\(\int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\) [107]

Optimal result

Integrand size = 28, antiderivative size = 391 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {3 a^5 \text {arctanh}(\sin (c+d x))}{8 d}-\frac {5 a^3 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {15 a b^4 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {a^4 b \sec ^5(c+d x)}{d}-\frac {2 a^2 b^3 \sec ^5(c+d x)}{d}+\frac {b^5 \sec ^5(c+d x)}{5 d}+\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac {2 b^5 \sec ^7(c+d x)}{7 d}+\frac {b^5 \sec ^9(c+d x)}{9 d}+\frac {3 a^5 \sec (c+d x) \tan (c+d x)}{8 d}-\frac {5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {15 a b^4 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {a^5 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{12 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{3 d}-\frac {5 a b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {5 a b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d} \]

[Out]

3/8*a^5*arctanh(sin(d*x+c))/d-5/8*a^3*b^2*arctanh(sin(d*x+c))/d+15/128*a*b^4*arctanh(sin(d*x+c))/d+a^4*b*sec(d
*x+c)^5/d-2*a^2*b^3*sec(d*x+c)^5/d+1/5*b^5*sec(d*x+c)^5/d+10/7*a^2*b^3*sec(d*x+c)^7/d-2/7*b^5*sec(d*x+c)^7/d+1
/9*b^5*sec(d*x+c)^9/d+3/8*a^5*sec(d*x+c)*tan(d*x+c)/d-5/8*a^3*b^2*sec(d*x+c)*tan(d*x+c)/d+15/128*a*b^4*sec(d*x
+c)*tan(d*x+c)/d+1/4*a^5*sec(d*x+c)^3*tan(d*x+c)/d-5/12*a^3*b^2*sec(d*x+c)^3*tan(d*x+c)/d+5/64*a*b^4*sec(d*x+c
)^3*tan(d*x+c)/d+5/3*a^3*b^2*sec(d*x+c)^5*tan(d*x+c)/d-5/16*a*b^4*sec(d*x+c)^5*tan(d*x+c)/d+5/8*a*b^4*sec(d*x+
c)^5*tan(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 22, 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} \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {3 a^5 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^5 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 a^5 \tan (c+d x) \sec (c+d x)}{8 d}+\frac {a^4 b \sec ^5(c+d x)}{d}-\frac {5 a^3 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 a^3 b^2 \tan (c+d x) \sec ^5(c+d x)}{3 d}-\frac {5 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{12 d}-\frac {5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac {2 a^2 b^3 \sec ^5(c+d x)}{d}+\frac {15 a b^4 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {5 a b^4 \tan ^3(c+d x) \sec ^5(c+d x)}{8 d}-\frac {5 a b^4 \tan (c+d x) \sec ^5(c+d x)}{16 d}+\frac {5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac {15 a b^4 \tan (c+d x) \sec (c+d x)}{128 d}+\frac {b^5 \sec ^9(c+d x)}{9 d}-\frac {2 b^5 \sec ^7(c+d x)}{7 d}+\frac {b^5 \sec ^5(c+d x)}{5 d} \]

[In]

Int[Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(3*a^5*ArcTanh[Sin[c + d*x]])/(8*d) - (5*a^3*b^2*ArcTanh[Sin[c + d*x]])/(8*d) + (15*a*b^4*ArcTanh[Sin[c + d*x]
])/(128*d) + (a^4*b*Sec[c + d*x]^5)/d - (2*a^2*b^3*Sec[c + d*x]^5)/d + (b^5*Sec[c + d*x]^5)/(5*d) + (10*a^2*b^
3*Sec[c + d*x]^7)/(7*d) - (2*b^5*Sec[c + d*x]^7)/(7*d) + (b^5*Sec[c + d*x]^9)/(9*d) + (3*a^5*Sec[c + d*x]*Tan[
c + d*x])/(8*d) - (5*a^3*b^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (15*a*b^4*Sec[c + d*x]*Tan[c + d*x])/(128*d) +
(a^5*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) - (5*a^3*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(12*d) + (5*a*b^4*Sec[c + d
*x]^3*Tan[c + d*x])/(64*d) + (5*a^3*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(3*d) - (5*a*b^4*Sec[c + d*x]^5*Tan[c + d
*x])/(16*d) + (5*a*b^4*Sec[c + d*x]^5*Tan[c + d*x]^3)/(8*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 ^5(c+d x)+5 a^4 b \sec ^5(c+d x) \tan (c+d x)+10 a^3 b^2 \sec ^5(c+d x) \tan ^2(c+d x)+10 a^2 b^3 \sec ^5(c+d x) \tan ^3(c+d x)+5 a b^4 \sec ^5(c+d x) \tan ^4(c+d x)+b^5 \sec ^5(c+d x) \tan ^5(c+d x)\right ) \, dx \\ & = a^5 \int \sec ^5(c+d x) \, dx+\left (5 a^4 b\right ) \int \sec ^5(c+d x) \tan (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \sec ^5(c+d x) \tan ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \sec ^5(c+d x) \tan ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \sec ^5(c+d x) \tan ^4(c+d x) \, dx+b^5 \int \sec ^5(c+d x) \tan ^5(c+d x) \, dx \\ & = \frac {a^5 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{3 d}+\frac {5 a b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d}+\frac {1}{4} \left (3 a^5\right ) \int \sec ^3(c+d x) \, dx-\frac {1}{3} \left (5 a^3 b^2\right ) \int \sec ^5(c+d x) \, dx-\frac {1}{8} \left (15 a b^4\right ) \int \sec ^5(c+d x) \tan ^2(c+d x) \, dx+\frac {\left (5 a^4 b\right ) \text {Subst}\left (\int x^4 \, dx,x,\sec (c+d x)\right )}{d}+\frac {\left (10 a^2 b^3\right ) \text {Subst}\left (\int x^4 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d}+\frac {b^5 \text {Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {a^4 b \sec ^5(c+d x)}{d}+\frac {3 a^5 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^5 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{12 d}+\frac {5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{3 d}-\frac {5 a b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {5 a b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d}+\frac {1}{8} \left (3 a^5\right ) \int \sec (c+d x) \, dx-\frac {1}{4} \left (5 a^3 b^2\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{16} \left (5 a b^4\right ) \int \sec ^5(c+d x) \, dx+\frac {\left (10 a^2 b^3\right ) \text {Subst}\left (\int \left (-x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{d}+\frac {b^5 \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {3 a^5 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^4 b \sec ^5(c+d x)}{d}-\frac {2 a^2 b^3 \sec ^5(c+d x)}{d}+\frac {b^5 \sec ^5(c+d x)}{5 d}+\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac {2 b^5 \sec ^7(c+d x)}{7 d}+\frac {b^5 \sec ^9(c+d x)}{9 d}+\frac {3 a^5 \sec (c+d x) \tan (c+d x)}{8 d}-\frac {5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^5 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{12 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{3 d}-\frac {5 a b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {5 a b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d}-\frac {1}{8} \left (5 a^3 b^2\right ) \int \sec (c+d x) \, dx+\frac {1}{64} \left (15 a b^4\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {3 a^5 \text {arctanh}(\sin (c+d x))}{8 d}-\frac {5 a^3 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^4 b \sec ^5(c+d x)}{d}-\frac {2 a^2 b^3 \sec ^5(c+d x)}{d}+\frac {b^5 \sec ^5(c+d x)}{5 d}+\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac {2 b^5 \sec ^7(c+d x)}{7 d}+\frac {b^5 \sec ^9(c+d x)}{9 d}+\frac {3 a^5 \sec (c+d x) \tan (c+d x)}{8 d}-\frac {5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {15 a b^4 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {a^5 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{12 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{3 d}-\frac {5 a b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {5 a b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d}+\frac {1}{128} \left (15 a b^4\right ) \int \sec (c+d x) \, dx \\ & = \frac {3 a^5 \text {arctanh}(\sin (c+d x))}{8 d}-\frac {5 a^3 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {15 a b^4 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {a^4 b \sec ^5(c+d x)}{d}-\frac {2 a^2 b^3 \sec ^5(c+d x)}{d}+\frac {b^5 \sec ^5(c+d x)}{5 d}+\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac {2 b^5 \sec ^7(c+d x)}{7 d}+\frac {b^5 \sec ^9(c+d x)}{9 d}+\frac {3 a^5 \sec (c+d x) \tan (c+d x)}{8 d}-\frac {5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {15 a b^4 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {a^5 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{12 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{3 d}-\frac {5 a b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {5 a b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.74 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.85 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {-40320 a \left (48 a^4-80 a^2 b^2+15 b^4\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec ^9(c+d x) \left (1935360 a^4 b-184320 a^2 b^3+223232 b^5+73728 \left (35 a^4 b-20 a^2 b^3-3 b^5\right ) \cos (2 (c+d x))+129024 \left (5 a^4 b-10 a^2 b^3+b^5\right ) \cos (4 (c+d x))+372960 a^5 \sin (4 (c+d x))+453600 a^3 b^2 \sin (4 (c+d x))-488250 a b^4 \sin (4 (c+d x))+131040 a^5 \sin (6 (c+d x))-218400 a^3 b^2 \sin (6 (c+d x))+40950 a b^4 \sin (6 (c+d x))+15120 a^5 \sin (8 (c+d x))-25200 a^3 b^2 \sin (8 (c+d x))+4725 a b^4 \sin (8 (c+d x))\right )+1260 a \left (656 a^4+2320 a^2 b^2+845 b^4\right ) \sec ^7(c+d x) \tan (c+d x)}{5160960 d} \]

[In]

Integrate[Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(-40320*a*(48*a^4 - 80*a^2*b^2 + 15*b^4)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Si
n[(c + d*x)/2]]) + Sec[c + d*x]^9*(1935360*a^4*b - 184320*a^2*b^3 + 223232*b^5 + 73728*(35*a^4*b - 20*a^2*b^3
- 3*b^5)*Cos[2*(c + d*x)] + 129024*(5*a^4*b - 10*a^2*b^3 + b^5)*Cos[4*(c + d*x)] + 372960*a^5*Sin[4*(c + d*x)]
+ 453600*a^3*b^2*Sin[4*(c + d*x)] - 488250*a*b^4*Sin[4*(c + d*x)] + 131040*a^5*Sin[6*(c + d*x)] - 218400*a^3*
b^2*Sin[6*(c + d*x)] + 40950*a*b^4*Sin[6*(c + d*x)] + 15120*a^5*Sin[8*(c + d*x)] - 25200*a^3*b^2*Sin[8*(c + d*
x)] + 4725*a*b^4*Sin[8*(c + d*x)]) + 1260*a*(656*a^4 + 2320*a^2*b^2 + 845*b^4)*Sec[c + d*x]^7*Tan[c + d*x])/(5
160960*d)

Maple [A] (verified)

Time = 2.93 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.88

method	result	size
parts	\(\frac {a^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {b^{5} \left (\frac {\sec \left (d x +c \right )^{9}}{9}-\frac {2 \sec \left (d x +c \right )^{7}}{7}+\frac {\sec \left (d x +c \right )^{5}}{5}\right )}{d}+\frac {10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}+\frac {10 a^{2} b^{3} \left (\frac {\sec \left (d x +c \right )^{7}}{7}-\frac {\sec \left (d x +c \right )^{5}}{5}\right )}{d}+\frac {a^{4} b \sec \left (d x +c \right )^{5}}{d}\)	\(345\)
derivativedivides	\(\frac {a^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{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 )+\frac {a^{4} b}{\cos \left (d x +c \right )^{5}}+10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+10 a^{2} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{35}\right )+5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{9 \cos \left (d x +c \right )^{9}}+\frac {\sin \left (d x +c \right )^{6}}{21 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{6}}{105 \cos \left (d x +c \right )^{5}}-\frac {\sin \left (d x +c \right )^{6}}{315 \cos \left (d x +c \right )^{3}}+\frac {\sin \left (d x +c \right )^{6}}{105 \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 )}{105}\right )}{d}\)	\(489\)
default	\(\frac {a^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{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 )+\frac {a^{4} b}{\cos \left (d x +c \right )^{5}}+10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+10 a^{2} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{35}\right )+5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{9 \cos \left (d x +c \right )^{9}}+\frac {\sin \left (d x +c \right )^{6}}{21 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{6}}{105 \cos \left (d x +c \right )^{5}}-\frac {\sin \left (d x +c \right )^{6}}{315 \cos \left (d x +c \right )^{3}}+\frac {\sin \left (d x +c \right )^{6}}{105 \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 )}{105}\right )}{d}\)	\(489\)
parallelrisch	\(\frac {-544320 \left (a^{4}-\frac {5}{3} a^{2} b^{2}+\frac {5}{16} b^{4}\right ) a \left (\frac {\cos \left (9 d x +9 c \right )}{36}+\frac {\cos \left (7 d x +7 c \right )}{4}+\cos \left (5 d x +5 c \right )+\frac {7 \cos \left (3 d x +3 c \right )}{3}+\frac {7 \cos \left (d x +c \right )}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+544320 \left (a^{4}-\frac {5}{3} a^{2} b^{2}+\frac {5}{16} b^{4}\right ) a \left (\frac {\cos \left (9 d x +9 c \right )}{36}+\frac {\cos \left (7 d x +7 c \right )}{4}+\cos \left (5 d x +5 c \right )+\frac {7 \cos \left (3 d x +3 c \right )}{3}+\frac {7 \cos \left (d x +c \right )}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (3386880 a^{4} b -1935360 a^{2} b^{3}+86016 b^{5}\right ) \cos \left (3 d x +3 c \right )+\left (1451520 a^{4} b -829440 a^{2} b^{3}+36864 b^{5}\right ) \cos \left (5 d x +5 c \right )+\left (362880 a^{4} b -207360 a^{2} b^{3}+9216 b^{5}\right ) \cos \left (7 d x +7 c \right )+\left (40320 a^{4} b -23040 a^{2} b^{3}+1024 b^{5}\right ) \cos \left (9 d x +9 c \right )+\left (5160960 a^{4} b -2949120 a^{2} b^{3}-442368 b^{5}\right ) \cos \left (2 d x +2 c \right )+1290240 \cos \left (4 d x +4 c \right ) b \left (a^{4}-2 a^{2} b^{2}+\frac {1}{5} b^{4}\right )+\left (826560 a^{5}+2923200 a^{3} b^{2}+1064700 a \,b^{4}\right ) \sin \left (2 d x +2 c \right )+\left (745920 a^{5}+907200 a^{3} b^{2}-976500 a \,b^{4}\right ) \sin \left (4 d x +4 c \right )+\left (262080 a^{5}-436800 a^{3} b^{2}+81900 a \,b^{4}\right ) \sin \left (6 d x +6 c \right )+\left (30240 a^{5}-50400 a^{3} b^{2}+9450 a \,b^{4}\right ) \sin \left (8 d x +8 c \right )+5080320 b \left (\left (a^{4}-\frac {4}{7} a^{2} b^{2}+\frac {8}{315} b^{4}\right ) \cos \left (d x +c \right )+\frac {16 a^{4}}{21}-\frac {32 a^{2} b^{2}}{441}+\frac {1744 b^{4}}{19845}\right )}{40320 d \left (\cos \left (9 d x +9 c \right )+9 \cos \left (7 d x +7 c \right )+36 \cos \left (5 d x +5 c \right )+84 \cos \left (3 d x +3 c \right )+126 \cos \left (d x +c \right )\right )}\)	\(571\)
risch	\(\frac {{\mathrm e}^{i \left (d x +c \right )} \left (-1290240 a^{2} b^{3} {\mathrm e}^{12 i \left (d x +c \right )}+645120 a^{4} b \,{\mathrm e}^{12 i \left (d x +c \right )}-15120 i a^{5} {\mathrm e}^{16 i \left (d x +c \right )}-131040 i a^{5} {\mathrm e}^{14 i \left (d x +c \right )}+1461600 i a^{3} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+40950 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-1461600 i a^{3} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+129024 b^{5} {\mathrm e}^{12 i \left (d x +c \right )}-532350 i a \,b^{4} {\mathrm e}^{10 i \left (d x +c \right )}+453600 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-488250 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-218400 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+532350 i a \,b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+25200 i a^{3} b^{2} {\mathrm e}^{16 i \left (d x +c \right )}-4725 i a \,b^{4} {\mathrm e}^{16 i \left (d x +c \right )}-221184 b^{5} {\mathrm e}^{10 i \left (d x +c \right )}-221184 b^{5} {\mathrm e}^{6 i \left (d x +c \right )}+15120 i a^{5}+129024 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+446464 b^{5} {\mathrm e}^{8 i \left (d x +c \right )}-372960 i a^{5} {\mathrm e}^{12 i \left (d x +c \right )}+218400 i a^{3} b^{2} {\mathrm e}^{14 i \left (d x +c \right )}-40950 i a \,b^{4} {\mathrm e}^{14 i \left (d x +c \right )}-413280 i a^{5} {\mathrm e}^{10 i \left (d x +c \right )}+413280 i a^{5} {\mathrm e}^{6 i \left (d x +c \right )}+372960 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}+131040 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}-25200 i a^{3} b^{2}+4725 i a \,b^{4}-1290240 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-1474560 a^{2} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+645120 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-368640 a^{2} b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+2580480 a^{4} b \,{\mathrm e}^{6 i \left (d x +c \right )}-1474560 a^{2} b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+3870720 a^{4} b \,{\mathrm e}^{8 i \left (d x +c \right )}+2580480 a^{4} b \,{\mathrm e}^{10 i \left (d x +c \right )}-453600 i a^{3} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}+488250 i a \,b^{4} {\mathrm e}^{12 i \left (d x +c \right )}\right )}{20160 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{9}}-\frac {3 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}+\frac {5 a^{3} b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {15 a \,b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d}+\frac {3 a^{5} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{8 d}-\frac {5 a^{3} b^{2} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{8 d}+\frac {15 a \,b^{4} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{128 d}\)	\(833\)

[In]

int(sec(d*x+c)^10*(cos(d*x+c)*a+b*sin(d*x+c))^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.66 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {315 \, {\left (48 \, a^{5} - 80 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \, {\left (48 \, a^{5} - 80 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 8960 \, b^{5} + 16128 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 23040 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 210 \, {\left (3 \, {\left (48 \, a^{5} - 80 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 240 \, a b^{4} \cos \left (d x + c\right ) + 2 \, {\left (48 \, a^{5} - 80 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 40 \, {\left (16 \, a^{3} b^{2} - 9 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{80640 \, d \cos \left (d x + c\right )^{9}} \]

[In]

integrate(sec(d*x+c)^10*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="fricas")

[Out]

1/80640*(315*(48*a^5 - 80*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^9*log(sin(d*x + c) + 1) - 315*(48*a^5 - 80*a^3*b^2
+ 15*a*b^4)*cos(d*x + c)^9*log(-sin(d*x + c) + 1) + 8960*b^5 + 16128*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x + c)
^4 + 23040*(5*a^2*b^3 - b^5)*cos(d*x + c)^2 + 210*(3*(48*a^5 - 80*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^7 + 240*a*b
^4*cos(d*x + c) + 2*(48*a^5 - 80*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5 + 40*(16*a^3*b^2 - 9*a*b^4)*cos(d*x + c)^3
)*sin(d*x + c))/(d*cos(d*x + c)^9)

Sympy [F(-1)]

Timed out. \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**10*(a*cos(d*x+c)+b*sin(d*x+c))**5,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.92 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {1575 \, a b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{7} - 11 \, \sin \left (d x + c\right )^{5} - 11 \, \sin \left (d x + c\right )^{3} + 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 8400 \, a^{3} b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 5040 \, a^{5} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {80640 \, a^{4} b}{\cos \left (d x + c\right )^{5}} + \frac {23040 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} a^{2} b^{3}}{\cos \left (d x + c\right )^{7}} - \frac {256 \, {\left (63 \, \cos \left (d x + c\right )^{4} - 90 \, \cos \left (d x + c\right )^{2} + 35\right )} b^{5}}{\cos \left (d x + c\right )^{9}}}{80640 \, d} \]

[In]

integrate(sec(d*x+c)^10*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/80640*(1575*a*b^4*(2*(3*sin(d*x + c)^7 - 11*sin(d*x + c)^5 - 11*sin(d*x + c)^3 + 3*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) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x
+ c) - 1)) - 8400*a^3*b^2*(2*(3*sin(d*x + c)^5 - 8*sin(d*x + c)^3 - 3*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*
x + c)^4 + 3*sin(d*x + c)^2 - 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) + 5040*a^5*(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)) - 80640*a^4*b/cos(d*x + c)^5 + 23040*(7*cos(d*x + c)^2 - 5)*a^2*b^3/cos(d*x + c)^7 - 256*(63*cos(d*x
+ c)^4 - 90*cos(d*x + c)^2 + 35)*b^5/cos(d*x + c)^9)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 888 vs. \(2 (359) = 718\).

Time = 0.62 (sec) , antiderivative size = 888, normalized size of antiderivative = 2.27 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^10*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="giac")

[Out]

1/40320*(315*(48*a^5 - 80*a^3*b^2 + 15*a*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 315*(48*a^5 - 80*a^3*b^2 +
15*a*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(25200*a^5*tan(1/2*d*x + 1/2*c)^17 + 25200*a^3*b^2*tan(1/2*d*
x + 1/2*c)^17 - 4725*a*b^4*tan(1/2*d*x + 1/2*c)^17 - 201600*a^4*b*tan(1/2*d*x + 1/2*c)^16 - 110880*a^5*tan(1/2
*d*x + 1/2*c)^15 + 319200*a^3*b^2*tan(1/2*d*x + 1/2*c)^15 + 40950*a*b^4*tan(1/2*d*x + 1/2*c)^15 + 806400*a^4*b
*tan(1/2*d*x + 1/2*c)^14 - 806400*a^2*b^3*tan(1/2*d*x + 1/2*c)^14 + 191520*a^5*tan(1/2*d*x + 1/2*c)^13 - 45360
0*a^3*b^2*tan(1/2*d*x + 1/2*c)^13 + 488250*a*b^4*tan(1/2*d*x + 1/2*c)^13 - 1612800*a^4*b*tan(1/2*d*x + 1/2*c)^
12 + 806400*a^2*b^3*tan(1/2*d*x + 1/2*c)^12 - 215040*b^5*tan(1/2*d*x + 1/2*c)^12 - 151200*a^5*tan(1/2*d*x + 1/
2*c)^11 - 151200*a^3*b^2*tan(1/2*d*x + 1/2*c)^11 + 532350*a*b^4*tan(1/2*d*x + 1/2*c)^11 + 2419200*a^4*b*tan(1/
2*d*x + 1/2*c)^10 - 806400*a^2*b^3*tan(1/2*d*x + 1/2*c)^10 - 322560*b^5*tan(1/2*d*x + 1/2*c)^10 - 2661120*a^4*
b*tan(1/2*d*x + 1/2*c)^8 + 2096640*a^2*b^3*tan(1/2*d*x + 1/2*c)^8 - 451584*b^5*tan(1/2*d*x + 1/2*c)^8 + 151200
*a^5*tan(1/2*d*x + 1/2*c)^7 + 151200*a^3*b^2*tan(1/2*d*x + 1/2*c)^7 - 532350*a*b^4*tan(1/2*d*x + 1/2*c)^7 + 17
74080*a^4*b*tan(1/2*d*x + 1/2*c)^6 - 1128960*a^2*b^3*tan(1/2*d*x + 1/2*c)^6 - 129024*b^5*tan(1/2*d*x + 1/2*c)^
6 - 191520*a^5*tan(1/2*d*x + 1/2*c)^5 + 453600*a^3*b^2*tan(1/2*d*x + 1/2*c)^5 - 488250*a*b^4*tan(1/2*d*x + 1/2
*c)^5 - 645120*a^4*b*tan(1/2*d*x + 1/2*c)^4 + 23040*a^2*b^3*tan(1/2*d*x + 1/2*c)^4 - 36864*b^5*tan(1/2*d*x + 1
/2*c)^4 + 110880*a^5*tan(1/2*d*x + 1/2*c)^3 - 319200*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 40950*a*b^4*tan(1/2*d*x
+ 1/2*c)^3 + 161280*a^4*b*tan(1/2*d*x + 1/2*c)^2 - 207360*a^2*b^3*tan(1/2*d*x + 1/2*c)^2 + 9216*b^5*tan(1/2*d*
x + 1/2*c)^2 - 25200*a^5*tan(1/2*d*x + 1/2*c) - 25200*a^3*b^2*tan(1/2*d*x + 1/2*c) + 4725*a*b^4*tan(1/2*d*x +
1/2*c) - 40320*a^4*b + 23040*a^2*b^3 - 1024*b^5)/(tan(1/2*d*x + 1/2*c)^2 - 1)^9)/d

Mupad [B] (verification not implemented)

Time = 27.63 (sec) , antiderivative size = 675, normalized size of antiderivative = 1.73 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^5}{4}-\frac {5\,a^3\,b^2}{4}+\frac {15\,a\,b^4}{64}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,a^5}{4}+\frac {5\,a^3\,b^2}{4}-\frac {15\,a\,b^4}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (40\,a^4\,b-40\,a^2\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\left (\frac {5\,a^5}{4}+\frac {5\,a^3\,b^2}{4}-\frac {15\,a\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (-\frac {11\,a^5}{2}+\frac {95\,a^3\,b^2}{6}+\frac {65\,a\,b^4}{32}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (-\frac {11\,a^5}{2}+\frac {95\,a^3\,b^2}{6}+\frac {65\,a\,b^4}{32}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {19\,a^5}{2}-\frac {45\,a^3\,b^2}{2}+\frac {775\,a\,b^4}{32}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {19\,a^5}{2}-\frac {45\,a^3\,b^2}{2}+\frac {775\,a\,b^4}{32}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {15\,a^5}{2}+\frac {15\,a^3\,b^2}{2}-\frac {845\,a\,b^4}{32}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {15\,a^5}{2}+\frac {15\,a^3\,b^2}{2}-\frac {845\,a\,b^4}{32}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (8\,a^4\,b-\frac {72\,a^2\,b^3}{7}+\frac {16\,b^5}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (32\,a^4\,b-\frac {8\,a^2\,b^3}{7}+\frac {64\,b^5}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (80\,a^4\,b-40\,a^2\,b^3+\frac {32\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (-120\,a^4\,b+40\,a^2\,b^3+16\,b^5\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (-88\,a^4\,b+56\,a^2\,b^3+\frac {32\,b^5}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (132\,a^4\,b-104\,a^2\,b^3+\frac {112\,b^5}{5}\right )+2\,a^4\,b+\frac {16\,b^5}{315}-\frac {8\,a^2\,b^3}{7}+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

[In]

int((a*cos(c + d*x) + b*sin(c + d*x))^5/cos(c + d*x)^10,x)

[Out]

(atanh(tan(c/2 + (d*x)/2))*((15*a*b^4)/64 + (3*a^5)/4 - (5*a^3*b^2)/4))/d - (tan(c/2 + (d*x)/2)*((5*a^5)/4 - (
15*a*b^4)/64 + (5*a^3*b^2)/4) - tan(c/2 + (d*x)/2)^14*(40*a^4*b - 40*a^2*b^3) - tan(c/2 + (d*x)/2)^17*((5*a^5)
/4 - (15*a*b^4)/64 + (5*a^3*b^2)/4) + tan(c/2 + (d*x)/2)^3*((65*a*b^4)/32 - (11*a^5)/2 + (95*a^3*b^2)/6) - tan
(c/2 + (d*x)/2)^15*((65*a*b^4)/32 - (11*a^5)/2 + (95*a^3*b^2)/6) + tan(c/2 + (d*x)/2)^5*((775*a*b^4)/32 + (19*
a^5)/2 - (45*a^3*b^2)/2) - tan(c/2 + (d*x)/2)^13*((775*a*b^4)/32 + (19*a^5)/2 - (45*a^3*b^2)/2) - tan(c/2 + (d
*x)/2)^7*((15*a^5)/2 - (845*a*b^4)/32 + (15*a^3*b^2)/2) + tan(c/2 + (d*x)/2)^11*((15*a^5)/2 - (845*a*b^4)/32 +
(15*a^3*b^2)/2) - tan(c/2 + (d*x)/2)^2*(8*a^4*b + (16*b^5)/35 - (72*a^2*b^3)/7) + tan(c/2 + (d*x)/2)^4*(32*a^
4*b + (64*b^5)/35 - (8*a^2*b^3)/7) + tan(c/2 + (d*x)/2)^12*(80*a^4*b + (32*b^5)/3 - 40*a^2*b^3) + tan(c/2 + (d
*x)/2)^10*(16*b^5 - 120*a^4*b + 40*a^2*b^3) + tan(c/2 + (d*x)/2)^6*((32*b^5)/5 - 88*a^4*b + 56*a^2*b^3) + tan(
c/2 + (d*x)/2)^8*(132*a^4*b + (112*b^5)/5 - 104*a^2*b^3) + 2*a^4*b + (16*b^5)/315 - (8*a^2*b^3)/7 + 10*a^4*b*t
an(c/2 + (d*x)/2)^16)/(d*(9*tan(c/2 + (d*x)/2)^2 - 36*tan(c/2 + (d*x)/2)^4 + 84*tan(c/2 + (d*x)/2)^6 - 126*tan
(c/2 + (d*x)/2)^8 + 126*tan(c/2 + (d*x)/2)^10 - 84*tan(c/2 + (d*x)/2)^12 + 36*tan(c/2 + (d*x)/2)^14 - 9*tan(c/
2 + (d*x)/2)^16 + tan(c/2 + (d*x)/2)^18 - 1))