3.108 \(\int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\)

Optimal. Leaf size=242 \[ \frac {a^5 \tan (c+d x)}{d}+\frac {5 a^4 b \tan ^2(c+d x)}{2 d}+\frac {10 a b^2 \left (a^2+b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {5 a^2 b \left (a^2+b^2\right ) \tan ^4(c+d x)}{2 d}+\frac {b^3 \left (5 a^2+b^2\right ) \tan ^8(c+d x)}{4 d}+\frac {b \left (5 a^4+20 a^2 b^2+b^4\right ) \tan ^6(c+d x)}{6 d}+\frac {a \left (a^4+20 a^2 b^2+5 b^4\right ) \tan ^5(c+d x)}{5 d}+\frac {2 a^3 \left (a^2+5 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {5 a b^4 \tan ^9(c+d x)}{9 d}+\frac {b^5 \tan ^{10}(c+d x)}{10 d} \]

[Out]

a^5*tan(d*x+c)/d+5/2*a^4*b*tan(d*x+c)^2/d+2/3*a^3*(a^2+5*b^2)*tan(d*x+c)^3/d+5/2*a^2*b*(a^2+b^2)*tan(d*x+c)^4/
d+1/5*a*(a^4+20*a^2*b^2+5*b^4)*tan(d*x+c)^5/d+1/6*b*(5*a^4+20*a^2*b^2+b^4)*tan(d*x+c)^6/d+10/7*a*b^2*(a^2+b^2)
*tan(d*x+c)^7/d+1/4*b^3*(5*a^2+b^2)*tan(d*x+c)^8/d+5/9*a*b^4*tan(d*x+c)^9/d+1/10*b^5*tan(d*x+c)^10/d

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Rubi [A]  time = 0.22, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3088, 948} \[ \frac {b^3 \left (5 a^2+b^2\right ) \tan ^8(c+d x)}{4 d}+\frac {10 a b^2 \left (a^2+b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {b \left (20 a^2 b^2+5 a^4+b^4\right ) \tan ^6(c+d x)}{6 d}+\frac {a \left (20 a^2 b^2+a^4+5 b^4\right ) \tan ^5(c+d x)}{5 d}+\frac {5 a^2 b \left (a^2+b^2\right ) \tan ^4(c+d x)}{2 d}+\frac {2 a^3 \left (a^2+5 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {5 a^4 b \tan ^2(c+d x)}{2 d}+\frac {a^5 \tan (c+d x)}{d}+\frac {5 a b^4 \tan ^9(c+d x)}{9 d}+\frac {b^5 \tan ^{10}(c+d x)}{10 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(a^5*Tan[c + d*x])/d + (5*a^4*b*Tan[c + d*x]^2)/(2*d) + (2*a^3*(a^2 + 5*b^2)*Tan[c + d*x]^3)/(3*d) + (5*a^2*b*
(a^2 + b^2)*Tan[c + d*x]^4)/(2*d) + (a*(a^4 + 20*a^2*b^2 + 5*b^4)*Tan[c + d*x]^5)/(5*d) + (b*(5*a^4 + 20*a^2*b
^2 + b^4)*Tan[c + d*x]^6)/(6*d) + (10*a*b^2*(a^2 + b^2)*Tan[c + d*x]^7)/(7*d) + (b^3*(5*a^2 + b^2)*Tan[c + d*x
]^8)/(4*d) + (5*a*b^4*Tan[c + d*x]^9)/(9*d) + (b^5*Tan[c + d*x]^10)/(10*d)

Rule 948

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0]))

Rule 3088

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symb
ol] :> -Dist[d^(-1), Subst[Int[(x^m*(b + a*x)^n)/(1 + x^2)^((m + n + 2)/2), x], x, Cot[c + d*x]], x] /; FreeQ[
{a, b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] &&  !(GtQ[n, 0] && GtQ[m, 1])

Rubi steps

\begin {align*} \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(b+a x)^5 \left (1+x^2\right )^2}{x^{11}} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {b^5}{x^{11}}+\frac {5 a b^4}{x^{10}}+\frac {2 \left (5 a^2 b^3+b^5\right )}{x^9}+\frac {10 a b^2 \left (a^2+b^2\right )}{x^8}+\frac {5 a^4 b+20 a^2 b^3+b^5}{x^7}+\frac {a^5+20 a^3 b^2+5 a b^4}{x^6}+\frac {10 a^2 b \left (a^2+b^2\right )}{x^5}+\frac {2 \left (a^5+5 a^3 b^2\right )}{x^4}+\frac {5 a^4 b}{x^3}+\frac {a^5}{x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a^5 \tan (c+d x)}{d}+\frac {5 a^4 b \tan ^2(c+d x)}{2 d}+\frac {2 a^3 \left (a^2+5 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {5 a^2 b \left (a^2+b^2\right ) \tan ^4(c+d x)}{2 d}+\frac {a \left (a^4+20 a^2 b^2+5 b^4\right ) \tan ^5(c+d x)}{5 d}+\frac {b \left (5 a^4+20 a^2 b^2+b^4\right ) \tan ^6(c+d x)}{6 d}+\frac {10 a b^2 \left (a^2+b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {b^3 \left (5 a^2+b^2\right ) \tan ^8(c+d x)}{4 d}+\frac {5 a b^4 \tan ^9(c+d x)}{9 d}+\frac {b^5 \tan ^{10}(c+d x)}{10 d}\\ \end {align*}

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Mathematica [A]  time = 1.22, size = 115, normalized size = 0.48 \[ \frac {\frac {1}{4} \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^8-\frac {4}{7} a \left (a^2+b^2\right ) (a+b \tan (c+d x))^7+\frac {1}{6} \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^6+\frac {1}{10} (a+b \tan (c+d x))^{10}-\frac {4}{9} a (a+b \tan (c+d x))^9}{b^5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(((a^2 + b^2)^2*(a + b*Tan[c + d*x])^6)/6 - (4*a*(a^2 + b^2)*(a + b*Tan[c + d*x])^7)/7 + ((3*a^2 + b^2)*(a + b
*Tan[c + d*x])^8)/4 - (4*a*(a + b*Tan[c + d*x])^9)/9 + (a + b*Tan[c + d*x])^10/10)/(b^5*d)

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fricas [A]  time = 0.53, size = 207, normalized size = 0.86 \[ \frac {126 \, b^{5} + 210 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 315 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (8 \, {\left (21 \, a^{5} - 30 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} + 4 \, {\left (21 \, a^{5} - 30 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 175 \, a b^{4} \cos \left (d x + c\right ) + 3 \, {\left (21 \, a^{5} - 30 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 50 \, {\left (9 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{1260 \, d \cos \left (d x + c\right )^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="fricas")

[Out]

1/1260*(126*b^5 + 210*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x + c)^4 + 315*(5*a^2*b^3 - b^5)*cos(d*x + c)^2 + 4*(
8*(21*a^5 - 30*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^9 + 4*(21*a^5 - 30*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^7 + 175*a*b^
4*cos(d*x + c) + 3*(21*a^5 - 30*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^5 + 50*(9*a^3*b^2 - 5*a*b^4)*cos(d*x + c)^3)*s
in(d*x + c))/(d*cos(d*x + c)^10)

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giac [A]  time = 1.18, size = 262, normalized size = 1.08 \[ \frac {126 \, b^{5} \tan \left (d x + c\right )^{10} + 700 \, a b^{4} \tan \left (d x + c\right )^{9} + 1575 \, a^{2} b^{3} \tan \left (d x + c\right )^{8} + 315 \, b^{5} \tan \left (d x + c\right )^{8} + 1800 \, a^{3} b^{2} \tan \left (d x + c\right )^{7} + 1800 \, a b^{4} \tan \left (d x + c\right )^{7} + 1050 \, a^{4} b \tan \left (d x + c\right )^{6} + 4200 \, a^{2} b^{3} \tan \left (d x + c\right )^{6} + 210 \, b^{5} \tan \left (d x + c\right )^{6} + 252 \, a^{5} \tan \left (d x + c\right )^{5} + 5040 \, a^{3} b^{2} \tan \left (d x + c\right )^{5} + 1260 \, a b^{4} \tan \left (d x + c\right )^{5} + 3150 \, a^{4} b \tan \left (d x + c\right )^{4} + 3150 \, a^{2} b^{3} \tan \left (d x + c\right )^{4} + 840 \, a^{5} \tan \left (d x + c\right )^{3} + 4200 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 3150 \, a^{4} b \tan \left (d x + c\right )^{2} + 1260 \, a^{5} \tan \left (d x + c\right )}{1260 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="giac")

[Out]

1/1260*(126*b^5*tan(d*x + c)^10 + 700*a*b^4*tan(d*x + c)^9 + 1575*a^2*b^3*tan(d*x + c)^8 + 315*b^5*tan(d*x + c
)^8 + 1800*a^3*b^2*tan(d*x + c)^7 + 1800*a*b^4*tan(d*x + c)^7 + 1050*a^4*b*tan(d*x + c)^6 + 4200*a^2*b^3*tan(d
*x + c)^6 + 210*b^5*tan(d*x + c)^6 + 252*a^5*tan(d*x + c)^5 + 5040*a^3*b^2*tan(d*x + c)^5 + 1260*a*b^4*tan(d*x
+ c)^5 + 3150*a^4*b*tan(d*x + c)^4 + 3150*a^2*b^3*tan(d*x + c)^4 + 840*a^5*tan(d*x + c)^3 + 4200*a^3*b^2*tan(
d*x + c)^3 + 3150*a^4*b*tan(d*x + c)^2 + 1260*a^5*tan(d*x + c))/d

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maple [A]  time = 0.27, size = 299, normalized size = 1.24 \[ \frac {-a^{5} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {5 a^{4} b}{6 \cos \left (d x +c \right )^{6}}+10 a^{3} b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+10 a^{2} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )+5 a \,b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )+b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{6}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{60 \cos \left (d x +c \right )^{6}}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^5,x)

[Out]

1/d*(-a^5*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+5/6*a^4*b/cos(d*x+c)^6+10*a^3*b^2*(1/7*sin(d*x
+c)^3/cos(d*x+c)^7+4/35*sin(d*x+c)^3/cos(d*x+c)^5+8/105*sin(d*x+c)^3/cos(d*x+c)^3)+10*a^2*b^3*(1/8*sin(d*x+c)^
4/cos(d*x+c)^8+1/12*sin(d*x+c)^4/cos(d*x+c)^6+1/24*sin(d*x+c)^4/cos(d*x+c)^4)+5*a*b^4*(1/9*sin(d*x+c)^5/cos(d*
x+c)^9+4/63*sin(d*x+c)^5/cos(d*x+c)^7+8/315*sin(d*x+c)^5/cos(d*x+c)^5)+b^5*(1/10*sin(d*x+c)^6/cos(d*x+c)^10+1/
20*sin(d*x+c)^6/cos(d*x+c)^8+1/60*sin(d*x+c)^6/cos(d*x+c)^6))

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maxima [A]  time = 0.78, size = 275, normalized size = 1.14 \[ \frac {84 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{5} + 120 \, {\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} a^{3} b^{2} + 20 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 90 \, \tan \left (d x + c\right )^{7} + 63 \, \tan \left (d x + c\right )^{5}\right )} a b^{4} + \frac {525 \, {\left (4 \, \sin \left (d x + c\right )^{2} - 1\right )} a^{2} b^{3}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - \frac {21 \, {\left (10 \, \sin \left (d x + c\right )^{4} - 5 \, \sin \left (d x + c\right )^{2} + 1\right )} b^{5}}{\sin \left (d x + c\right )^{10} - 5 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} - 1} - \frac {1050 \, a^{4} b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{3}}}{1260 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="maxima")

[Out]

1/1260*(84*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*a^5 + 120*(15*tan(d*x + c)^7 + 42*tan(d*x
+ c)^5 + 35*tan(d*x + c)^3)*a^3*b^2 + 20*(35*tan(d*x + c)^9 + 90*tan(d*x + c)^7 + 63*tan(d*x + c)^5)*a*b^4 + 5
25*(4*sin(d*x + c)^2 - 1)*a^2*b^3/(sin(d*x + c)^8 - 4*sin(d*x + c)^6 + 6*sin(d*x + c)^4 - 4*sin(d*x + c)^2 + 1
) - 21*(10*sin(d*x + c)^4 - 5*sin(d*x + c)^2 + 1)*b^5/(sin(d*x + c)^10 - 5*sin(d*x + c)^8 + 10*sin(d*x + c)^6
- 10*sin(d*x + c)^4 + 5*sin(d*x + c)^2 - 1) - 1050*a^4*b/(sin(d*x + c)^2 - 1)^3)/d

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mupad [B]  time = 4.44, size = 548, normalized size = 2.26 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {616\,a^5}{15}-\frac {176\,a^3\,b^2}{3}+32\,a\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (40\,a^4\,b-40\,a^2\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (40\,a^4\,b-40\,a^2\,b^3\right )-2\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {616\,a^5}{15}-\frac {176\,a^3\,b^2}{3}+32\,a\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (-88\,a^5+\frac {720\,a^3\,b^2}{7}+\frac {160\,a\,b^4}{7}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (-88\,a^5+\frac {720\,a^3\,b^2}{7}+\frac {160\,a\,b^4}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {388\,a^5}{3}-\frac {4240\,a^3\,b^2}{21}+\frac {3520\,a\,b^4}{63}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {388\,a^5}{3}-\frac {4240\,a^3\,b^2}{21}+\frac {3520\,a\,b^4}{63}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {280\,a^4\,b}{3}-\frac {80\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {280\,a^4\,b}{3}-\frac {80\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (220\,a^4\,b-160\,a^2\,b^3+\frac {192\,b^5}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (-\frac {520\,a^4\,b}{3}+\frac {200\,a^2\,b^3}{3}+\frac {64\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (-\frac {520\,a^4\,b}{3}+\frac {200\,a^2\,b^3}{3}+\frac {64\,b^5}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {38\,a^5}{3}-\frac {80\,a^3\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\left (\frac {38\,a^5}{3}-\frac {80\,a^3\,b^2}{3}\right )+2\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cos(c + d*x) + b*sin(c + d*x))^5/cos(c + d*x)^11,x)

[Out]

(tan(c/2 + (d*x)/2)^5*(32*a*b^4 + (616*a^5)/15 - (176*a^3*b^2)/3) - tan(c/2 + (d*x)/2)^4*(40*a^4*b - 40*a^2*b^
3) - tan(c/2 + (d*x)/2)^16*(40*a^4*b - 40*a^2*b^3) - 2*a^5*tan(c/2 + (d*x)/2)^19 - tan(c/2 + (d*x)/2)^15*(32*a
*b^4 + (616*a^5)/15 - (176*a^3*b^2)/3) + tan(c/2 + (d*x)/2)^7*((160*a*b^4)/7 - 88*a^5 + (720*a^3*b^2)/7) - tan
(c/2 + (d*x)/2)^13*((160*a*b^4)/7 - 88*a^5 + (720*a^3*b^2)/7) + tan(c/2 + (d*x)/2)^9*((3520*a*b^4)/63 + (388*a
^5)/3 - (4240*a^3*b^2)/21) - tan(c/2 + (d*x)/2)^11*((3520*a*b^4)/63 + (388*a^5)/3 - (4240*a^3*b^2)/21) + tan(c
/2 + (d*x)/2)^6*((280*a^4*b)/3 + (32*b^5)/3 - (80*a^2*b^3)/3) + tan(c/2 + (d*x)/2)^14*((280*a^4*b)/3 + (32*b^5
)/3 - (80*a^2*b^3)/3) + tan(c/2 + (d*x)/2)^10*(220*a^4*b + (192*b^5)/5 - 160*a^2*b^3) + tan(c/2 + (d*x)/2)^8*(
(64*b^5)/3 - (520*a^4*b)/3 + (200*a^2*b^3)/3) + tan(c/2 + (d*x)/2)^12*((64*b^5)/3 - (520*a^4*b)/3 + (200*a^2*b
^3)/3) - tan(c/2 + (d*x)/2)^3*((38*a^5)/3 - (80*a^3*b^2)/3) + tan(c/2 + (d*x)/2)^17*((38*a^5)/3 - (80*a^3*b^2)
/3) + 2*a^5*tan(c/2 + (d*x)/2) + 10*a^4*b*tan(c/2 + (d*x)/2)^2 + 10*a^4*b*tan(c/2 + (d*x)/2)^18)/(d*(tan(c/2 +
(d*x)/2)^2 - 1)^10)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**11*(a*cos(d*x+c)+b*sin(d*x+c))**5,x)

[Out]

Timed out

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