\(\int (a+b \sec ^2(e+f x))^2 \tan ^4(e+f x) \, dx\) [331]

Optimal result

Integrand size = 23, antiderivative size = 77 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^4(e+f x) \, dx=a^2 x-\frac {a^2 \tan (e+f x)}{f}+\frac {a^2 \tan ^3(e+f x)}{3 f}+\frac {b (2 a+b) \tan ^5(e+f x)}{5 f}+\frac {b^2 \tan ^7(e+f x)}{7 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 77, 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} \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^4(e+f x) \, dx=\frac {a^2 \tan ^3(e+f x)}{3 f}-\frac {a^2 \tan (e+f x)}{f}+a^2 x+\frac {b (2 a+b) \tan ^5(e+f x)}{5 f}+\frac {b^2 \tan ^7(e+f x)}{7 f} \]

[In]

Int[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^4,x]

[Out]

a^2*x - (a^2*Tan[e + f*x])/f + (a^2*Tan[e + f*x]^3)/(3*f) + (b*(2*a + b)*Tan[e + f*x]^5)/(5*f) + (b^2*Tan[e +
f*x]^7)/(7*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^4 \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+b (2 a+b) x^4+b^2 x^6+\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 {b (2 a+b) \tan ^5(e+f x)}{5 f}+\frac {b^2 \tan ^7(e+f x)}{7 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 {b (2 a+b) \tan ^5(e+f x)}{5 f}+\frac {b^2 \tan ^7(e+f x)}{7 f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(395\) vs. \(2(77)=154\).

Time = 2.30 (sec) , antiderivative size = 395, normalized size of antiderivative = 5.13 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^4(e+f x) \, dx=\frac {\sec (e) \sec ^7(e+f x) \left (3675 a^2 f x \cos (f x)+3675 a^2 f x \cos (2 e+f x)+2205 a^2 f x \cos (2 e+3 f x)+2205 a^2 f x \cos (4 e+3 f x)+735 a^2 f x \cos (4 e+5 f x)+735 a^2 f x \cos (6 e+5 f x)+105 a^2 f x \cos (6 e+7 f x)+105 a^2 f x \cos (8 e+7 f x)-5320 a^2 \sin (f x)+1680 a b \sin (f x)+840 b^2 \sin (f x)+4480 a^2 \sin (2 e+f x)-1260 a b \sin (2 e+f x)+420 b^2 \sin (2 e+f x)-3780 a^2 \sin (2 e+3 f x)+924 a b \sin (2 e+3 f x)-168 b^2 \sin (2 e+3 f x)+2100 a^2 \sin (4 e+3 f x)-840 a b \sin (4 e+3 f x)-420 b^2 \sin (4 e+3 f x)-1540 a^2 \sin (4 e+5 f x)+168 a b \sin (4 e+5 f x)+84 b^2 \sin (4 e+5 f x)+420 a^2 \sin (6 e+5 f x)-420 a b \sin (6 e+5 f x)-280 a^2 \sin (6 e+7 f x)+84 a b \sin (6 e+7 f x)+12 b^2 \sin (6 e+7 f x)\right )}{13440 f} \]

[In]

Integrate[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^4,x]

[Out]

(Sec[e]*Sec[e + f*x]^7*(3675*a^2*f*x*Cos[f*x] + 3675*a^2*f*x*Cos[2*e + f*x] + 2205*a^2*f*x*Cos[2*e + 3*f*x] +
2205*a^2*f*x*Cos[4*e + 3*f*x] + 735*a^2*f*x*Cos[4*e + 5*f*x] + 735*a^2*f*x*Cos[6*e + 5*f*x] + 105*a^2*f*x*Cos[
6*e + 7*f*x] + 105*a^2*f*x*Cos[8*e + 7*f*x] - 5320*a^2*Sin[f*x] + 1680*a*b*Sin[f*x] + 840*b^2*Sin[f*x] + 4480*
a^2*Sin[2*e + f*x] - 1260*a*b*Sin[2*e + f*x] + 420*b^2*Sin[2*e + f*x] - 3780*a^2*Sin[2*e + 3*f*x] + 924*a*b*Si
n[2*e + 3*f*x] - 168*b^2*Sin[2*e + 3*f*x] + 2100*a^2*Sin[4*e + 3*f*x] - 840*a*b*Sin[4*e + 3*f*x] - 420*b^2*Sin
[4*e + 3*f*x] - 1540*a^2*Sin[4*e + 5*f*x] + 168*a*b*Sin[4*e + 5*f*x] + 84*b^2*Sin[4*e + 5*f*x] + 420*a^2*Sin[6
*e + 5*f*x] - 420*a*b*Sin[6*e + 5*f*x] - 280*a^2*Sin[6*e + 7*f*x] + 84*a*b*Sin[6*e + 7*f*x] + 12*b^2*Sin[6*e +
7*f*x]))/(13440*f)

Maple [A] (verified)

Time = 6.53 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01

method	result	size
parts	\(\frac {a^{2} \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {b^{2} \left (\frac {\tan \left (f x +e \right )^{7}}{7}+\frac {\tan \left (f x +e \right )^{5}}{5}\right )}{f}+\frac {2 a b \tan \left (f x +e \right )^{5}}{5 f}\)	\(78\)
derivativedivides	\(\frac {a^{2} \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+f x +e \right )+\frac {2 a b \sin \left (f x +e \right )^{5}}{5 \cos \left (f x +e \right )^{5}}+b^{2} \left (\frac {\sin \left (f x +e \right )^{5}}{7 \cos \left (f x +e \right )^{7}}+\frac {2 \sin \left (f x +e \right )^{5}}{35 \cos \left (f x +e \right )^{5}}\right )}{f}\)	\(94\)
default	\(\frac {a^{2} \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+f x +e \right )+\frac {2 a b \sin \left (f x +e \right )^{5}}{5 \cos \left (f x +e \right )^{5}}+b^{2} \left (\frac {\sin \left (f x +e \right )^{5}}{7 \cos \left (f x +e \right )^{7}}+\frac {2 \sin \left (f x +e \right )^{5}}{35 \cos \left (f x +e \right )^{5}}\right )}{f}\)	\(94\)
risch	\(a^{2} x +\frac {4 i \left (-105 a^{2} {\mathrm e}^{12 i \left (f x +e \right )}+105 a b \,{\mathrm e}^{12 i \left (f x +e \right )}-525 a^{2} {\mathrm e}^{10 i \left (f x +e \right )}+210 a b \,{\mathrm e}^{10 i \left (f x +e \right )}+105 b^{2} {\mathrm e}^{10 i \left (f x +e \right )}-1120 a^{2} {\mathrm e}^{8 i \left (f x +e \right )}+315 a b \,{\mathrm e}^{8 i \left (f x +e \right )}-105 b^{2} {\mathrm e}^{8 i \left (f x +e \right )}-1330 a^{2} {\mathrm e}^{6 i \left (f x +e \right )}+420 a b \,{\mathrm e}^{6 i \left (f x +e \right )}+210 b^{2} {\mathrm e}^{6 i \left (f x +e \right )}-945 a^{2} {\mathrm e}^{4 i \left (f x +e \right )}+231 a b \,{\mathrm e}^{4 i \left (f x +e \right )}-42 b^{2} {\mathrm e}^{4 i \left (f x +e \right )}-385 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}+42 a b \,{\mathrm e}^{2 i \left (f x +e \right )}+21 b^{2} {\mathrm e}^{2 i \left (f x +e \right )}-70 a^{2}+21 a b +3 b^{2}\right )}{105 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{7}}\)	\(273\)

[In]

int((a+b*sec(f*x+e)^2)^2*tan(f*x+e)^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.47 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^4(e+f x) \, dx=\frac {105 \, a^{2} f x \cos \left (f x + e\right )^{7} - {\left (2 \, {\left (70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right )} \cos \left (f x + e\right )^{6} - {\left (35 \, a^{2} - 84 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 6 \, {\left (7 \, a b - 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 15 \, b^{2}\right )} \sin \left (f x + e\right )}{105 \, f \cos \left (f x + e\right )^{7}} \]

[In]

integrate((a+b*sec(f*x+e)^2)^2*tan(f*x+e)^4,x, algorithm="fricas")

[Out]

1/105*(105*a^2*f*x*cos(f*x + e)^7 - (2*(70*a^2 - 21*a*b - 3*b^2)*cos(f*x + e)^6 - (35*a^2 - 84*a*b + 3*b^2)*co
s(f*x + e)^4 - 6*(7*a*b - 4*b^2)*cos(f*x + e)^2 - 15*b^2)*sin(f*x + e))/(f*cos(f*x + e)^7)

Sympy [F]

\[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^4(e+f x) \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \tan ^{4}{\left (e + f x \right )}\, dx \]

[In]

integrate((a+b*sec(f*x+e)**2)**2*tan(f*x+e)**4,x)

[Out]

Integral((a + b*sec(e + f*x)**2)**2*tan(e + f*x)**4, x)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^4(e+f x) \, dx=\frac {15 \, b^{2} \tan \left (f x + e\right )^{7} + 21 \, {\left (2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{5} + 35 \, a^{2} \tan \left (f x + e\right )^{3} + 105 \, {\left (f x + e\right )} a^{2} - 105 \, a^{2} \tan \left (f x + e\right )}{105 \, f} \]

[In]

integrate((a+b*sec(f*x+e)^2)^2*tan(f*x+e)^4,x, algorithm="maxima")

[Out]

1/105*(15*b^2*tan(f*x + e)^7 + 21*(2*a*b + b^2)*tan(f*x + e)^5 + 35*a^2*tan(f*x + e)^3 + 105*(f*x + e)*a^2 - 1
05*a^2*tan(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 1.20 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^4(e+f x) \, dx=\frac {15 \, b^{2} \tan \left (f x + e\right )^{7} + 42 \, a b \tan \left (f x + e\right )^{5} + 21 \, b^{2} \tan \left (f x + e\right )^{5} + 35 \, a^{2} \tan \left (f x + e\right )^{3} + 105 \, {\left (f x + e\right )} a^{2} - 105 \, a^{2} \tan \left (f x + e\right )}{105 \, f} \]

[In]

integrate((a+b*sec(f*x+e)^2)^2*tan(f*x+e)^4,x, algorithm="giac")

[Out]

1/105*(15*b^2*tan(f*x + e)^7 + 42*a*b*tan(f*x + e)^5 + 21*b^2*tan(f*x + e)^5 + 35*a^2*tan(f*x + e)^3 + 105*(f*
x + e)*a^2 - 105*a^2*tan(f*x + e))/f

Mupad [B] (verification not implemented)

Time = 19.43 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.26 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \tan ^4(e+f x) \, dx=\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {{\left (a+b\right )}^2}{3}+\frac {b^2}{3}-\frac {2\,b\,\left (a+b\right )}{3}\right )-\mathrm {tan}\left (e+f\,x\right )\,\left ({\left (a+b\right )}^2+b^2-2\,b\,\left (a+b\right )\right )-{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (\frac {b^2}{5}-\frac {2\,b\,\left (a+b\right )}{5}\right )+\frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^7}{7}+a^2\,f\,x}{f} \]

[In]

int(tan(e + f*x)^4*(a + b/cos(e + f*x)^2)^2,x)

[Out]

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