\(\int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx\) [90]

Optimal result

Integrand size = 28, antiderivative size = 408 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {5 a^4 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {15 a^2 b^2 \text {arctanh}(\sin (c+d x))}{64 d}+\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{256 d}+\frac {4 a^3 b \sec ^7(c+d x)}{7 d}-\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {4 a b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {15 a^2 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac {3 b^4 \sec (c+d x) \tan (c+d x)}{256 d}+\frac {5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d} \]

[Out]

5/16*a^4*arctanh(sin(d*x+c))/d-15/64*a^2*b^2*arctanh(sin(d*x+c))/d+3/256*b^4*arctanh(sin(d*x+c))/d+4/7*a^3*b*s
ec(d*x+c)^7/d-4/7*a*b^3*sec(d*x+c)^7/d+4/9*a*b^3*sec(d*x+c)^9/d+5/16*a^4*sec(d*x+c)*tan(d*x+c)/d-15/64*a^2*b^2
*sec(d*x+c)*tan(d*x+c)/d+3/256*b^4*sec(d*x+c)*tan(d*x+c)/d+5/24*a^4*sec(d*x+c)^3*tan(d*x+c)/d-5/32*a^2*b^2*sec
(d*x+c)^3*tan(d*x+c)/d+1/128*b^4*sec(d*x+c)^3*tan(d*x+c)/d+1/6*a^4*sec(d*x+c)^5*tan(d*x+c)/d-1/8*a^2*b^2*sec(d
*x+c)^5*tan(d*x+c)/d+1/160*b^4*sec(d*x+c)^5*tan(d*x+c)/d+3/4*a^2*b^2*sec(d*x+c)^7*tan(d*x+c)/d-3/80*b^4*sec(d*
x+c)^7*tan(d*x+c)/d+1/10*b^4*sec(d*x+c)^7*tan(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3169, 3853, 3855, 2686, 30, 2691, 14} \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {5 a^4 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^4 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac {5 a^4 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {5 a^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac {4 a^3 b \sec ^7(c+d x)}{7 d}-\frac {15 a^2 b^2 \text {arctanh}(\sin (c+d x))}{64 d}+\frac {3 a^2 b^2 \tan (c+d x) \sec ^7(c+d x)}{4 d}-\frac {a^2 b^2 \tan (c+d x) \sec ^5(c+d x)}{8 d}-\frac {5 a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{32 d}-\frac {15 a^2 b^2 \tan (c+d x) \sec (c+d x)}{64 d}+\frac {4 a b^3 \sec ^9(c+d x)}{9 d}-\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{256 d}+\frac {b^4 \tan ^3(c+d x) \sec ^7(c+d x)}{10 d}-\frac {3 b^4 \tan (c+d x) \sec ^7(c+d x)}{80 d}+\frac {b^4 \tan (c+d x) \sec ^5(c+d x)}{160 d}+\frac {b^4 \tan (c+d x) \sec ^3(c+d x)}{128 d}+\frac {3 b^4 \tan (c+d x) \sec (c+d x)}{256 d} \]

[In]

Int[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]

[Out]

(5*a^4*ArcTanh[Sin[c + d*x]])/(16*d) - (15*a^2*b^2*ArcTanh[Sin[c + d*x]])/(64*d) + (3*b^4*ArcTanh[Sin[c + d*x]
])/(256*d) + (4*a^3*b*Sec[c + d*x]^7)/(7*d) - (4*a*b^3*Sec[c + d*x]^7)/(7*d) + (4*a*b^3*Sec[c + d*x]^9)/(9*d)
+ (5*a^4*Sec[c + d*x]*Tan[c + d*x])/(16*d) - (15*a^2*b^2*Sec[c + d*x]*Tan[c + d*x])/(64*d) + (3*b^4*Sec[c + d*
x]*Tan[c + d*x])/(256*d) + (5*a^4*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) - (5*a^2*b^2*Sec[c + d*x]^3*Tan[c + d*x]
)/(32*d) + (b^4*Sec[c + d*x]^3*Tan[c + d*x])/(128*d) + (a^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) - (a^2*b^2*Sec[
c + d*x]^5*Tan[c + d*x])/(8*d) + (b^4*Sec[c + d*x]^5*Tan[c + d*x])/(160*d) + (3*a^2*b^2*Sec[c + d*x]^7*Tan[c +
d*x])/(4*d) - (3*b^4*Sec[c + d*x]^7*Tan[c + d*x])/(80*d) + (b^4*Sec[c + d*x]^7*Tan[c + d*x]^3)/(10*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

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

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 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 3169

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_.), x_Sym
bol] :> Int[ExpandTrig[cos[c + d*x]^m*(a*cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] &&
IntegerQ[m] && IGtQ[n, 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}& = \int \left (a^4 \sec ^7(c+d x)+4 a^3 b \sec ^7(c+d x) \tan (c+d x)+6 a^2 b^2 \sec ^7(c+d x) \tan ^2(c+d x)+4 a b^3 \sec ^7(c+d x) \tan ^3(c+d x)+b^4 \sec ^7(c+d x) \tan ^4(c+d x)\right ) \, dx \\ & = a^4 \int \sec ^7(c+d x) \, dx+\left (4 a^3 b\right ) \int \sec ^7(c+d x) \tan (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \sec ^7(c+d x) \tan ^3(c+d x) \, dx+b^4 \int \sec ^7(c+d x) \tan ^4(c+d x) \, dx \\ & = \frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}+\frac {1}{6} \left (5 a^4\right ) \int \sec ^5(c+d x) \, dx-\frac {1}{4} \left (3 a^2 b^2\right ) \int \sec ^7(c+d x) \, dx-\frac {1}{10} \left (3 b^4\right ) \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx+\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int x^6 \, dx,x,\sec (c+d x)\right )}{d}+\frac {\left (4 a b^3\right ) \text {Subst}\left (\int x^6 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {4 a^3 b \sec ^7(c+d x)}{7 d}+\frac {5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}+\frac {1}{8} \left (5 a^4\right ) \int \sec ^3(c+d x) \, dx-\frac {1}{8} \left (5 a^2 b^2\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{80} \left (3 b^4\right ) \int \sec ^7(c+d x) \, dx+\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \left (-x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {4 a^3 b \sec ^7(c+d x)}{7 d}-\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {4 a b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}+\frac {1}{16} \left (5 a^4\right ) \int \sec (c+d x) \, dx-\frac {1}{32} \left (15 a^2 b^2\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{32} b^4 \int \sec ^5(c+d x) \, dx \\ & = \frac {5 a^4 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^3 b \sec ^7(c+d x)}{7 d}-\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {4 a b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {15 a^2 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac {5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}-\frac {1}{64} \left (15 a^2 b^2\right ) \int \sec (c+d x) \, dx+\frac {1}{128} \left (3 b^4\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {5 a^4 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {15 a^2 b^2 \text {arctanh}(\sin (c+d x))}{64 d}+\frac {4 a^3 b \sec ^7(c+d x)}{7 d}-\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {4 a b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {15 a^2 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac {3 b^4 \sec (c+d x) \tan (c+d x)}{256 d}+\frac {5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}+\frac {1}{256} \left (3 b^4\right ) \int \sec (c+d x) \, dx \\ & = \frac {5 a^4 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {15 a^2 b^2 \text {arctanh}(\sin (c+d x))}{64 d}+\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{256 d}+\frac {4 a^3 b \sec ^7(c+d x)}{7 d}-\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {4 a b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {15 a^2 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac {3 b^4 \sec (c+d x) \tan (c+d x)}{256 d}+\frac {5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.16 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.59 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {-80640 \left (80 a^4-60 a^2 b^2+3 b^4\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+3 \sec ^{10}(c+d x) \left (983040 a b \left (a^2-b^2\right ) \cos (3 (c+d x))+420 \left (1552 a^4+1908 a^2 b^2-505 b^4\right ) \sin (3 (c+d x))+7 \left (80 a^4-60 a^2 b^2+3 b^4\right ) (628 \sin (5 (c+d x))+145 \sin (7 (c+d x))+15 \sin (9 (c+d x)))\right )+10 \sec ^9(c+d x) \left (32768 a b \left (27 a^2+b^2\right )+189 \left (592 a^4+1604 a^2 b^2+739 b^4\right ) \tan (c+d x)\right )}{20643840 d} \]

[In]

Integrate[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]

[Out]

(-80640*(80*a^4 - 60*a^2*b^2 + 3*b^4)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(
c + d*x)/2]]) + 3*Sec[c + d*x]^10*(983040*a*b*(a^2 - b^2)*Cos[3*(c + d*x)] + 420*(1552*a^4 + 1908*a^2*b^2 - 50
5*b^4)*Sin[3*(c + d*x)] + 7*(80*a^4 - 60*a^2*b^2 + 3*b^4)*(628*Sin[5*(c + d*x)] + 145*Sin[7*(c + d*x)] + 15*Si
n[9*(c + d*x)])) + 10*Sec[c + d*x]^9*(32768*a*b*(27*a^2 + b^2) + 189*(592*a^4 + 1604*a^2*b^2 + 739*b^4)*Tan[c
+ d*x]))/(20643840*d)

Maple [A] (verified)

Time = 2.51 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.86

method	result	size
parts	\(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{32 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{256 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{256}-\frac {3 \sin \left (d x +c \right )}{256}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{256}\right )}{d}+\frac {4 a \,b^{3} \left (\frac {\sec \left (d x +c \right )^{9}}{9}-\frac {\sec \left (d x +c \right )^{7}}{7}\right )}{d}+\frac {4 a^{3} b \sec \left (d x +c \right )^{7}}{7 d}+\frac {6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \sin \left (d x +c \right )^{3}}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \sin \left (d x +c \right )^{3}}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \sin \left (d x +c \right )^{3}}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}\)	\(350\)
derivativedivides	\(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {4 a^{3} b}{7 \cos \left (d x +c \right )^{7}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \sin \left (d x +c \right )^{3}}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \sin \left (d x +c \right )^{3}}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \sin \left (d x +c \right )^{3}}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{63}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{32 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{256 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{256}-\frac {3 \sin \left (d x +c \right )}{256}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{256}\right )}{d}\)	\(427\)
default	\(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {4 a^{3} b}{7 \cos \left (d x +c \right )^{7}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \sin \left (d x +c \right )^{3}}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \sin \left (d x +c \right )^{3}}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \sin \left (d x +c \right )^{3}}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{63}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{32 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{256 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{256}-\frac {3 \sin \left (d x +c \right )}{256}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{256}\right )}{d}\)	\(427\)
parallelrisch	\(\frac {-1134000 \left (\frac {\cos \left (10 d x +10 c \right )}{45}+\frac {2 \cos \left (8 d x +8 c \right )}{9}+\cos \left (6 d x +6 c \right )+\frac {8 \cos \left (4 d x +4 c \right )}{3}+\frac {14 \cos \left (2 d x +2 c \right )}{3}+\frac {14}{5}\right ) \left (a^{4}-\frac {3}{4} a^{2} b^{2}+\frac {3}{80} b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+1134000 \left (\frac {\cos \left (10 d x +10 c \right )}{45}+\frac {2 \cos \left (8 d x +8 c \right )}{9}+\cos \left (6 d x +6 c \right )+\frac {8 \cos \left (4 d x +4 c \right )}{3}+\frac {14 \cos \left (2 d x +2 c \right )}{3}+\frac {14}{5}\right ) \left (a^{4}-\frac {3}{4} a^{2} b^{2}+\frac {3}{80} b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (46080 a^{3} b -10240 a \,b^{3}\right ) \cos \left (10 d x +10 c \right )+\left (9676800 a^{3} b -2150400 a \,b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (5529600 a^{3} b -1228800 a \,b^{3}\right ) \cos \left (4 d x +4 c \right )+\left (2073600 a^{3} b -460800 a \,b^{3}\right ) \cos \left (6 d x +6 c \right )+\left (460800 a^{3} b -102400 a \,b^{3}\right ) \cos \left (8 d x +8 c \right )+\left (3911040 a^{4}+4808160 a^{2} b^{2}-1272600 b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (2110080 a^{4}-1582560 a^{2} b^{2}+79128 b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (487200 a^{4}-365400 a^{2} b^{2}+18270 b^{4}\right ) \sin \left (7 d x +7 c \right )+\left (50400 a^{4}-37800 a^{2} b^{2}+1890 b^{4}\right ) \sin \left (9 d x +9 c \right )+\left (5898240 a^{3} b -5898240 a \,b^{3}\right ) \cos \left (3 d x +3 c \right )+\left (2237760 a^{4}+6063120 a^{2} b^{2}+2793420 b^{4}\right ) \sin \left (d x +c \right )[phone] b \left (\left (a^{2}+\frac {b^{2}}{27}\right ) \cos \left (d x +c \right )+\frac {21 a^{2}}{64}-\frac {7 b^{2}}{96}\right ) a}{80640 d \left (\cos \left (10 d x +10 c \right )+10 \cos \left (8 d x +8 c \right )+45 \cos \left (6 d x +6 c \right )+120 \cos \left (4 d x +4 c \right )+210 \cos \left (2 d x +2 c \right )+126\right )}\)	\(547\)
risch	\(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (-25200 a^{4}-945 b^{4}+18900 a^{2} b^{2}+1118880 a^{4} {\mathrm e}^{10 i \left (d x +c \right )}-9135 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+636300 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+1396710 b^{4} {\mathrm e}^{10 i \left (d x +c \right )}+2404080 a^{2} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-791280 a^{2} b^{2} {\mathrm e}^{14 i \left (d x +c \right )}+327680 i a \,b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+8847360 i a^{3} b \,{\mathrm e}^{10 i \left (d x +c \right )}+3031560 a^{2} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+8847360 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+2949120 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}+327680 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-2949120 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-2949120 i a \,b^{3} {\mathrm e}^{12 i \left (d x +c \right )}+2949120 i a^{3} b \,{\mathrm e}^{12 i \left (d x +c \right )}+243600 a^{4} {\mathrm e}^{16 i \left (d x +c \right )}+9135 b^{4} {\mathrm e}^{16 i \left (d x +c \right )}+25200 a^{4} {\mathrm e}^{18 i \left (d x +c \right )}+945 b^{4} {\mathrm e}^{18 i \left (d x +c \right )}+1055040 a^{4} {\mathrm e}^{14 i \left (d x +c \right )}-1118880 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}-1396710 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-1955520 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-1055040 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-39564 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-243600 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-3031560 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-2404080 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+791280 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+182700 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+1955520 a^{4} {\mathrm e}^{12 i \left (d x +c \right )}+39564 b^{4} {\mathrm e}^{14 i \left (d x +c \right )}-636300 b^{4} {\mathrm e}^{12 i \left (d x +c \right )}-182700 a^{2} b^{2} {\mathrm e}^{16 i \left (d x +c \right )}-18900 a^{2} b^{2} {\mathrm e}^{18 i \left (d x +c \right )}\right )}{40320 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{10}}+\frac {5 \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right ) a^{4}}{16 d}-\frac {15 b^{2} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right ) a^{2}}{64 d}+\frac {3 b^{4} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{256 d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{4}}{16 d}+\frac {15 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{64 d}-\frac {3 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d}\)	\(786\)

[In]

int(sec(d*x+c)^11*(cos(d*x+c)*a+b*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

a^4/d*(-(-1/6*sec(d*x+c)^5-5/24*sec(d*x+c)^3-5/16*sec(d*x+c))*tan(d*x+c)+5/16*ln(sec(d*x+c)+tan(d*x+c)))+b^4/d
*(1/10*sin(d*x+c)^5/cos(d*x+c)^10+1/16*sin(d*x+c)^5/cos(d*x+c)^8+1/32*sin(d*x+c)^5/cos(d*x+c)^6+1/128*sin(d*x+
c)^5/cos(d*x+c)^4-1/256*sin(d*x+c)^5/cos(d*x+c)^2-1/256*sin(d*x+c)^3-3/256*sin(d*x+c)+3/256*ln(sec(d*x+c)+tan(
d*x+c)))+4*a*b^3/d*(1/9*sec(d*x+c)^9-1/7*sec(d*x+c)^7)+4/7*a^3*b*sec(d*x+c)^7/d+6*a^2*b^2/d*(1/8*sin(d*x+c)^3/
cos(d*x+c)^8+5/48*sin(d*x+c)^3/cos(d*x+c)^6+5/64*sin(d*x+c)^3/cos(d*x+c)^4+5/128*sin(d*x+c)^3/cos(d*x+c)^2+5/1
28*sin(d*x+c)-5/128*ln(sec(d*x+c)+tan(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.62 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {315 \, {\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{10} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \, {\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{10} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 71680 \, a b^{3} \cos \left (d x + c\right ) + 92160 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} + 42 \, {\left (15 \, {\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{6} + 8 \, {\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 384 \, b^{4} + 48 \, {\left (60 \, a^{2} b^{2} - 11 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{161280 \, d \cos \left (d x + c\right )^{10}} \]

[In]

integrate(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/161280*(315*(80*a^4 - 60*a^2*b^2 + 3*b^4)*cos(d*x + c)^10*log(sin(d*x + c) + 1) - 315*(80*a^4 - 60*a^2*b^2 +
3*b^4)*cos(d*x + c)^10*log(-sin(d*x + c) + 1) + 71680*a*b^3*cos(d*x + c) + 92160*(a^3*b - a*b^3)*cos(d*x + c)
^3 + 42*(15*(80*a^4 - 60*a^2*b^2 + 3*b^4)*cos(d*x + c)^8 + 10*(80*a^4 - 60*a^2*b^2 + 3*b^4)*cos(d*x + c)^6 + 8
*(80*a^4 - 60*a^2*b^2 + 3*b^4)*cos(d*x + c)^4 + 384*b^4 + 48*(60*a^2*b^2 - 11*b^4)*cos(d*x + c)^2)*sin(d*x + c
))/(d*cos(d*x + c)^10)

Sympy [F(-1)]

Timed out. \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**11*(a*cos(d*x+c)+b*sin(d*x+c))**4,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.94 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {63 \, b^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{9} - 70 \, \sin \left (d x + c\right )^{7} + 128 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{10} - 5 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 1260 \, a^{2} b^{2} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{7} - 55 \, \sin \left (d x + c\right )^{5} + 73 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 1680 \, a^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {92160 \, a^{3} b}{\cos \left (d x + c\right )^{7}} + \frac {10240 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} a b^{3}}{\cos \left (d x + c\right )^{9}}}{161280 \, d} \]

[In]

integrate(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/161280*(63*b^4*(2*(15*sin(d*x + c)^9 - 70*sin(d*x + c)^7 + 128*sin(d*x + c)^5 + 70*sin(d*x + c)^3 - 15*sin(
d*x + c))/(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)
- 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 1260*a^2*b^2*(2*(15*sin(d*x + c)^7 - 55*sin(d*x + c)^
5 + 73*sin(d*x + c)^3 + 15*sin(d*x + c))/(sin(d*x + c)^8 - 4*sin(d*x + c)^6 + 6*sin(d*x + c)^4 - 4*sin(d*x + c
)^2 + 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) + 1680*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)) - 92160*a^3*b/cos(d*x + c)^7 + 10240*(9*cos(d*x + c)^2 - 7)*a*b^3/cos(d*x + c)^9
)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 880 vs. \(2 (372) = 744\).

Time = 0.50 (sec) , antiderivative size = 880, normalized size of antiderivative = 2.16 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="giac")

[Out]

1/80640*(315*(80*a^4 - 60*a^2*b^2 + 3*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 315*(80*a^4 - 60*a^2*b^2 + 3*b
^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(55440*a^4*tan(1/2*d*x + 1/2*c)^19 + 18900*a^2*b^2*tan(1/2*d*x + 1/
2*c)^19 - 945*b^4*tan(1/2*d*x + 1/2*c)^19 - 322560*a^3*b*tan(1/2*d*x + 1/2*c)^18 - 213360*a^4*tan(1/2*d*x + 1/
2*c)^17 + 462420*a^2*b^2*tan(1/2*d*x + 1/2*c)^17 + 9135*b^4*tan(1/2*d*x + 1/2*c)^17 + 967680*a^3*b*tan(1/2*d*x
+ 1/2*c)^16 - 645120*a*b^3*tan(1/2*d*x + 1/2*c)^16 + 450240*a^4*tan(1/2*d*x + 1/2*c)^15 + 146160*a^2*b^2*tan(
1/2*d*x + 1/2*c)^15 + 218484*b^4*tan(1/2*d*x + 1/2*c)^15 - 2580480*a^3*b*tan(1/2*d*x + 1/2*c)^14 - 430080*a*b^
3*tan(1/2*d*x + 1/2*c)^14 - 624960*a^4*tan(1/2*d*x + 1/2*c)^13 + 468720*a^2*b^2*tan(1/2*d*x + 1/2*c)^13 + 6539
40*b^4*tan(1/2*d*x + 1/2*c)^13 + 5160960*a^3*b*tan(1/2*d*x + 1/2*c)^12 - 2150400*a*b^3*tan(1/2*d*x + 1/2*c)^12
+ 332640*a^4*tan(1/2*d*x + 1/2*c)^11 - 1096200*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 1183770*b^4*tan(1/2*d*x + 1/
2*c)^11 - 5806080*a^3*b*tan(1/2*d*x + 1/2*c)^10 + 1290240*a*b^3*tan(1/2*d*x + 1/2*c)^10 + 332640*a^4*tan(1/2*d
*x + 1/2*c)^9 - 1096200*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 1183770*b^4*tan(1/2*d*x + 1/2*c)^9 + 4515840*a^3*b*ta
n(1/2*d*x + 1/2*c)^8 - 624960*a^4*tan(1/2*d*x + 1/2*c)^7 + 468720*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 653940*b^4*
tan(1/2*d*x + 1/2*c)^7 - 2949120*a^3*b*tan(1/2*d*x + 1/2*c)^6 + 1658880*a*b^3*tan(1/2*d*x + 1/2*c)^6 + 450240*
a^4*tan(1/2*d*x + 1/2*c)^5 + 146160*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 218484*b^4*tan(1/2*d*x + 1/2*c)^5 + 11059
20*a^3*b*tan(1/2*d*x + 1/2*c)^4 + 184320*a*b^3*tan(1/2*d*x + 1/2*c)^4 - 213360*a^4*tan(1/2*d*x + 1/2*c)^3 + 46
2420*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 9135*b^4*tan(1/2*d*x + 1/2*c)^3 - 138240*a^3*b*tan(1/2*d*x + 1/2*c)^2 +
102400*a*b^3*tan(1/2*d*x + 1/2*c)^2 + 55440*a^4*tan(1/2*d*x + 1/2*c) + 18900*a^2*b^2*tan(1/2*d*x + 1/2*c) - 94
5*b^4*tan(1/2*d*x + 1/2*c) + 46080*a^3*b - 10240*a*b^3)/(tan(1/2*d*x + 1/2*c)^2 - 1)^10)/d

Mupad [B] (verification not implemented)

Time = 27.36 (sec) , antiderivative size = 703, normalized size of antiderivative = 1.72 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Too large to display} \]

[In]

int((a*cos(c + d*x) + b*sin(c + d*x))^4/cos(c + d*x)^11,x)

[Out]

(atanh(tan(c/2 + (d*x)/2))*((5*a^4)/8 + (3*b^4)/128 - (15*a^2*b^2)/32))/d + (tan(c/2 + (d*x)/2)^19*((11*a^4)/8
- (3*b^4)/128 + (15*a^2*b^2)/32) + tan(c/2 + (d*x)/2)^7*((519*b^4)/32 - (31*a^4)/2 + (93*a^2*b^2)/8) + tan(c/
2 + (d*x)/2)^13*((519*b^4)/32 - (31*a^4)/2 + (93*a^2*b^2)/8) + tan(c/2 + (d*x)/2)^3*((29*b^4)/128 - (127*a^4)/
24 + (367*a^2*b^2)/32) + tan(c/2 + (d*x)/2)^17*((29*b^4)/128 - (127*a^4)/24 + (367*a^2*b^2)/32) + tan(c/2 + (d
*x)/2)^5*((67*a^4)/6 + (867*b^4)/160 + (29*a^2*b^2)/8) + tan(c/2 + (d*x)/2)^15*((67*a^4)/6 + (867*b^4)/160 + (
29*a^2*b^2)/8) + tan(c/2 + (d*x)/2)^9*((33*a^4)/4 + (1879*b^4)/64 - (435*a^2*b^2)/16) + tan(c/2 + (d*x)/2)^11*
((33*a^4)/4 + (1879*b^4)/64 - (435*a^2*b^2)/16) - (16*a*b^3)/63 + (8*a^3*b)/7 + tan(c/2 + (d*x)/2)*((11*a^4)/8
- (3*b^4)/128 + (15*a^2*b^2)/32) - tan(c/2 + (d*x)/2)^16*(16*a*b^3 - 24*a^3*b) - tan(c/2 + (d*x)/2)^14*((32*a
*b^3)/3 + 64*a^3*b) + tan(c/2 + (d*x)/2)^10*(32*a*b^3 - 144*a^3*b) + tan(c/2 + (d*x)/2)^4*((32*a*b^3)/7 + (192
*a^3*b)/7) + tan(c/2 + (d*x)/2)^2*((160*a*b^3)/63 - (24*a^3*b)/7) - tan(c/2 + (d*x)/2)^12*((160*a*b^3)/3 - 128
*a^3*b) + tan(c/2 + (d*x)/2)^6*((288*a*b^3)/7 - (512*a^3*b)/7) + 112*a^3*b*tan(c/2 + (d*x)/2)^8 - 8*a^3*b*tan(
c/2 + (d*x)/2)^18)/(d*(45*tan(c/2 + (d*x)/2)^4 - 10*tan(c/2 + (d*x)/2)^2 - 120*tan(c/2 + (d*x)/2)^6 + 210*tan(
c/2 + (d*x)/2)^8 - 252*tan(c/2 + (d*x)/2)^10 + 210*tan(c/2 + (d*x)/2)^12 - 120*tan(c/2 + (d*x)/2)^14 + 45*tan(
c/2 + (d*x)/2)^16 - 10*tan(c/2 + (d*x)/2)^18 + tan(c/2 + (d*x)/2)^20 + 1))