*b)/3 + b^2/3))
/f + (tan(e + f*x)^5*(a^2/5 - (2*a*b)/5 + b^2/5))/f