\(\int (a+b \sec ^2(e+f x))^2 \tan ^6(e+f x) \, dx\) [330]

Optimal result

Integrand size = 23, antiderivative size = 95 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^6(e+f x) \, dx=-a^2 x+\frac {a^2 \tan (e+f x)}{f}-\frac {a^2 \tan ^3(e+f x)}{3 f}+\frac {a^2 \tan ^5(e+f x)}{5 f}+\frac {b (2 a+b) \tan ^7(e+f x)}{7 f}+\frac {b^2 \tan ^9(e+f x)}{9 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 95, 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 = {4226, 1816, 209} \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^6(e+f x) \, dx=\frac {a^2 \tan ^5(e+f x)}{5 f}-\frac {a^2 \tan ^3(e+f x)}{3 f}+\frac {a^2 \tan (e+f x)}{f}-a^2 x+\frac {b (2 a+b) \tan ^7(e+f x)}{7 f}+\frac {b^2 \tan ^9(e+f x)}{9 f} \]

[In]

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

[Out]

-(a^2*x) + (a^2*Tan[e + f*x])/f - (a^2*Tan[e + f*x]^3)/(3*f) + (a^2*Tan[e + f*x]^5)/(5*f) + (b*(2*a + 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 1816

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 4226

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2
*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] ||
EqQ[n, 2])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^6 \left (a+b \left (1+x^2\right )\right )^2}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (a^2-a^2 x^2+a^2 x^4+b (2 a+b) x^6+b^2 x^8-\frac {a^2}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {a^2 \tan (e+f x)}{f}-\frac {a^2 \tan ^3(e+f x)}{3 f}+\frac {a^2 \tan ^5(e+f x)}{5 f}+\frac {b (2 a+b) \tan ^7(e+f x)}{7 f}+\frac {b^2 \tan ^9(e+f x)}{9 f}-\frac {a^2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -a^2 x+\frac {a^2 \tan (e+f x)}{f}-\frac {a^2 \tan ^3(e+f x)}{3 f}+\frac {a^2 \tan ^5(e+f x)}{5 f}+\frac {b (2 a+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. \(275\) vs. \(2(95)=190\).

Time = 3.71 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.89 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^6(e+f x) \, dx=-\frac {4 \left (b+a \cos ^2(e+f x)\right )^2 \sec ^9(e+f x) \left (315 a^2 f x \cos ^9(e+f x)-35 b^2 \sec (e) \sin (f x)-5 (18 a-19 b) b \cos ^2(e+f x) \sec (e) \sin (f x)-3 \left (21 a^2-90 a b+25 b^2\right ) \cos ^4(e+f x) \sec (e) \sin (f x)+\left (231 a^2-270 a b+5 b^2\right ) \cos ^6(e+f x) \sec (e) \sin (f x)-\left (483 a^2-90 a b-10 b^2\right ) \cos ^8(e+f x) \sec (e) \sin (f x)-35 b^2 \cos (e+f x) \tan (e)-5 (18 a-19 b) b \cos ^3(e+f x) \tan (e)-3 \left (21 a^2-90 a b+25 b^2\right ) \cos ^5(e+f x) \tan (e)+\left (231 a^2-270 a b+5 b^2\right ) \cos ^7(e+f x) \tan (e)\right )}{315 f (a+2 b+a \cos (2 (e+f x)))^2} \]

[In]

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

[Out]

(-4*(b + a*Cos[e + f*x]^2)^2*Sec[e + f*x]^9*(315*a^2*f*x*Cos[e + f*x]^9 - 35*b^2*Sec[e]*Sin[f*x] - 5*(18*a - 1
9*b)*b*Cos[e + f*x]^2*Sec[e]*Sin[f*x] - 3*(21*a^2 - 90*a*b + 25*b^2)*Cos[e + f*x]^4*Sec[e]*Sin[f*x] + (231*a^2
- 270*a*b + 5*b^2)*Cos[e + f*x]^6*Sec[e]*Sin[f*x] - (483*a^2 - 90*a*b - 10*b^2)*Cos[e + f*x]^8*Sec[e]*Sin[f*x
] - 35*b^2*Cos[e + f*x]*Tan[e] - 5*(18*a - 19*b)*b*Cos[e + f*x]^3*Tan[e] - 3*(21*a^2 - 90*a*b + 25*b^2)*Cos[e
+ f*x]^5*Tan[e] + (231*a^2 - 270*a*b + 5*b^2)*Cos[e + f*x]^7*Tan[e]))/(315*f*(a + 2*b + a*Cos[2*(e + f*x)])^2)

Maple [A] (verified)

Time = 12.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93

method	result	size
parts	\(\frac {a^{2} \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {b^{2} \left (\frac {\tan \left (f x +e \right )^{9}}{9}+\frac {\tan \left (f x +e \right )^{7}}{7}\right )}{f}+\frac {2 a b \tan \left (f x +e \right )^{7}}{7 f}\)	\(88\)
derivativedivides	\(\frac {a^{2} \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-f x -e \right )+\frac {2 a b \sin \left (f x +e \right )^{7}}{7 \cos \left (f x +e \right )^{7}}+b^{2} \left (\frac {\sin \left (f x +e \right )^{7}}{9 \cos \left (f x +e \right )^{9}}+\frac {2 \sin \left (f x +e \right )^{7}}{63 \cos \left (f x +e \right )^{7}}\right )}{f}\)	\(105\)
default	\(\frac {a^{2} \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-f x -e \right )+\frac {2 a b \sin \left (f x +e \right )^{7}}{7 \cos \left (f x +e \right )^{7}}+b^{2} \left (\frac {\sin \left (f x +e \right )^{7}}{9 \cos \left (f x +e \right )^{9}}+\frac {2 \sin \left (f x +e \right )^{7}}{63 \cos \left (f x +e \right )^{7}}\right )}{f}\)	\(105\)
risch	\(-a^{2} x -\frac {2 i \left (90 a b -483 a^{2}+10 b^{2}-28350 a^{2} {\mathrm e}^{10 i \left (f x +e \right )}+5040 a b \,{\mathrm e}^{8 i \left (f x +e \right )}+3780 a b \,{\mathrm e}^{6 i \left (f x +e \right )}+1980 a b \,{\mathrm e}^{4 i \left (f x +e \right )}+180 a b \,{\mathrm e}^{2 i \left (f x +e \right )}-32508 a^{2} {\mathrm e}^{8 i \left (f x +e \right )}-1890 b^{2} {\mathrm e}^{8 i \left (f x +e \right )}-24402 a^{2} {\mathrm e}^{6 i \left (f x +e \right )}+1890 b^{2} {\mathrm e}^{6 i \left (f x +e \right )}-11718 a^{2} {\mathrm e}^{4 i \left (f x +e \right )}-270 b^{2} {\mathrm e}^{4 i \left (f x +e \right )}-3402 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}+90 b^{2} {\mathrm e}^{2 i \left (f x +e \right )}-5670 a^{2} {\mathrm e}^{14 i \left (f x +e \right )}-16170 a^{2} {\mathrm e}^{12 i \left (f x +e \right )}-1050 b^{2} {\mathrm e}^{12 i \left (f x +e \right )}+630 a b \,{\mathrm e}^{16 i \left (f x +e \right )}+1260 a b \,{\mathrm e}^{14 i \left (f x +e \right )}+3780 a b \,{\mathrm e}^{12 i \left (f x +e \right )}-945 a^{2} {\mathrm e}^{16 i \left (f x +e \right )}+630 b^{2} {\mathrm e}^{14 i \left (f x +e \right )}+6300 a b \,{\mathrm e}^{10 i \left (f x +e \right )}+3150 b^{2} {\mathrm e}^{10 i \left (f x +e \right )}\right )}{315 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{9}}\)	\(356\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.44 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^6(e+f x) \, dx=-\frac {315 \, a^{2} f x \cos \left (f x + e\right )^{9} - {\left ({\left (483 \, a^{2} - 90 \, a b - 10 \, b^{2}\right )} \cos \left (f x + e\right )^{8} - {\left (231 \, a^{2} - 270 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{6} + 3 \, {\left (21 \, a^{2} - 90 \, a b + 25 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 5 \, {\left (18 \, a b - 19 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 35 \, b^{2}\right )} \sin \left (f x + e\right )}{315 \, f \cos \left (f x + e\right )^{9}} \]

[In]

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

[Out]

-1/315*(315*a^2*f*x*cos(f*x + e)^9 - ((483*a^2 - 90*a*b - 10*b^2)*cos(f*x + e)^8 - (231*a^2 - 270*a*b + 5*b^2)
*cos(f*x + e)^6 + 3*(21*a^2 - 90*a*b + 25*b^2)*cos(f*x + e)^4 + 5*(18*a*b - 19*b^2)*cos(f*x + e)^2 + 35*b^2)*s
in(f*x + e))/(f*cos(f*x + e)^9)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.88 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^6(e+f x) \, dx=\frac {35 \, b^{2} \tan \left (f x + e\right )^{9} + 45 \, {\left (2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{7} + 63 \, a^{2} \tan \left (f x + e\right )^{5} - 105 \, a^{2} \tan \left (f x + e\right )^{3} - 315 \, {\left (f x + e\right )} a^{2} + 315 \, a^{2} \tan \left (f x + e\right )}{315 \, f} \]

[In]

integrate((a+b*sec(f*x+e)^2)^2*tan(f*x+e)^6,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*tan(f*x + e)^5 - 105*a^2*tan(f*x + e)^
3 - 315*(f*x + e)*a^2 + 315*a^2*tan(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 2.52 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.96 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^6(e+f x) \, dx=\frac {35 \, b^{2} \tan \left (f x + e\right )^{9} + 90 \, a b \tan \left (f x + e\right )^{7} + 45 \, b^{2} \tan \left (f x + e\right )^{7} + 63 \, a^{2} \tan \left (f x + e\right )^{5} - 105 \, a^{2} \tan \left (f x + e\right )^{3} - 315 \, {\left (f x + e\right )} a^{2} + 315 \, a^{2} \tan \left (f x + e\right )}{315 \, f} \]

[In]

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

[Out]

1/315*(35*b^2*tan(f*x + e)^9 + 90*a*b*tan(f*x + e)^7 + 45*b^2*tan(f*x + e)^7 + 63*a^2*tan(f*x + e)^5 - 105*a^2
*tan(f*x + e)^3 - 315*(f*x + e)*a^2 + 315*a^2*tan(f*x + e))/f

Mupad [B] (verification not implemented)

Time = 19.69 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.33 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^6(e+f x) \, dx=\frac {\mathrm {tan}\left (e+f\,x\right )\,\left ({\left (a+b\right )}^2+b^2-2\,b\,\left (a+b\right )\right )-{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {{\left (a+b\right )}^2}{3}+\frac {b^2}{3}-\frac {2\,b\,\left (a+b\right )}{3}\right )+{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (\frac {{\left (a+b\right )}^2}{5}+\frac {b^2}{5}-\frac {2\,b\,\left (a+b\right )}{5}\right )-{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (\frac {b^2}{7}-\frac {2\,b\,\left (a+b\right )}{7}\right )+\frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^9}{9}-a^2\,f\,x}{f} \]

[In]

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

[Out]

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