\(\int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\) [109]

Optimal result

Integrand size = 28, antiderivative size = 472 \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {5 a^5 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {25 a^3 b^2 \text {arctanh}(\sin (c+d x))}{64 d}+\frac {15 a b^4 \text {arctanh}(\sin (c+d x))}{256 d}+\frac {5 a^4 b \sec ^7(c+d x)}{7 d}-\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}+\frac {b^5 \sec ^7(c+d x)}{7 d}+\frac {10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac {2 b^5 \sec ^9(c+d x)}{9 d}+\frac {b^5 \sec ^{11}(c+d x)}{11 d}+\frac {5 a^5 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {25 a^3 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac {15 a b^4 \sec (c+d x) \tan (c+d x)}{256 d}+\frac {5 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {25 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{96 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac {a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{24 d}+\frac {a b^4 \sec ^5(c+d x) \tan (c+d x)}{32 d}+\frac {5 a^3 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 a b^4 \sec ^7(c+d x) \tan (c+d x)}{16 d}+\frac {a b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{2 d} \]

[Out]

5/16*a^5*arctanh(sin(d*x+c))/d-25/64*a^3*b^2*arctanh(sin(d*x+c))/d+15/256*a*b^4*arctanh(sin(d*x+c))/d+5/7*a^4*
b*sec(d*x+c)^7/d-10/7*a^2*b^3*sec(d*x+c)^7/d+1/7*b^5*sec(d*x+c)^7/d+10/9*a^2*b^3*sec(d*x+c)^9/d-2/9*b^5*sec(d*
x+c)^9/d+1/11*b^5*sec(d*x+c)^11/d+5/16*a^5*sec(d*x+c)*tan(d*x+c)/d-25/64*a^3*b^2*sec(d*x+c)*tan(d*x+c)/d+15/25
6*a*b^4*sec(d*x+c)*tan(d*x+c)/d+5/24*a^5*sec(d*x+c)^3*tan(d*x+c)/d-25/96*a^3*b^2*sec(d*x+c)^3*tan(d*x+c)/d+5/1
28*a*b^4*sec(d*x+c)^3*tan(d*x+c)/d+1/6*a^5*sec(d*x+c)^5*tan(d*x+c)/d-5/24*a^3*b^2*sec(d*x+c)^5*tan(d*x+c)/d+1/
32*a*b^4*sec(d*x+c)^5*tan(d*x+c)/d+5/4*a^3*b^2*sec(d*x+c)^7*tan(d*x+c)/d-3/16*a*b^4*sec(d*x+c)^7*tan(d*x+c)/d+
1/2*a*b^4*sec(d*x+c)^7*tan(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3169, 3853, 3855, 2686, 30, 2691, 14, 276} \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {5 a^5 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac {5 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {5 a^5 \tan (c+d x) \sec (c+d x)}{16 d}+\frac {5 a^4 b \sec ^7(c+d x)}{7 d}-\frac {25 a^3 b^2 \text {arctanh}(\sin (c+d x))}{64 d}+\frac {5 a^3 b^2 \tan (c+d x) \sec ^7(c+d x)}{4 d}-\frac {5 a^3 b^2 \tan (c+d x) \sec ^5(c+d x)}{24 d}-\frac {25 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{96 d}-\frac {25 a^3 b^2 \tan (c+d x) \sec (c+d x)}{64 d}+\frac {10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}+\frac {15 a b^4 \text {arctanh}(\sin (c+d x))}{256 d}+\frac {a b^4 \tan ^3(c+d x) \sec ^7(c+d x)}{2 d}-\frac {3 a b^4 \tan (c+d x) \sec ^7(c+d x)}{16 d}+\frac {a b^4 \tan (c+d x) \sec ^5(c+d x)}{32 d}+\frac {5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{128 d}+\frac {15 a b^4 \tan (c+d x) \sec (c+d x)}{256 d}+\frac {b^5 \sec ^{11}(c+d x)}{11 d}-\frac {2 b^5 \sec ^9(c+d x)}{9 d}+\frac {b^5 \sec ^7(c+d x)}{7 d} \]

[In]

Int[Sec[c + d*x]^12*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(5*a^5*ArcTanh[Sin[c + d*x]])/(16*d) - (25*a^3*b^2*ArcTanh[Sin[c + d*x]])/(64*d) + (15*a*b^4*ArcTanh[Sin[c + d
*x]])/(256*d) + (5*a^4*b*Sec[c + d*x]^7)/(7*d) - (10*a^2*b^3*Sec[c + d*x]^7)/(7*d) + (b^5*Sec[c + d*x]^7)/(7*d
) + (10*a^2*b^3*Sec[c + d*x]^9)/(9*d) - (2*b^5*Sec[c + d*x]^9)/(9*d) + (b^5*Sec[c + d*x]^11)/(11*d) + (5*a^5*S
ec[c + d*x]*Tan[c + d*x])/(16*d) - (25*a^3*b^2*Sec[c + d*x]*Tan[c + d*x])/(64*d) + (15*a*b^4*Sec[c + d*x]*Tan[
c + d*x])/(256*d) + (5*a^5*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) - (25*a^3*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(96*
d) + (5*a*b^4*Sec[c + d*x]^3*Tan[c + d*x])/(128*d) + (a^5*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) - (5*a^3*b^2*Sec[
c + d*x]^5*Tan[c + d*x])/(24*d) + (a*b^4*Sec[c + d*x]^5*Tan[c + d*x])/(32*d) + (5*a^3*b^2*Sec[c + d*x]^7*Tan[c
+ d*x])/(4*d) - (3*a*b^4*Sec[c + d*x]^7*Tan[c + d*x])/(16*d) + (a*b^4*Sec[c + d*x]^7*Tan[c + d*x]^3)/(2*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 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

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^5 \sec ^7(c+d x)+5 a^4 b \sec ^7(c+d x) \tan (c+d x)+10 a^3 b^2 \sec ^7(c+d x) \tan ^2(c+d x)+10 a^2 b^3 \sec ^7(c+d x) \tan ^3(c+d x)+5 a b^4 \sec ^7(c+d x) \tan ^4(c+d x)+b^5 \sec ^7(c+d x) \tan ^5(c+d x)\right ) \, dx \\ & = a^5 \int \sec ^7(c+d x) \, dx+\left (5 a^4 b\right ) \int \sec ^7(c+d x) \tan (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \sec ^7(c+d x) \tan ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \sec ^7(c+d x) \tan ^4(c+d x) \, dx+b^5 \int \sec ^7(c+d x) \tan ^5(c+d x) \, dx \\ & = \frac {a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {5 a^3 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}+\frac {a b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{2 d}+\frac {1}{6} \left (5 a^5\right ) \int \sec ^5(c+d x) \, dx-\frac {1}{4} \left (5 a^3 b^2\right ) \int \sec ^7(c+d x) \, dx-\frac {1}{2} \left (3 a b^4\right ) \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx+\frac {\left (5 a^4 b\right ) \text {Subst}\left (\int x^6 \, dx,x,\sec (c+d x)\right )}{d}+\frac {\left (10 a^2 b^3\right ) \text {Subst}\left (\int x^6 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d}+\frac {b^5 \text {Subst}\left (\int x^6 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {5 a^4 b \sec ^7(c+d x)}{7 d}+\frac {5 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{24 d}+\frac {5 a^3 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 a b^4 \sec ^7(c+d x) \tan (c+d x)}{16 d}+\frac {a b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{2 d}+\frac {1}{8} \left (5 a^5\right ) \int \sec ^3(c+d x) \, dx-\frac {1}{24} \left (25 a^3 b^2\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{16} \left (3 a b^4\right ) \int \sec ^7(c+d x) \, dx+\frac {\left (10 a^2 b^3\right ) \text {Subst}\left (\int \left (-x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{d}+\frac {b^5 \text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {5 a^4 b \sec ^7(c+d x)}{7 d}-\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}+\frac {b^5 \sec ^7(c+d x)}{7 d}+\frac {10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac {2 b^5 \sec ^9(c+d x)}{9 d}+\frac {b^5 \sec ^{11}(c+d x)}{11 d}+\frac {5 a^5 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {5 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {25 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{96 d}+\frac {a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{24 d}+\frac {a b^4 \sec ^5(c+d x) \tan (c+d x)}{32 d}+\frac {5 a^3 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 a b^4 \sec ^7(c+d x) \tan (c+d x)}{16 d}+\frac {a b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{2 d}+\frac {1}{16} \left (5 a^5\right ) \int \sec (c+d x) \, dx-\frac {1}{32} \left (25 a^3 b^2\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{32} \left (5 a b^4\right ) \int \sec ^5(c+d x) \, dx \\ & = \frac {5 a^5 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {5 a^4 b \sec ^7(c+d x)}{7 d}-\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}+\frac {b^5 \sec ^7(c+d x)}{7 d}+\frac {10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac {2 b^5 \sec ^9(c+d x)}{9 d}+\frac {b^5 \sec ^{11}(c+d x)}{11 d}+\frac {5 a^5 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {25 a^3 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac {5 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {25 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{96 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac {a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{24 d}+\frac {a b^4 \sec ^5(c+d x) \tan (c+d x)}{32 d}+\frac {5 a^3 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 a b^4 \sec ^7(c+d x) \tan (c+d x)}{16 d}+\frac {a b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{2 d}-\frac {1}{64} \left (25 a^3 b^2\right ) \int \sec (c+d x) \, dx+\frac {1}{128} \left (15 a b^4\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {5 a^5 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {25 a^3 b^2 \text {arctanh}(\sin (c+d x))}{64 d}+\frac {5 a^4 b \sec ^7(c+d x)}{7 d}-\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}+\frac {b^5 \sec ^7(c+d x)}{7 d}+\frac {10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac {2 b^5 \sec ^9(c+d x)}{9 d}+\frac {b^5 \sec ^{11}(c+d x)}{11 d}+\frac {5 a^5 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {25 a^3 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac {15 a b^4 \sec (c+d x) \tan (c+d x)}{256 d}+\frac {5 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {25 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{96 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac {a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{24 d}+\frac {a b^4 \sec ^5(c+d x) \tan (c+d x)}{32 d}+\frac {5 a^3 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 a b^4 \sec ^7(c+d x) \tan (c+d x)}{16 d}+\frac {a b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{2 d}+\frac {1}{256} \left (15 a b^4\right ) \int \sec (c+d x) \, dx \\ & = \frac {5 a^5 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {25 a^3 b^2 \text {arctanh}(\sin (c+d x))}{64 d}+\frac {15 a b^4 \text {arctanh}(\sin (c+d x))}{256 d}+\frac {5 a^4 b \sec ^7(c+d x)}{7 d}-\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}+\frac {b^5 \sec ^7(c+d x)}{7 d}+\frac {10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac {2 b^5 \sec ^9(c+d x)}{9 d}+\frac {b^5 \sec ^{11}(c+d x)}{11 d}+\frac {5 a^5 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {25 a^3 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac {15 a b^4 \sec (c+d x) \tan (c+d x)}{256 d}+\frac {5 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {25 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{96 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac {a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{24 d}+\frac {a b^4 \sec ^5(c+d x) \tan (c+d x)}{32 d}+\frac {5 a^3 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 a b^4 \sec ^7(c+d x) \tan (c+d x)}{16 d}+\frac {a b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.52 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.79 \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {-1774080 a \left (16 a^4-20 a^2 b^2+3 b^4\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec ^{11}(c+d x) \left (24330240 a^4 b+1802240 a^2 b^3+3031040 b^5+3604480 \left (9 a^4 b-4 a^2 b^3-b^5\right ) \cos (2 (c+d x))+1622016 \left (5 a^4 b-10 a^2 b^3+b^5\right ) \cos (4 (c+d x))+6623232 a^5 \sin (4 (c+d x))+5913600 a^3 b^2 \sin (4 (c+d x))-6564096 a b^4 \sin (4 (c+d x))+2857008 a^5 \sin (6 (c+d x))-3571260 a^3 b^2 \sin (6 (c+d x))+535689 a b^4 \sin (6 (c+d x))+591360 a^5 \sin (8 (c+d x))-739200 a^3 b^2 \sin (8 (c+d x))+110880 a b^4 \sin (8 (c+d x))+55440 a^5 \sin (10 (c+d x))-69300 a^3 b^2 \sin (10 (c+d x))+10395 a b^4 \sin (10 (c+d x))\right )+13860 a \left (976 a^4+2876 a^2 b^2+1207 b^4\right ) \sec ^9(c+d x) \tan (c+d x)}{90832896 d} \]

[In]

Integrate[Sec[c + d*x]^12*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(-1774080*a*(16*a^4 - 20*a^2*b^2 + 3*b^4)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + S
in[(c + d*x)/2]]) + Sec[c + d*x]^11*(24330240*a^4*b + 1802240*a^2*b^3 + 3031040*b^5 + 3604480*(9*a^4*b - 4*a^2
*b^3 - b^5)*Cos[2*(c + d*x)] + 1622016*(5*a^4*b - 10*a^2*b^3 + b^5)*Cos[4*(c + d*x)] + 6623232*a^5*Sin[4*(c +
d*x)] + 5913600*a^3*b^2*Sin[4*(c + d*x)] - 6564096*a*b^4*Sin[4*(c + d*x)] + 2857008*a^5*Sin[6*(c + d*x)] - 357
1260*a^3*b^2*Sin[6*(c + d*x)] + 535689*a*b^4*Sin[6*(c + d*x)] + 591360*a^5*Sin[8*(c + d*x)] - 739200*a^3*b^2*S
in[8*(c + d*x)] + 110880*a*b^4*Sin[8*(c + d*x)] + 55440*a^5*Sin[10*(c + d*x)] - 69300*a^3*b^2*Sin[10*(c + d*x)
] + 10395*a*b^4*Sin[10*(c + d*x)]) + 13860*a*(976*a^4 + 2876*a^2*b^2 + 1207*b^4)*Sec[c + d*x]^9*Tan[c + d*x])/
(90832896*d)

Maple [A] (verified)

Time = 3.30 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.83

method	result	size
parts	\(\frac {a^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {b^{5} \left (\frac {\sec \left (d x +c \right )^{11}}{11}-\frac {2 \sec \left (d x +c \right )^{9}}{9}+\frac {\sec \left (d x +c \right )^{7}}{7}\right )}{d}+\frac {5 a^{4} b \sec \left (d x +c \right )^{7}}{7 d}+\frac {10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \sin \left (d x +c \right )^{3}}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \sin \left (d x +c \right )^{3}}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \sin \left (d x +c \right )^{3}}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}+\frac {5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{32 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{256 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{256}-\frac {3 \sin \left (d x +c \right )}{256}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{256}\right )}{d}+\frac {10 a^{2} b^{3} \left (\frac {\sec \left (d x +c \right )^{9}}{9}-\frac {\sec \left (d x +c \right )^{7}}{7}\right )}{d}\)	\(392\)
derivativedivides	\(\frac {a^{5} \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 {5 a^{4} b}{7 \cos \left (d x +c \right )^{7}}+10 a^{3} 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 )+10 a^{2} 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 )+5 a \,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 )+b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{11 \cos \left (d x +c \right )^{11}}+\frac {5 \sin \left (d x +c \right )^{6}}{99 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{6}}{231 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{6}}{231 \cos \left (d x +c \right )^{5}}-\frac {\sin \left (d x +c \right )^{6}}{693 \cos \left (d x +c \right )^{3}}+\frac {\sin \left (d x +c \right )^{6}}{231 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{231}\right )}{d}\)	\(572\)
default	\(\frac {a^{5} \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 {5 a^{4} b}{7 \cos \left (d x +c \right )^{7}}+10 a^{3} 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 )+10 a^{2} 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 )+5 a \,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 )+b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{11 \cos \left (d x +c \right )^{11}}+\frac {5 \sin \left (d x +c \right )^{6}}{99 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{6}}{231 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{6}}{231 \cos \left (d x +c \right )^{5}}-\frac {\sin \left (d x +c \right )^{6}}{693 \cos \left (d x +c \right )^{3}}+\frac {\sin \left (d x +c \right )^{6}}{231 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{231}\right )}{d}\)	\(572\)
parallelrisch	\(\frac {-9147600 \left (\frac {\cos \left (11 d x +11 c \right )}{165}+\frac {\cos \left (9 d x +9 c \right )}{15}+\frac {\cos \left (7 d x +7 c \right )}{3}+\cos \left (5 d x +5 c \right )+2 \cos \left (3 d x +3 c \right )+\frac {14 \cos \left (d x +c \right )}{5}\right ) a \left (a^{4}-\frac {5}{4} a^{2} b^{2}+\frac {3}{16} b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+9147600 \left (\frac {\cos \left (11 d x +11 c \right )}{165}+\frac {\cos \left (9 d x +9 c \right )}{15}+\frac {\cos \left (7 d x +7 c \right )}{3}+\cos \left (5 d x +5 c \right )+2 \cos \left (3 d x +3 c \right )+\frac {14 \cos \left (d x +c \right )}{5}\right ) a \left (a^{4}-\frac {5}{4} a^{2} b^{2}+\frac {3}{16} b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (126720 a^{4} b -56320 a^{2} b^{3}+2048 b^{5}\right ) \cos \left (11 d x +11 c \right )[phone] b \left (a^{4}-\frac {4}{9} a^{2} b^{2}+\frac {8}{495} b^{4}\right ) \cos \left (3 d x +3 c \right )+\left (20908800 a^{4} b -9292800 a^{2} b^{3}+337920 b^{5}\right ) \cos \left (5 d x +5 c \right )+\left (6969600 a^{4} b -3097600 a^{2} b^{3}+112640 b^{5}\right ) \cos \left (7 d x +7 c \right )+\left (1393920 a^{4} b -619520 a^{2} b^{3}+22528 b^{5}\right ) \cos \left (9 d x +9 c \right )+\left (110880 a^{5}-138600 a^{3} b^{2}+20790 a \,b^{4}\right ) \sin \left (10 d x +10 c \right )[phone] b \left (a^{4}-\frac {4}{9} a^{2} b^{2}-\frac {1}{9} b^{4}\right ) \cos \left (2 d x +2 c \right )[phone] \cos \left (4 d x +4 c \right ) b \left (a^{4}-2 a^{2} b^{2}+\frac {1}{5} b^{4}\right )+\left (13527360 a^{5}[phone] a^{3} b^{2}[phone] a \,b^{4}\right ) \sin \left (2 d x +2 c \right )+\left (13246464 a^{5}[phone] a^{3} b^{2}-13128192 a \,b^{4}\right ) \sin \left (4 d x +4 c \right )+5714016 a \left (a^{4}-\frac {5}{4} a^{2} b^{2}+\frac {3}{16} b^{4}\right ) \sin \left (6 d x +6 c \right )+\left (1182720 a^{5}-1478400 a^{3} b^{2}+221760 a \,b^{4}\right ) \sin \left (8 d x +8 c \right )[phone] b \left (\left (a^{4}-\frac {4}{9} a^{2} b^{2}+\frac {8}{495} b^{4}\right ) \cos \left (d x +c \right )+\frac {64 a^{4}}{77}+\frac {128 a^{2} b^{2}}{2079}+\frac {2368 b^{4}}{22869}\right )}{177408 d \left (\cos \left (11 d x +11 c \right )+11 \cos \left (9 d x +9 c \right )+55 \cos \left (7 d x +7 c \right )+165 \cos \left (5 d x +5 c \right )+330 \cos \left (3 d x +3 c \right )+462 \cos \left (d x +c \right )\right )}\)	\(661\)
risch	\(\text {Expression too large to display}\)	\(943\)

[In]

int(sec(d*x+c)^12*(cos(d*x+c)*a+b*sin(d*x+c))^5,x,method=_RETURNVERBOSE)

[Out]

a^5/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^5/d
*(1/11*sec(d*x+c)^11-2/9*sec(d*x+c)^9+1/7*sec(d*x+c)^7)+5/7*a^4*b*sec(d*x+c)^7/d+10*a^3*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
/128*sin(d*x+c)-5/128*ln(sec(d*x+c)+tan(d*x+c)))+5*a*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)))+10*a^2*b^3/d*(1/9*sec(d*x+c)^9-1/7*sec(d*x+
c)^7)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.61 \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {3465 \, {\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{11} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3465 \, {\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{11} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 32256 \, b^{5} + 50688 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 78848 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 462 \, {\left (15 \, {\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} + 10 \, {\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 384 \, a b^{4} \cos \left (d x + c\right ) + 8 \, {\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 48 \, {\left (20 \, a^{3} b^{2} - 11 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{354816 \, d \cos \left (d x + c\right )^{11}} \]

[In]

integrate(sec(d*x+c)^12*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="fricas")

[Out]

1/354816*(3465*(16*a^5 - 20*a^3*b^2 + 3*a*b^4)*cos(d*x + c)^11*log(sin(d*x + c) + 1) - 3465*(16*a^5 - 20*a^3*b
^2 + 3*a*b^4)*cos(d*x + c)^11*log(-sin(d*x + c) + 1) + 32256*b^5 + 50688*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x
+ c)^4 + 78848*(5*a^2*b^3 - b^5)*cos(d*x + c)^2 + 462*(15*(16*a^5 - 20*a^3*b^2 + 3*a*b^4)*cos(d*x + c)^9 + 10*
(16*a^5 - 20*a^3*b^2 + 3*a*b^4)*cos(d*x + c)^7 + 384*a*b^4*cos(d*x + c) + 8*(16*a^5 - 20*a^3*b^2 + 3*a*b^4)*co
s(d*x + c)^5 + 48*(20*a^3*b^2 - 11*a*b^4)*cos(d*x + c)^3)*sin(d*x + c))/(d*cos(d*x + c)^11)

Sympy [F(-1)]

Timed out. \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**12*(a*cos(d*x+c)+b*sin(d*x+c))**5,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 420, normalized size of antiderivative = 0.89 \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {693 \, a b^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{9} - 70 \, \sin \left (d x + c\right )^{7} + 128 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{10} - 5 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4620 \, a^{3} b^{2} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{7} - 55 \, \sin \left (d x + c\right )^{5} + 73 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 3696 \, a^{5} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {253440 \, a^{4} b}{\cos \left (d x + c\right )^{7}} + \frac {56320 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} a^{2} b^{3}}{\cos \left (d x + c\right )^{9}} - \frac {512 \, {\left (99 \, \cos \left (d x + c\right )^{4} - 154 \, \cos \left (d x + c\right )^{2} + 63\right )} b^{5}}{\cos \left (d x + c\right )^{11}}}{354816 \, d} \]

[In]

integrate(sec(d*x+c)^12*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/354816*(693*a*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*s
in(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)) - 4620*a^3*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)) + 3696*a^5*(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)) - 253440*a^4*b/cos(d*x + c)^7 + 56320*(9*cos(d*x + c)^2 - 7)*a^2*b^3/cos(d*x
+ c)^9 - 512*(99*cos(d*x + c)^4 - 154*cos(d*x + c)^2 + 63)*b^5/cos(d*x + c)^11)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1096 vs. \(2 (430) = 860\).

Time = 0.65 (sec) , antiderivative size = 1096, normalized size of antiderivative = 2.32 \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^12*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="giac")

[Out]

1/177408*(3465*(16*a^5 - 20*a^3*b^2 + 3*a*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3465*(16*a^5 - 20*a^3*b^2
+ 3*a*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(121968*a^5*tan(1/2*d*x + 1/2*c)^21 + 69300*a^3*b^2*tan(1/2*
d*x + 1/2*c)^21 - 10395*a*b^4*tan(1/2*d*x + 1/2*c)^21 - 887040*a^4*b*tan(1/2*d*x + 1/2*c)^20 - 591360*a^5*tan(
1/2*d*x + 1/2*c)^19 + 1626240*a^3*b^2*tan(1/2*d*x + 1/2*c)^19 + 110880*a*b^4*tan(1/2*d*x + 1/2*c)^19 + 3548160
*a^4*b*tan(1/2*d*x + 1/2*c)^18 - 3548160*a^2*b^3*tan(1/2*d*x + 1/2*c)^18 + 1459920*a^5*tan(1/2*d*x + 1/2*c)^17
- 1159620*a^3*b^2*tan(1/2*d*x + 1/2*c)^17 + 2302839*a*b^4*tan(1/2*d*x + 1/2*c)^17 - 9757440*a^4*b*tan(1/2*d*x
+ 1/2*c)^16 + 1182720*a^2*b^3*tan(1/2*d*x + 1/2*c)^16 - 946176*b^5*tan(1/2*d*x + 1/2*c)^16 - 2365440*a^5*tan(
1/2*d*x + 1/2*c)^15 + 1182720*a^3*b^2*tan(1/2*d*x + 1/2*c)^15 + 4790016*a*b^4*tan(1/2*d*x + 1/2*c)^15 + 212889
60*a^4*b*tan(1/2*d*x + 1/2*c)^14 - 9461760*a^2*b^3*tan(1/2*d*x + 1/2*c)^14 - 2365440*b^5*tan(1/2*d*x + 1/2*c)^
14 + 2106720*a^5*tan(1/2*d*x + 1/2*c)^13 - 5738040*a^3*b^2*tan(1/2*d*x + 1/2*c)^13 + 5828130*a*b^4*tan(1/2*d*x
+ 1/2*c)^13 - 30159360*a^4*b*tan(1/2*d*x + 1/2*c)^12 + 18923520*a^2*b^3*tan(1/2*d*x + 1/2*c)^12 - 5203968*b^5
*tan(1/2*d*x + 1/2*c)^12 + 28385280*a^4*b*tan(1/2*d*x + 1/2*c)^10 - 7096320*a^2*b^3*tan(1/2*d*x + 1/2*c)^10 -
4257792*b^5*tan(1/2*d*x + 1/2*c)^10 - 2106720*a^5*tan(1/2*d*x + 1/2*c)^9 + 5738040*a^3*b^2*tan(1/2*d*x + 1/2*c
)^9 - 5828130*a*b^4*tan(1/2*d*x + 1/2*c)^9 - 20528640*a^4*b*tan(1/2*d*x + 1/2*c)^8 + 9123840*a^2*b^3*tan(1/2*d
*x + 1/2*c)^8 - 3041280*b^5*tan(1/2*d*x + 1/2*c)^8 + 2365440*a^5*tan(1/2*d*x + 1/2*c)^7 - 1182720*a^3*b^2*tan(
1/2*d*x + 1/2*c)^7 - 4790016*a*b^4*tan(1/2*d*x + 1/2*c)^7 + 11151360*a^4*b*tan(1/2*d*x + 1/2*c)^6 - 8110080*a^
2*b^3*tan(1/2*d*x + 1/2*c)^6 - 608256*b^5*tan(1/2*d*x + 1/2*c)^6 - 1459920*a^5*tan(1/2*d*x + 1/2*c)^5 + 115962
0*a^3*b^2*tan(1/2*d*x + 1/2*c)^5 - 2302839*a*b^4*tan(1/2*d*x + 1/2*c)^5 - 3421440*a^4*b*tan(1/2*d*x + 1/2*c)^4
- 450560*a^2*b^3*tan(1/2*d*x + 1/2*c)^4 - 112640*b^5*tan(1/2*d*x + 1/2*c)^4 + 591360*a^5*tan(1/2*d*x + 1/2*c)
^3 - 1626240*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 110880*a*b^4*tan(1/2*d*x + 1/2*c)^3 + 506880*a^4*b*tan(1/2*d*x +
1/2*c)^2 - 619520*a^2*b^3*tan(1/2*d*x + 1/2*c)^2 + 22528*b^5*tan(1/2*d*x + 1/2*c)^2 - 121968*a^5*tan(1/2*d*x
+ 1/2*c) - 69300*a^3*b^2*tan(1/2*d*x + 1/2*c) + 10395*a*b^4*tan(1/2*d*x + 1/2*c) - 126720*a^4*b + 56320*a^2*b^
3 - 2048*b^5)/(tan(1/2*d*x + 1/2*c)^2 - 1)^11)/d

Mupad [B] (verification not implemented)

Time = 29.27 (sec) , antiderivative size = 831, normalized size of antiderivative = 1.76 \[ \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\text {Too large to display} \]

[In]

int((a*cos(c + d*x) + b*sin(c + d*x))^5/cos(c + d*x)^12,x)

[Out]

(5*a*atanh(tan(c/2 + (d*x)/2))*(16*a^4 + 3*b^4 - 20*a^2*b^2))/(128*d) - (tan(c/2 + (d*x)/2)*((11*a^5)/8 - (15*
a*b^4)/128 + (25*a^3*b^2)/32) - tan(c/2 + (d*x)/2)^18*(40*a^4*b - 40*a^2*b^3) + tan(c/2 + (d*x)/2)^3*((5*a*b^4
)/4 - (20*a^5)/3 + (55*a^3*b^2)/3) - tan(c/2 + (d*x)/2)^19*((5*a*b^4)/4 - (20*a^5)/3 + (55*a^3*b^2)/3) + tan(c
/2 + (d*x)/2)^7*(54*a*b^4 - (80*a^5)/3 + (40*a^3*b^2)/3) - tan(c/2 + (d*x)/2)^15*(54*a*b^4 - (80*a^5)/3 + (40*
a^3*b^2)/3) - tan(c/2 + (d*x)/2)^21*((11*a^5)/8 - (15*a*b^4)/128 + (25*a^3*b^2)/32) + tan(c/2 + (d*x)/2)^5*((3
323*a*b^4)/128 + (395*a^5)/24 - (1255*a^3*b^2)/96) - tan(c/2 + (d*x)/2)^17*((3323*a*b^4)/128 + (395*a^5)/24 -
(1255*a^3*b^2)/96) + tan(c/2 + (d*x)/2)^9*((4205*a*b^4)/64 + (95*a^5)/4 - (1035*a^3*b^2)/16) - tan(c/2 + (d*x)
/2)^13*((4205*a*b^4)/64 + (95*a^5)/4 - (1035*a^3*b^2)/16) + tan(c/2 + (d*x)/2)^16*(110*a^4*b + (32*b^5)/3 - (4
0*a^2*b^3)/3) + tan(c/2 + (d*x)/2)^10*(48*b^5 - 320*a^4*b + 80*a^2*b^3) - tan(c/2 + (d*x)/2)^2*((40*a^4*b)/7 +
(16*b^5)/63 - (440*a^2*b^3)/63) + tan(c/2 + (d*x)/2)^14*((80*b^5)/3 - 240*a^4*b + (320*a^2*b^3)/3) + tan(c/2
+ (d*x)/2)^4*((270*a^4*b)/7 + (80*b^5)/63 + (320*a^2*b^3)/63) + tan(c/2 + (d*x)/2)^12*(340*a^4*b + (176*b^5)/3
- (640*a^2*b^3)/3) + tan(c/2 + (d*x)/2)^6*((48*b^5)/7 - (880*a^4*b)/7 + (640*a^2*b^3)/7) + tan(c/2 + (d*x)/2)
^8*((1620*a^4*b)/7 + (240*b^5)/7 - (720*a^2*b^3)/7) + (10*a^4*b)/7 + (16*b^5)/693 - (40*a^2*b^3)/63 + 10*a^4*b
*tan(c/2 + (d*x)/2)^20)/(d*(11*tan(c/2 + (d*x)/2)^2 - 55*tan(c/2 + (d*x)/2)^4 + 165*tan(c/2 + (d*x)/2)^6 - 330
*tan(c/2 + (d*x)/2)^8 + 462*tan(c/2 + (d*x)/2)^10 - 462*tan(c/2 + (d*x)/2)^12 + 330*tan(c/2 + (d*x)/2)^14 - 16
5*tan(c/2 + (d*x)/2)^16 + 55*tan(c/2 + (d*x)/2)^18 - 11*tan(c/2 + (d*x)/2)^20 + tan(c/2 + (d*x)/2)^22 - 1))