\(\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx\) [1]

Optimal result

Integrand size = 26, antiderivative size = 196 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=a^2 c^5 x-\frac {19 a^2 c^5 \text {arctanh}(\sin (e+f x))}{16 f}-\frac {a^2 c^5 \tan (e+f x)}{f}+\frac {17 a^2 c^5 \sec (e+f x) \tan (e+f x)}{16 f}+\frac {a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f} \]

[Out]

a^2*c^5*x-19/16*a^2*c^5*arctanh(sin(f*x+e))/f-a^2*c^5*tan(f*x+e)/f+17/16*a^2*c^5*sec(f*x+e)*tan(f*x+e)/f+1/8*a
^2*c^5*sec(f*x+e)^3*tan(f*x+e)/f+1/3*a^2*c^5*tan(f*x+e)^3/f-3/4*a^2*c^5*sec(f*x+e)*tan(f*x+e)^3/f-1/6*a^2*c^5*
sec(f*x+e)^3*tan(f*x+e)^3/f+3/5*a^2*c^5*tan(f*x+e)^5/f

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3989, 3971, 3554, 8, 2691, 3855, 2687, 30, 3853} \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=-\frac {19 a^2 c^5 \text {arctanh}(\sin (e+f x))}{16 f}+\frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {a^2 c^5 \tan (e+f x)}{f}-\frac {a^2 c^5 \tan ^3(e+f x) \sec ^3(e+f x)}{6 f}+\frac {a^2 c^5 \tan (e+f x) \sec ^3(e+f x)}{8 f}-\frac {3 a^2 c^5 \tan ^3(e+f x) \sec (e+f x)}{4 f}+\frac {17 a^2 c^5 \tan (e+f x) \sec (e+f x)}{16 f}+a^2 c^5 x \]

[In]

Int[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5,x]

[Out]

a^2*c^5*x - (19*a^2*c^5*ArcTanh[Sin[e + f*x]])/(16*f) - (a^2*c^5*Tan[e + f*x])/f + (17*a^2*c^5*Sec[e + f*x]*Ta
n[e + f*x])/(16*f) + (a^2*c^5*Sec[e + f*x]^3*Tan[e + f*x])/(8*f) + (a^2*c^5*Tan[e + f*x]^3)/(3*f) - (3*a^2*c^5
*Sec[e + f*x]*Tan[e + f*x]^3)/(4*f) - (a^2*c^5*Sec[e + f*x]^3*Tan[e + f*x]^3)/(6*f) + (3*a^2*c^5*Tan[e + f*x]^
5)/(5*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3971

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandI
ntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]

Rule 3989

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Di
st[((-a)*c)^m, Int[Cot[e + f*x]^(2*m)*(c + d*Csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
&& EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] &&  !(IntegerQ[n] && GtQ[m - n, 0])

Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int (c-c \sec (e+f x))^3 \tan ^4(e+f x) \, dx \\ & = \left (a^2 c^2\right ) \int \left (c^3 \tan ^4(e+f x)-3 c^3 \sec (e+f x) \tan ^4(e+f x)+3 c^3 \sec ^2(e+f x) \tan ^4(e+f x)-c^3 \sec ^3(e+f x) \tan ^4(e+f x)\right ) \, dx \\ & = \left (a^2 c^5\right ) \int \tan ^4(e+f x) \, dx-\left (a^2 c^5\right ) \int \sec ^3(e+f x) \tan ^4(e+f x) \, dx-\left (3 a^2 c^5\right ) \int \sec (e+f x) \tan ^4(e+f x) \, dx+\left (3 a^2 c^5\right ) \int \sec ^2(e+f x) \tan ^4(e+f x) \, dx \\ & = \frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {1}{2} \left (a^2 c^5\right ) \int \sec ^3(e+f x) \tan ^2(e+f x) \, dx-\left (a^2 c^5\right ) \int \tan ^2(e+f x) \, dx+\frac {1}{4} \left (9 a^2 c^5\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx+\frac {\left (3 a^2 c^5\right ) \text {Subst}\left (\int x^4 \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {a^2 c^5 \tan (e+f x)}{f}+\frac {9 a^2 c^5 \sec (e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f}-\frac {1}{8} \left (a^2 c^5\right ) \int \sec ^3(e+f x) \, dx+\left (a^2 c^5\right ) \int 1 \, dx-\frac {1}{8} \left (9 a^2 c^5\right ) \int \sec (e+f x) \, dx \\ & = a^2 c^5 x-\frac {9 a^2 c^5 \text {arctanh}(\sin (e+f x))}{8 f}-\frac {a^2 c^5 \tan (e+f x)}{f}+\frac {17 a^2 c^5 \sec (e+f x) \tan (e+f x)}{16 f}+\frac {a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f}-\frac {1}{16} \left (a^2 c^5\right ) \int \sec (e+f x) \, dx \\ & = a^2 c^5 x-\frac {19 a^2 c^5 \text {arctanh}(\sin (e+f x))}{16 f}-\frac {a^2 c^5 \tan (e+f x)}{f}+\frac {17 a^2 c^5 \sec (e+f x) \tan (e+f x)}{16 f}+\frac {a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.74 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.84 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {a^2 c^5 \sec ^6(e+f x) \left (1200 e+1200 f x-4560 \text {arctanh}(\sin (e+f x)) \cos ^6(e+f x)+1800 (e+f x) \cos (2 (e+f x))+720 e \cos (4 (e+f x))+720 f x \cos (4 (e+f x))+120 e \cos (6 (e+f x))+120 f x \cos (6 (e+f x))-210 \sin (e+f x)-120 \sin (2 (e+f x))+865 \sin (3 (e+f x))-768 \sin (4 (e+f x))+435 \sin (5 (e+f x))-88 \sin (6 (e+f x))\right )}{3840 f} \]

[In]

Integrate[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5,x]

[Out]

(a^2*c^5*Sec[e + f*x]^6*(1200*e + 1200*f*x - 4560*ArcTanh[Sin[e + f*x]]*Cos[e + f*x]^6 + 1800*(e + f*x)*Cos[2*
(e + f*x)] + 720*e*Cos[4*(e + f*x)] + 720*f*x*Cos[4*(e + f*x)] + 120*e*Cos[6*(e + f*x)] + 120*f*x*Cos[6*(e + f
*x)] - 210*Sin[e + f*x] - 120*Sin[2*(e + f*x)] + 865*Sin[3*(e + f*x)] - 768*Sin[4*(e + f*x)] + 435*Sin[5*(e +
f*x)] - 88*Sin[6*(e + f*x)]))/(3840*f)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.91 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.05

method	result	size
risch	\(a^{2} c^{5} x -\frac {i c^{5} a^{2} \left (435 \,{\mathrm e}^{11 i \left (f x +e \right )}-240 \,{\mathrm e}^{10 i \left (f x +e \right )}+865 \,{\mathrm e}^{9 i \left (f x +e \right )}+1200 \,{\mathrm e}^{8 i \left (f x +e \right )}-210 \,{\mathrm e}^{7 i \left (f x +e \right )}+1760 \,{\mathrm e}^{6 i \left (f x +e \right )}+210 \,{\mathrm e}^{5 i \left (f x +e \right )}+1440 \,{\mathrm e}^{4 i \left (f x +e \right )}-865 \,{\mathrm e}^{3 i \left (f x +e \right )}+1296 \,{\mathrm e}^{2 i \left (f x +e \right )}-435 \,{\mathrm e}^{i \left (f x +e \right )}+176\right )}{120 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{6}}+\frac {19 c^{5} a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{16 f}-\frac {19 c^{5} a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{16 f}\)	\(206\)
parallelrisch	\(-\frac {a^{2} c^{5} \left (\frac {19 \left (-5-\frac {15 \cos \left (2 f x +2 e \right )}{2}-3 \cos \left (4 f x +4 e \right )-\frac {\cos \left (6 f x +6 e \right )}{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8}+\frac {19 \left (5+\frac {\cos \left (6 f x +6 e \right )}{2}+3 \cos \left (4 f x +4 e \right )+\frac {15 \cos \left (2 f x +2 e \right )}{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8}-15 f x \cos \left (2 f x +2 e \right )-6 f x \cos \left (4 f x +4 e \right )-f x \cos \left (6 f x +6 e \right )-10 f x +\sin \left (2 f x +2 e \right )-\frac {173 \sin \left (3 f x +3 e \right )}{24}+\frac {32 \sin \left (4 f x +4 e \right )}{5}-\frac {29 \sin \left (5 f x +5 e \right )}{8}+\frac {11 \sin \left (6 f x +6 e \right )}{15}+\frac {7 \sin \left (f x +e \right )}{4}\right )}{f \left (6 \cos \left (4 f x +4 e \right )+10+15 \cos \left (2 f x +2 e \right )+\cos \left (6 f x +6 e \right )\right )}\)	\(250\)
derivativedivides	\(\frac {-c^{5} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )-3 c^{5} a^{2} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )-c^{5} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+5 c^{5} a^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+5 c^{5} a^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+c^{5} a^{2} \tan \left (f x +e \right )-3 c^{5} a^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{5} a^{2} \left (f x +e \right )}{f}\)	\(268\)
default	\(\frac {-c^{5} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )-3 c^{5} a^{2} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )-c^{5} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+5 c^{5} a^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+5 c^{5} a^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+c^{5} a^{2} \tan \left (f x +e \right )-3 c^{5} a^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{5} a^{2} \left (f x +e \right )}{f}\)	\(268\)
parts	\(a^{2} c^{5} x +\frac {a^{2} c^{5} \tan \left (f x +e \right )}{f}-\frac {3 c^{5} a^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+\frac {5 c^{5} a^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}+\frac {5 c^{5} a^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}-\frac {c^{5} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}-\frac {3 c^{5} a^{2} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )}{f}-\frac {c^{5} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}\)	\(281\)
norman	\(\frac {a^{2} c^{5} x +a^{2} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}-6 a^{2} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+15 a^{2} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-20 a^{2} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+15 a^{2} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-6 a^{2} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}+\frac {3 c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}-\frac {19 c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{24 f}-\frac {61 c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{20 f}+\frac {291 c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{20 f}-\frac {209 c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{8 f}+\frac {35 c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{8 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{6}}+\frac {19 c^{5} a^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16 f}-\frac {19 c^{5} a^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16 f}\)	\(322\)

[In]

int((a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^5,x,method=_RETURNVERBOSE)

[Out]

a^2*c^5*x-1/120*I*c^5*a^2*(435*exp(11*I*(f*x+e))-240*exp(10*I*(f*x+e))+865*exp(9*I*(f*x+e))+1200*exp(8*I*(f*x+
e))-210*exp(7*I*(f*x+e))+1760*exp(6*I*(f*x+e))+210*exp(5*I*(f*x+e))+1440*exp(4*I*(f*x+e))-865*exp(3*I*(f*x+e))
+1296*exp(2*I*(f*x+e))-435*exp(I*(f*x+e))+176)/f/(1+exp(2*I*(f*x+e)))^6+19/16*c^5*a^2/f*ln(exp(I*(f*x+e))-I)-1
9/16*c^5*a^2/f*ln(exp(I*(f*x+e))+I)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.91 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {480 \, a^{2} c^{5} f x \cos \left (f x + e\right )^{6} - 285 \, a^{2} c^{5} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) + 285 \, a^{2} c^{5} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (176 \, a^{2} c^{5} \cos \left (f x + e\right )^{5} - 435 \, a^{2} c^{5} \cos \left (f x + e\right )^{4} + 208 \, a^{2} c^{5} \cos \left (f x + e\right )^{3} + 110 \, a^{2} c^{5} \cos \left (f x + e\right )^{2} - 144 \, a^{2} c^{5} \cos \left (f x + e\right ) + 40 \, a^{2} c^{5}\right )} \sin \left (f x + e\right )}{480 \, f \cos \left (f x + e\right )^{6}} \]

[In]

integrate((a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^5,x, algorithm="fricas")

[Out]

1/480*(480*a^2*c^5*f*x*cos(f*x + e)^6 - 285*a^2*c^5*cos(f*x + e)^6*log(sin(f*x + e) + 1) + 285*a^2*c^5*cos(f*x
+ e)^6*log(-sin(f*x + e) + 1) - 2*(176*a^2*c^5*cos(f*x + e)^5 - 435*a^2*c^5*cos(f*x + e)^4 + 208*a^2*c^5*cos(
f*x + e)^3 + 110*a^2*c^5*cos(f*x + e)^2 - 144*a^2*c^5*cos(f*x + e) + 40*a^2*c^5)*sin(f*x + e))/(f*cos(f*x + e)
^6)

Sympy [F]

\[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=- a^{2} c^{5} \left (\int \left (-1\right )\, dx + \int 3 \sec {\left (e + f x \right )}\, dx + \int \left (- \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- 5 \sec ^{3}{\left (e + f x \right )}\right )\, dx + \int 5 \sec ^{4}{\left (e + f x \right )}\, dx + \int \sec ^{5}{\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{6}{\left (e + f x \right )}\right )\, dx + \int \sec ^{7}{\left (e + f x \right )}\, dx\right ) \]

[In]

integrate((a+a*sec(f*x+e))**2*(c-c*sec(f*x+e))**5,x)

[Out]

-a**2*c**5*(Integral(-1, x) + Integral(3*sec(e + f*x), x) + Integral(-sec(e + f*x)**2, x) + Integral(-5*sec(e
+ f*x)**3, x) + Integral(5*sec(e + f*x)**4, x) + Integral(sec(e + f*x)**5, x) + Integral(-3*sec(e + f*x)**6, x
) + Integral(sec(e + f*x)**7, x))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.70 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {96 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{2} c^{5} - 800 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c^{5} + 480 \, {\left (f x + e\right )} a^{2} c^{5} + 5 \, a^{2} c^{5} {\left (\frac {2 \, {\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 30 \, a^{2} c^{5} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 600 \, a^{2} c^{5} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 1440 \, a^{2} c^{5} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 480 \, a^{2} c^{5} \tan \left (f x + e\right )}{480 \, f} \]

[In]

integrate((a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^5,x, algorithm="maxima")

[Out]

1/480*(96*(3*tan(f*x + e)^5 + 10*tan(f*x + e)^3 + 15*tan(f*x + e))*a^2*c^5 - 800*(tan(f*x + e)^3 + 3*tan(f*x +
e))*a^2*c^5 + 480*(f*x + e)*a^2*c^5 + 5*a^2*c^5*(2*(15*sin(f*x + e)^5 - 40*sin(f*x + e)^3 + 33*sin(f*x + e))/
(sin(f*x + e)^6 - 3*sin(f*x + e)^4 + 3*sin(f*x + e)^2 - 1) - 15*log(sin(f*x + e) + 1) + 15*log(sin(f*x + e) -
1)) + 30*a^2*c^5*(2*(3*sin(f*x + e)^3 - 5*sin(f*x + e))/(sin(f*x + e)^4 - 2*sin(f*x + e)^2 + 1) - 3*log(sin(f*
x + e) + 1) + 3*log(sin(f*x + e) - 1)) - 600*a^2*c^5*(2*sin(f*x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) +
1) + log(sin(f*x + e) - 1)) - 1440*a^2*c^5*log(sec(f*x + e) + tan(f*x + e)) + 480*a^2*c^5*tan(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.97 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {240 \, {\left (f x + e\right )} a^{2} c^{5} - 285 \, a^{2} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) + 285 \, a^{2} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) + \frac {2 \, {\left (525 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} - 3135 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 1746 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 366 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 95 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 45 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{6}}}{240 \, f} \]

[In]

integrate((a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^5,x, algorithm="giac")

[Out]

1/240*(240*(f*x + e)*a^2*c^5 - 285*a^2*c^5*log(abs(tan(1/2*f*x + 1/2*e) + 1)) + 285*a^2*c^5*log(abs(tan(1/2*f*
x + 1/2*e) - 1)) + 2*(525*a^2*c^5*tan(1/2*f*x + 1/2*e)^11 - 3135*a^2*c^5*tan(1/2*f*x + 1/2*e)^9 + 1746*a^2*c^5
*tan(1/2*f*x + 1/2*e)^7 - 366*a^2*c^5*tan(1/2*f*x + 1/2*e)^5 - 95*a^2*c^5*tan(1/2*f*x + 1/2*e)^3 + 45*a^2*c^5*
tan(1/2*f*x + 1/2*e))/(tan(1/2*f*x + 1/2*e)^2 - 1)^6)/f

Mupad [B] (verification not implemented)

Time = 15.54 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.16 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=a^2\,c^5\,x-\frac {-\frac {35\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{8}+\frac {209\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{8}-\frac {291\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{20}+\frac {61\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{20}+\frac {19\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{24}-\frac {3\,a^2\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {19\,a^2\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{8\,f} \]

[In]

int((a + a/cos(e + f*x))^2*(c - c/cos(e + f*x))^5,x)

[Out]

a^2*c^5*x - ((19*a^2*c^5*tan(e/2 + (f*x)/2)^3)/24 + (61*a^2*c^5*tan(e/2 + (f*x)/2)^5)/20 - (291*a^2*c^5*tan(e/
2 + (f*x)/2)^7)/20 + (209*a^2*c^5*tan(e/2 + (f*x)/2)^9)/8 - (35*a^2*c^5*tan(e/2 + (f*x)/2)^11)/8 - (3*a^2*c^5*
tan(e/2 + (f*x)/2))/8)/(f*(15*tan(e/2 + (f*x)/2)^4 - 6*tan(e/2 + (f*x)/2)^2 - 20*tan(e/2 + (f*x)/2)^6 + 15*tan
(e/2 + (f*x)/2)^8 - 6*tan(e/2 + (f*x)/2)^10 + tan(e/2 + (f*x)/2)^12 + 1)) - (19*a^2*c^5*atanh(tan(e/2 + (f*x)/
2)))/(8*f)