\(\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx\) [37]

Optimal result

Integrand size = 31, antiderivative size = 229 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\frac {7 a^4 (7 A+8 B) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^4 (72 A+83 B) \tan (c+d x)}{15 d}+\frac {7 a^4 (7 A+8 B) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^4 (159 A+176 B) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac {(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d} \]

[Out]

7/16*a^4*(7*A+8*B)*arctanh(sin(d*x+c))/d+1/15*a^4*(72*A+83*B)*tan(d*x+c)/d+7/16*a^4*(7*A+8*B)*sec(d*x+c)*tan(d
*x+c)/d+1/120*a^4*(159*A+176*B)*sec(d*x+c)^2*tan(d*x+c)/d+1/120*(73*A+72*B)*(a^4+a^4*cos(d*x+c))*sec(d*x+c)^3*
tan(d*x+c)/d+1/10*(3*A+2*B)*(a^2+a^2*cos(d*x+c))^2*sec(d*x+c)^4*tan(d*x+c)/d+1/6*a*A*(a+a*cos(d*x+c))^3*sec(d*
x+c)^5*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {3054, 3047, 3100, 2827, 3853, 3855, 3852, 8} \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\frac {7 a^4 (7 A+8 B) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^4 (72 A+83 B) \tan (c+d x)}{15 d}+\frac {a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac {7 a^4 (7 A+8 B) \tan (c+d x) \sec (c+d x)}{16 d}+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{120 d}+\frac {(3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{10 d}+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d} \]

[In]

Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^7,x]

[Out]

(7*a^4*(7*A + 8*B)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^4*(72*A + 83*B)*Tan[c + d*x])/(15*d) + (7*a^4*(7*A + 8*B
)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^4*(159*A + 176*B)*Sec[c + d*x]^2*Tan[c + d*x])/(120*d) + ((73*A + 72*
B)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + ((3*A + 2*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec
[c + d*x]^4*Tan[c + d*x])/(10*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)

Rule 8

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

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3054

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d
*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x
])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n
+ 1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a
*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*
n] || EqQ[c, 0])

Rule 3100

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m
+ 1)*(a^2 - b^2))), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B +
a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b,
e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{6} \int (a+a \cos (c+d x))^3 (3 a (3 A+2 B)+2 a (A+3 B) \cos (c+d x)) \sec ^6(c+d x) \, dx \\ & = \frac {(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{30} \int (a+a \cos (c+d x))^2 \left (a^2 (73 A+72 B)+14 a^2 (2 A+3 B) \cos (c+d x)\right ) \sec ^5(c+d x) \, dx \\ & = \frac {(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{120} \int (a+a \cos (c+d x)) \left (3 a^3 (159 A+176 B)+6 a^3 (43 A+52 B) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx \\ & = \frac {(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{120} \int \left (3 a^4 (159 A+176 B)+\left (6 a^4 (43 A+52 B)+3 a^4 (159 A+176 B)\right ) \cos (c+d x)+6 a^4 (43 A+52 B) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx \\ & = \frac {a^4 (159 A+176 B) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac {(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{360} \int \left (315 a^4 (7 A+8 B)+24 a^4 (72 A+83 B) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx \\ & = \frac {a^4 (159 A+176 B) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac {(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{8} \left (7 a^4 (7 A+8 B)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{15} \left (a^4 (72 A+83 B)\right ) \int \sec ^2(c+d x) \, dx \\ & = \frac {7 a^4 (7 A+8 B) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^4 (159 A+176 B) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac {(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{16} \left (7 a^4 (7 A+8 B)\right ) \int \sec (c+d x) \, dx-\frac {\left (a^4 (72 A+83 B)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d} \\ & = \frac {7 a^4 (7 A+8 B) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^4 (72 A+83 B) \tan (c+d x)}{15 d}+\frac {7 a^4 (7 A+8 B) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^4 (159 A+176 B) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac {(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.11 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.13 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\frac {49 a^4 A \text {arctanh}(\sin (c+d x))}{16 d}+\frac {7 a^4 B \text {arctanh}(\sin (c+d x))}{2 d}+\frac {8 a^4 A \tan (c+d x)}{d}+\frac {8 a^4 B \tan (c+d x)}{d}+\frac {49 a^4 A \sec (c+d x) \tan (c+d x)}{16 d}+\frac {7 a^4 B \sec (c+d x) \tan (c+d x)}{2 d}+\frac {41 a^4 A \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^4 B \sec ^3(c+d x) \tan (c+d x)}{d}+\frac {a^4 A \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {4 a^4 A \tan ^3(c+d x)}{d}+\frac {8 a^4 B \tan ^3(c+d x)}{3 d}+\frac {4 a^4 A \tan ^5(c+d x)}{5 d}+\frac {a^4 B \tan ^5(c+d x)}{5 d} \]

[In]

Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^7,x]

[Out]

(49*a^4*A*ArcTanh[Sin[c + d*x]])/(16*d) + (7*a^4*B*ArcTanh[Sin[c + d*x]])/(2*d) + (8*a^4*A*Tan[c + d*x])/d + (
8*a^4*B*Tan[c + d*x])/d + (49*a^4*A*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (7*a^4*B*Sec[c + d*x]*Tan[c + d*x])/(2
*d) + (41*a^4*A*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (a^4*B*Sec[c + d*x]^3*Tan[c + d*x])/d + (a^4*A*Sec[c + d
*x]^5*Tan[c + d*x])/(6*d) + (4*a^4*A*Tan[c + d*x]^3)/d + (8*a^4*B*Tan[c + d*x]^3)/(3*d) + (4*a^4*A*Tan[c + d*x
]^5)/(5*d) + (a^4*B*Tan[c + d*x]^5)/(5*d)

Maple [A] (verified)

Time = 6.18 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.10

method	result	size
parallelrisch	\(\frac {125 \left (-\frac {147 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (A +\frac {8 B}{7}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{100}+\frac {147 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (A +\frac {8 B}{7}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{100}+\frac {28 \left (8 A +7 B \right ) \sin \left (2 d x +2 c \right )}{125}+\frac {\left (116 B +\frac {769 A}{6}\right ) \sin \left (3 d x +3 c \right )}{125}+\frac {48 \left (12 A +13 B \right ) \sin \left (4 d x +4 c \right )}{625}+\frac {7 \left (\frac {7 A}{2}+4 B \right ) \sin \left (5 d x +5 c \right )}{125}+\frac {4 \left (24 A +\frac {83 B}{3}\right ) \sin \left (6 d x +6 c \right )}{625}+\sin \left (d x +c \right ) \left (\frac {88 B}{125}+A \right )\right ) a^{4}}{4 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\)	\(251\)
parts	\(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {\left (a^{4} A +4 B \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}-\frac {\left (4 a^{4} A +6 B \,a^{4}\right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}\right ) \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {B \,a^{4} \tan \left (d x +c \right )}{d}\)	\(267\)
derivativedivides	\(\frac {a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,a^{4} \tan \left (d x +c \right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 B \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )-4 a^{4} A \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} A \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-B \,a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\)	\(365\)
default	\(\frac {a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,a^{4} \tan \left (d x +c \right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 B \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )-4 a^{4} A \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} A \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-B \,a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\)	\(365\)
risch	\(-\frac {i a^{4} \left (735 A \,{\mathrm e}^{11 i \left (d x +c \right )}+840 B \,{\mathrm e}^{11 i \left (d x +c \right )}-240 B \,{\mathrm e}^{10 i \left (d x +c \right )}+3845 A \,{\mathrm e}^{9 i \left (d x +c \right )}+3480 B \,{\mathrm e}^{9 i \left (d x +c \right )}-1920 A \,{\mathrm e}^{8 i \left (d x +c \right )}-4080 B \,{\mathrm e}^{8 i \left (d x +c \right )}+3750 A \,{\mathrm e}^{7 i \left (d x +c \right )}+2640 B \,{\mathrm e}^{7 i \left (d x +c \right )}-11520 A \,{\mathrm e}^{6 i \left (d x +c \right )}-13280 B \,{\mathrm e}^{6 i \left (d x +c \right )}-3750 A \,{\mathrm e}^{5 i \left (d x +c \right )}-2640 B \,{\mathrm e}^{5 i \left (d x +c \right )}-15360 A \,{\mathrm e}^{4 i \left (d x +c \right )}-15840 B \,{\mathrm e}^{4 i \left (d x +c \right )}-3845 A \,{\mathrm e}^{3 i \left (d x +c \right )}-3480 B \,{\mathrm e}^{3 i \left (d x +c \right )}-6912 A \,{\mathrm e}^{2 i \left (d x +c \right )}-7728 B \,{\mathrm e}^{2 i \left (d x +c \right )}-735 A \,{\mathrm e}^{i \left (d x +c \right )}-840 B \,{\mathrm e}^{i \left (d x +c \right )}-1152 A -1328 B \right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {49 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}-\frac {49 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}-\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}\)	\(371\)

[In]

int((a+cos(d*x+c)*a)^4*(A+B*cos(d*x+c))*sec(d*x+c)^7,x,method=_RETURNVERBOSE)

[Out]

125/4*(-147/100*(1/15*cos(6*d*x+6*c)+2/5*cos(4*d*x+4*c)+cos(2*d*x+2*c)+2/3)*(A+8/7*B)*ln(tan(1/2*d*x+1/2*c)-1)
+147/100*(1/15*cos(6*d*x+6*c)+2/5*cos(4*d*x+4*c)+cos(2*d*x+2*c)+2/3)*(A+8/7*B)*ln(tan(1/2*d*x+1/2*c)+1)+28/125
*(8*A+7*B)*sin(2*d*x+2*c)+1/125*(116*B+769/6*A)*sin(3*d*x+3*c)+48/625*(12*A+13*B)*sin(4*d*x+4*c)+7/125*(7/2*A+
4*B)*sin(5*d*x+5*c)+4/625*(24*A+83/3*B)*sin(6*d*x+6*c)+sin(d*x+c)*(88/125*B+A))*a^4/d/(cos(6*d*x+6*c)+6*cos(4*
d*x+4*c)+15*cos(2*d*x+2*c)+10)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.81 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\frac {105 \, {\left (7 \, A + 8 \, B\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (7 \, A + 8 \, B\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (72 \, A + 83 \, B\right )} a^{4} \cos \left (d x + c\right )^{5} + 105 \, {\left (7 \, A + 8 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} + 32 \, {\left (18 \, A + 17 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 10 \, {\left (41 \, A + 24 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 48 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 40 \, A a^{4}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]

[In]

integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c)^7,x, algorithm="fricas")

[Out]

1/480*(105*(7*A + 8*B)*a^4*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 105*(7*A + 8*B)*a^4*cos(d*x + c)^6*log(-sin(
d*x + c) + 1) + 2*(16*(72*A + 83*B)*a^4*cos(d*x + c)^5 + 105*(7*A + 8*B)*a^4*cos(d*x + c)^4 + 32*(18*A + 17*B)
*a^4*cos(d*x + c)^3 + 10*(41*A + 24*B)*a^4*cos(d*x + c)^2 + 48*(4*A + B)*a^4*cos(d*x + c) + 40*A*a^4)*sin(d*x
+ c))/(d*cos(d*x + c)^6)

Sympy [F(-1)]

Timed out. \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+a*cos(d*x+c))**4*(A+B*cos(d*x+c))*sec(d*x+c)**7,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (215) = 430\).

Time = 0.30 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.03 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\frac {128 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{4} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 32 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a^{4} + 960 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} - 5 \, A a^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, B a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 480 \, B a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, B a^{4} \tan \left (d x + c\right )}{480 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c)^7,x, algorithm="maxima")

[Out]

1/480*(128*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^4 + 640*(tan(d*x + c)^3 + 3*tan(d*x +
c))*A*a^4 + 32*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*B*a^4 + 960*(tan(d*x + c)^3 + 3*tan(d*
x + c))*B*a^4 - 5*A*a^4*(2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d
*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 180*A*a^4*(2*(3*sin
(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*
x + c) - 1)) - 120*B*a^4*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*lo
g(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 120*A*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x
+ c) + 1) + log(sin(d*x + c) - 1)) - 480*B*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) +
log(sin(d*x + c) - 1)) + 480*B*a^4*tan(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.22 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\frac {105 \, {\left (7 \, A a^{4} + 8 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (7 \, A a^{4} + 8 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (735 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 840 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 4165 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 4760 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9702 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11088 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 11802 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 13488 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7355 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9320 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3000 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c)^7,x, algorithm="giac")

[Out]

1/240*(105*(7*A*a^4 + 8*B*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*(7*A*a^4 + 8*B*a^4)*log(abs(tan(1/2*d*
x + 1/2*c) - 1)) - 2*(735*A*a^4*tan(1/2*d*x + 1/2*c)^11 + 840*B*a^4*tan(1/2*d*x + 1/2*c)^11 - 4165*A*a^4*tan(1
/2*d*x + 1/2*c)^9 - 4760*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 9702*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 11088*B*a^4*tan(1/
2*d*x + 1/2*c)^7 - 11802*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 13488*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 7355*A*a^4*tan(1/
2*d*x + 1/2*c)^3 + 9320*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 3105*A*a^4*tan(1/2*d*x + 1/2*c) - 3000*B*a^4*tan(1/2*d*
x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^6)/d

Mupad [B] (verification not implemented)

Time = 2.95 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.14 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\frac {\left (-\frac {49\,A\,a^4}{8}-7\,B\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {833\,A\,a^4}{24}+\frac {119\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {1617\,A\,a^4}{20}-\frac {462\,B\,a^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1967\,A\,a^4}{20}+\frac {562\,B\,a^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {1471\,A\,a^4}{24}-\frac {233\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {207\,A\,a^4}{8}+25\,B\,a^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {7\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (7\,A+8\,B\right )}{8\,d} \]

[In]

int(((A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4)/cos(c + d*x)^7,x)

[Out]

(tan(c/2 + (d*x)/2)*((207*A*a^4)/8 + 25*B*a^4) - tan(c/2 + (d*x)/2)^11*((49*A*a^4)/8 + 7*B*a^4) + tan(c/2 + (d
*x)/2)^9*((833*A*a^4)/24 + (119*B*a^4)/3) - tan(c/2 + (d*x)/2)^3*((1471*A*a^4)/24 + (233*B*a^4)/3) - tan(c/2 +
(d*x)/2)^7*((1617*A*a^4)/20 + (462*B*a^4)/5) + tan(c/2 + (d*x)/2)^5*((1967*A*a^4)/20 + (562*B*a^4)/5))/(d*(15
*tan(c/2 + (d*x)/2)^4 - 6*tan(c/2 + (d*x)/2)^2 - 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 - 6*tan(c/2
+ (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (7*a^4*atanh(tan(c/2 + (d*x)/2))*(7*A + 8*B))/(8*d)