\(\int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\) [105]

Optimal result

Integrand size = 28, antiderivative size = 318 \[ \int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {a^5 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {5 a^3 b^2 \text {arctanh}(\sin (c+d x))}{4 d}+\frac {5 a b^4 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {5 a^4 b \sec ^3(c+d x)}{3 d}-\frac {10 a^2 b^3 \sec ^3(c+d x)}{3 d}+\frac {b^5 \sec ^3(c+d x)}{3 d}+\frac {2 a^2 b^3 \sec ^5(c+d x)}{d}-\frac {2 b^5 \sec ^5(c+d x)}{5 d}+\frac {b^5 \sec ^7(c+d x)}{7 d}+\frac {a^5 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{4 d}+\frac {5 a b^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{2 d}-\frac {5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan ^3(c+d x)}{6 d} \]

[Out]

1/2*a^5*arctanh(sin(d*x+c))/d-5/4*a^3*b^2*arctanh(sin(d*x+c))/d+5/16*a*b^4*arctanh(sin(d*x+c))/d+5/3*a^4*b*sec
(d*x+c)^3/d-10/3*a^2*b^3*sec(d*x+c)^3/d+1/3*b^5*sec(d*x+c)^3/d+2*a^2*b^3*sec(d*x+c)^5/d-2/5*b^5*sec(d*x+c)^5/d
+1/7*b^5*sec(d*x+c)^7/d+1/2*a^5*sec(d*x+c)*tan(d*x+c)/d-5/4*a^3*b^2*sec(d*x+c)*tan(d*x+c)/d+5/16*a*b^4*sec(d*x
+c)*tan(d*x+c)/d+5/2*a^3*b^2*sec(d*x+c)^3*tan(d*x+c)/d-5/8*a*b^4*sec(d*x+c)^3*tan(d*x+c)/d+5/6*a*b^4*sec(d*x+c
)^3*tan(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 19, 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} \[ \int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {a^5 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a^5 \tan (c+d x) \sec (c+d x)}{2 d}+\frac {5 a^4 b \sec ^3(c+d x)}{3 d}-\frac {5 a^3 b^2 \text {arctanh}(\sin (c+d x))}{4 d}+\frac {5 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{2 d}-\frac {5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{4 d}+\frac {2 a^2 b^3 \sec ^5(c+d x)}{d}-\frac {10 a^2 b^3 \sec ^3(c+d x)}{3 d}+\frac {5 a b^4 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {5 a b^4 \tan ^3(c+d x) \sec ^3(c+d x)}{6 d}-\frac {5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac {5 a b^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac {b^5 \sec ^7(c+d x)}{7 d}-\frac {2 b^5 \sec ^5(c+d x)}{5 d}+\frac {b^5 \sec ^3(c+d x)}{3 d} \]

[In]

Int[Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1677\) vs. \(2(318)=636\).

Time = 8.72 (sec) , antiderivative size = 1677, normalized size of antiderivative = 5.27 \[ \int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {b \left (1400 a^4-1540 a^2 b^2+103 b^4\right ) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{1680 d (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\left (-8 a^5+20 a^3 b^2-5 a b^4\right ) \cos ^5(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^5}{16 d (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\left (8 a^5-20 a^3 b^2+5 a b^4\right ) \cos ^5(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^5}{16 d (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\left (35 a b^4+3 b^5\right ) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{336 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\left (350 a^3 b^2+140 a^2 b^3-175 a b^4-18 b^5\right ) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{560 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\left (840 a^5+1400 a^4 b-2100 a^3 b^2-1540 a^2 b^3+525 a b^4+103 b^5\right ) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{3360 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {b^5 \cos ^5(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^5}{56 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^7 (a \cos (c+d x)+b \sin (c+d x))^5}-\frac {b^5 \cos ^5(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^5}{56 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^7 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\left (-35 a b^4+3 b^5\right ) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{336 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\left (-350 a^3 b^2+140 a^2 b^3+175 a b^4-18 b^5\right ) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{560 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\left (-840 a^5+1400 a^4 b+2100 a^3 b^2-1540 a^2 b^3-525 a b^4+103 b^5\right ) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{3360 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\cos ^5(c+d x) \left (-1400 a^4 b \sin \left (\frac {1}{2} (c+d x)\right )+1540 a^2 b^3 \sin \left (\frac {1}{2} (c+d x)\right )-103 b^5 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^5}{1680 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\cos ^5(c+d x) \left (-1400 a^4 b \sin \left (\frac {1}{2} (c+d x)\right )+1540 a^2 b^3 \sin \left (\frac {1}{2} (c+d x)\right )-103 b^5 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^5}{1680 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\cos ^5(c+d x) \left (70 a^2 b^3 \sin \left (\frac {1}{2} (c+d x)\right )-9 b^5 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^5}{140 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\cos ^5(c+d x) \left (-70 a^2 b^3 \sin \left (\frac {1}{2} (c+d x)\right )+9 b^5 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^5}{140 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\cos ^5(c+d x) \left (1400 a^4 b \sin \left (\frac {1}{2} (c+d x)\right )-1540 a^2 b^3 \sin \left (\frac {1}{2} (c+d x)\right )+103 b^5 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^5}{1680 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac {\cos ^5(c+d x) \left (1400 a^4 b \sin \left (\frac {1}{2} (c+d x)\right )-1540 a^2 b^3 \sin \left (\frac {1}{2} (c+d x)\right )+103 b^5 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^5}{1680 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^5} \]

[In]

Integrate[Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(b*(1400*a^4 - 1540*a^2*b^2 + 103*b^4)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(1680*d*(a*Cos[c + d*x] + b*Sin[
c + d*x])^5) + ((-8*a^5 + 20*a^3*b^2 - 5*a*b^4)*Cos[c + d*x]^5*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b
*Tan[c + d*x])^5)/(16*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((8*a^5 - 20*a^3*b^2 + 5*a*b^4)*Cos[c + d*x]^5*
Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^5)/(16*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) +
((35*a*b^4 + 3*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(336*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^6*(a*C
os[c + d*x] + b*Sin[c + d*x])^5) + ((350*a^3*b^2 + 140*a^2*b^3 - 175*a*b^4 - 18*b^5)*Cos[c + d*x]^5*(a + b*Tan
[c + d*x])^5)/(560*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((840*a^5
+ 1400*a^4*b - 2100*a^3*b^2 - 1540*a^2*b^3 + 525*a*b^4 + 103*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(3360
*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (b^5*Cos[c + d*x]^5*Sin[(c +
d*x)/2]*(a + b*Tan[c + d*x])^5)/(56*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^7*(a*Cos[c + d*x] + b*Sin[c + d*x
])^5) - (b^5*Cos[c + d*x]^5*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^5)/(56*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2
])^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((-35*a*b^4 + 3*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(336*d
*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((-350*a^3*b^2 + 140*a^2*b^3 +
175*a*b^4 - 18*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(560*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(a*
Cos[c + d*x] + b*Sin[c + d*x])^5) + ((-840*a^5 + 1400*a^4*b + 2100*a^3*b^2 - 1540*a^2*b^3 - 525*a*b^4 + 103*b^
5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(3360*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*
Sin[c + d*x])^5) + (Cos[c + d*x]^5*(-1400*a^4*b*Sin[(c + d*x)/2] + 1540*a^2*b^3*Sin[(c + d*x)/2] - 103*b^5*Sin
[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(1680*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin
[c + d*x])^5) + (Cos[c + d*x]^5*(-1400*a^4*b*Sin[(c + d*x)/2] + 1540*a^2*b^3*Sin[(c + d*x)/2] - 103*b^5*Sin[(c
+ d*x)/2])*(a + b*Tan[c + d*x])^5)/(1680*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c +
d*x])^5) + (Cos[c + d*x]^5*(70*a^2*b^3*Sin[(c + d*x)/2] - 9*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(140
*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (Cos[c + d*x]^5*(-70*a^2*b^3
*Sin[(c + d*x)/2] + 9*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(140*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2
])^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (Cos[c + d*x]^5*(1400*a^4*b*Sin[(c + d*x)/2] - 1540*a^2*b^3*Sin[(c
+ d*x)/2] + 103*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(1680*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3
*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (Cos[c + d*x]^5*(1400*a^4*b*Sin[(c + d*x)/2] - 1540*a^2*b^3*Sin[(c + d
*x)/2] + 103*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(1680*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Co
s[c + d*x] + b*Sin[c + d*x])^5)

Maple [A] (verified)

Time = 2.39 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.93

method	result	size
parts	\(\frac {a^{5} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {b^{5} \left (\frac {\sec \left (d x +c \right )^{7}}{7}-\frac {2 \sec \left (d x +c \right )^{5}}{5}+\frac {\sec \left (d x +c \right )^{3}}{3}\right )}{d}+\frac {10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{24 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{48 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{48}-\frac {\sin \left (d x +c \right )}{16}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {10 a^{2} b^{3} \left (\frac {\sec \left (d x +c \right )^{5}}{5}-\frac {\sec \left (d x +c \right )^{3}}{3}\right )}{d}+\frac {5 a^{4} b \sec \left (d x +c \right )^{3}}{3 d}\)	\(297\)
derivativedivides	\(\frac {a^{5} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {5 a^{4} b}{3 \cos \left (d x +c \right )^{3}}+10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+10 a^{2} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{15}\right )+5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{24 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{48 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{48}-\frac {\sin \left (d x +c \right )}{16}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{7 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{6}}{35 \cos \left (d x +c \right )^{5}}-\frac {\sin \left (d x +c \right )^{6}}{105 \cos \left (d x +c \right )^{3}}+\frac {\sin \left (d x +c \right )^{6}}{35 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{35}\right )}{d}\)	\(405\)
default	\(\frac {a^{5} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {5 a^{4} b}{3 \cos \left (d x +c \right )^{3}}+10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+10 a^{2} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{15}\right )+5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{24 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{48 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{48}-\frac {\sin \left (d x +c \right )}{16}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{7 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{6}}{35 \cos \left (d x +c \right )^{5}}-\frac {\sin \left (d x +c \right )^{6}}{105 \cos \left (d x +c \right )^{3}}+\frac {\sin \left (d x +c \right )^{6}}{35 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{35}\right )}{d}\)	\(405\)
parallelrisch	\(\frac {-5880 a \left (\frac {\cos \left (7 d x +7 c \right )}{7}+\cos \left (5 d x +5 c \right )+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )\right ) \left (a^{4}-\frac {5}{2} a^{2} b^{2}+\frac {5}{8} b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+5880 a \left (\frac {\cos \left (7 d x +7 c \right )}{7}+\cos \left (5 d x +5 c \right )+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )\right ) \left (a^{4}-\frac {5}{2} a^{2} b^{2}+\frac {5}{8} b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (58800 a^{4} b -47040 a^{2} b^{3}+2688 b^{5}\right ) \cos \left (3 d x +3 c \right )+\left (19600 a^{4} b -15680 a^{2} b^{3}+896 b^{5}\right ) \cos \left (5 d x +5 c \right )+\left (2800 a^{4} b -2240 a^{2} b^{3}+128 b^{5}\right ) \cos \left (7 d x +7 c \right )+\left (89600 a^{4} b -71680 a^{2} b^{3}-3584 b^{5}\right ) \cos \left (2 d x +2 c \right )+22400 \cos \left (4 d x +4 c \right ) b \left (a^{4}-2 a^{2} b^{2}+\frac {1}{5} b^{4}\right )+\left (8400 a^{5}+46200 a^{3} b^{2}+10850 a \,b^{4}\right ) \sin \left (2 d x +2 c \right )+\left (6720 a^{5}+16800 a^{3} b^{2}-15400 a \,b^{4}\right ) \sin \left (4 d x +4 c \right )+\left (1680 a^{5}-4200 a^{3} b^{2}+1050 a \,b^{4}\right ) \sin \left (6 d x +6 c \right )+98000 b \left (\left (a^{4}-\frac {4}{5} a^{2} b^{2}+\frac {8}{175} b^{4}\right ) \cos \left (d x +c \right )+\frac {24 a^{4}}{35}-\frac {48 a^{2} b^{2}}{175}+\frac {456 b^{4}}{6125}\right )}{11760 d \left (\frac {\cos \left (7 d x +7 c \right )}{7}+\cos \left (5 d x +5 c \right )+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )\right )}\)	\(478\)
risch	\(\frac {{\mathrm e}^{i \left (d x +c \right )} \left (-7700 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-8400 i a^{3} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+7700 i a \,b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-23100 i a^{3} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-5425 i a \,b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+23100 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+5425 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+8400 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+2240 b^{5} {\mathrm e}^{10 i \left (d x +c \right )}+2240 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+7296 b^{5} {\mathrm e}^{6 i \left (d x +c \right )}+840 i a^{5}-1792 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}-1792 b^{5} {\mathrm e}^{8 i \left (d x +c \right )}-840 i a^{5} {\mathrm e}^{12 i \left (d x +c \right )}-3360 i a^{5} {\mathrm e}^{10 i \left (d x +c \right )}-4200 i a^{5} {\mathrm e}^{8 i \left (d x +c \right )}+4200 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}+3360 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}-2100 i a^{3} b^{2}+525 i a \,b^{4}-22400 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-35840 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+11200 a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}-26880 a^{2} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+44800 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-35840 a^{2} b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+67200 a^{4} b \,{\mathrm e}^{6 i \left (d x +c \right )}-22400 a^{2} b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+44800 a^{4} b \,{\mathrm e}^{8 i \left (d x +c \right )}+11200 a^{4} b \,{\mathrm e}^{10 i \left (d x +c \right )}+2100 i a^{3} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-525 i a \,b^{4} {\mathrm e}^{12 i \left (d x +c \right )}\right )}{840 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}+\frac {a^{5} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{2 d}-\frac {5 a^{3} b^{2} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{4 d}+\frac {5 a \,b^{4} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{16 d}-\frac {a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}+\frac {5 a^{3} b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{4 d}-\frac {5 a \,b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}\)	\(723\)

[In]

int(sec(d*x+c)^8*(cos(d*x+c)*a+b*sin(d*x+c))^5,x,method=_RETURNVERBOSE)

[Out]

a^5/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+b^5/d*(1/7*sec(d*x+c)^7-2/5*sec(d*x+c)^5+1/3*s
ec(d*x+c)^3)+10*a^3*b^2/d*(1/4*sin(d*x+c)^3/cos(d*x+c)^4+1/8*sin(d*x+c)^3/cos(d*x+c)^2+1/8*sin(d*x+c)-1/8*ln(s
ec(d*x+c)+tan(d*x+c)))+5*a*b^4/d*(1/6*sin(d*x+c)^5/cos(d*x+c)^6+1/24*sin(d*x+c)^5/cos(d*x+c)^4-1/48*sin(d*x+c)
^5/cos(d*x+c)^2-1/48*sin(d*x+c)^3-1/16*sin(d*x+c)+1/16*ln(sec(d*x+c)+tan(d*x+c)))+10*a^2*b^3/d*(1/5*sec(d*x+c)
^5-1/3*sec(d*x+c)^3)+5/3*a^4*b*sec(d*x+c)^3/d

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.71 \[ \int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {105 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 480 \, b^{5} + 1120 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 1344 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 70 \, {\left (40 \, a b^{4} \cos \left (d x + c\right ) + 3 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (12 \, a^{3} b^{2} - 7 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \]

[In]

integrate(sec(d*x+c)^8*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="fricas")

[Out]

1/3360*(105*(8*a^5 - 20*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^7*log(sin(d*x + c) + 1) - 105*(8*a^5 - 20*a^3*b^2 + 5*
a*b^4)*cos(d*x + c)^7*log(-sin(d*x + c) + 1) + 480*b^5 + 1120*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x + c)^4 + 13
44*(5*a^2*b^3 - b^5)*cos(d*x + c)^2 + 70*(40*a*b^4*cos(d*x + c) + 3*(8*a^5 - 20*a^3*b^2 + 5*a*b^4)*cos(d*x + c
)^5 + 10*(12*a^3*b^2 - 7*a*b^4)*cos(d*x + c)^3)*sin(d*x + c))/(d*cos(d*x + c)^7)

Sympy [F(-1)]

Timed out. \[ \int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**8*(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 = 289, normalized size of antiderivative = 0.91 \[ \int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {175 \, a b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{5} + 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 2100 \, a^{3} b^{2} {\left (\frac {2 \, {\left (\sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 840 \, a^{5} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {5600 \, a^{4} b}{\cos \left (d x + c\right )^{3}} + \frac {2240 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} a^{2} b^{3}}{\cos \left (d x + c\right )^{5}} - \frac {32 \, {\left (35 \, \cos \left (d x + c\right )^{4} - 42 \, \cos \left (d x + c\right )^{2} + 15\right )} b^{5}}{\cos \left (d x + c\right )^{7}}}{3360 \, d} \]

[In]

integrate(sec(d*x+c)^8*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/3360*(175*a*b^4*(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)) - 2100*a^3*b^2*(2*(sin(d*x + c)
^3 + sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) +
840*a^5*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 5600*a^4*b/cos
(d*x + c)^3 + 2240*(5*cos(d*x + c)^2 - 3)*a^2*b^3/cos(d*x + c)^5 - 32*(35*cos(d*x + c)^4 - 42*cos(d*x + c)^2 +
15)*b^5/cos(d*x + c)^7)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 680 vs. \(2 (290) = 580\).

Time = 0.57 (sec) , antiderivative size = 680, normalized size of antiderivative = 2.14 \[ \int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^8*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="giac")

[Out]

1/1680*(105*(8*a^5 - 20*a^3*b^2 + 5*a*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*(8*a^5 - 20*a^3*b^2 + 5*a*
b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(840*a^5*tan(1/2*d*x + 1/2*c)^13 + 2100*a^3*b^2*tan(1/2*d*x + 1/2*
c)^13 - 525*a*b^4*tan(1/2*d*x + 1/2*c)^13 - 8400*a^4*b*tan(1/2*d*x + 1/2*c)^12 - 3360*a^5*tan(1/2*d*x + 1/2*c)
^11 + 8400*a^3*b^2*tan(1/2*d*x + 1/2*c)^11 + 3500*a*b^4*tan(1/2*d*x + 1/2*c)^11 + 33600*a^4*b*tan(1/2*d*x + 1/
2*c)^10 - 33600*a^2*b^3*tan(1/2*d*x + 1/2*c)^10 + 4200*a^5*tan(1/2*d*x + 1/2*c)^9 - 23100*a^3*b^2*tan(1/2*d*x
+ 1/2*c)^9 + 16975*a*b^4*tan(1/2*d*x + 1/2*c)^9 - 53200*a^4*b*tan(1/2*d*x + 1/2*c)^8 + 56000*a^2*b^3*tan(1/2*d
*x + 1/2*c)^8 - 8960*b^5*tan(1/2*d*x + 1/2*c)^8 + 44800*a^4*b*tan(1/2*d*x + 1/2*c)^6 - 22400*a^2*b^3*tan(1/2*d
*x + 1/2*c)^6 - 4480*b^5*tan(1/2*d*x + 1/2*c)^6 - 4200*a^5*tan(1/2*d*x + 1/2*c)^5 + 23100*a^3*b^2*tan(1/2*d*x
+ 1/2*c)^5 - 16975*a*b^4*tan(1/2*d*x + 1/2*c)^5 - 25200*a^4*b*tan(1/2*d*x + 1/2*c)^4 + 13440*a^2*b^3*tan(1/2*d
*x + 1/2*c)^4 - 2688*b^5*tan(1/2*d*x + 1/2*c)^4 + 3360*a^5*tan(1/2*d*x + 1/2*c)^3 - 8400*a^3*b^2*tan(1/2*d*x +
1/2*c)^3 - 3500*a*b^4*tan(1/2*d*x + 1/2*c)^3 + 11200*a^4*b*tan(1/2*d*x + 1/2*c)^2 - 15680*a^2*b^3*tan(1/2*d*x
+ 1/2*c)^2 + 896*b^5*tan(1/2*d*x + 1/2*c)^2 - 840*a^5*tan(1/2*d*x + 1/2*c) - 2100*a^3*b^2*tan(1/2*d*x + 1/2*c
) + 525*a*b^4*tan(1/2*d*x + 1/2*c) - 2800*a^4*b + 2240*a^2*b^3 - 128*b^5)/(tan(1/2*d*x + 1/2*c)^2 - 1)^7)/d

Mupad [B] (verification not implemented)

Time = 27.40 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.62 \[ \int \sec ^8(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 (a^5-\frac {5\,a^3\,b^2}{2}+\frac {5\,a\,b^4}{8}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (-4\,a^5+10\,a^3\,b^2+\frac {25\,a\,b^4}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (40\,a^4\,b-40\,a^2\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (a^5+\frac {5\,a^3\,b^2}{2}-\frac {5\,a\,b^4}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (-4\,a^5+10\,a^3\,b^2+\frac {25\,a\,b^4}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (5\,a^5-\frac {55\,a^3\,b^2}{2}+\frac {485\,a\,b^4}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (5\,a^5-\frac {55\,a^3\,b^2}{2}+\frac {485\,a\,b^4}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (30\,a^4\,b-16\,a^2\,b^3+\frac {16\,b^5}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {40\,a^4\,b}{3}-\frac {56\,a^2\,b^3}{3}+\frac {16\,b^5}{15}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (-\frac {160\,a^4\,b}{3}+\frac {80\,a^2\,b^3}{3}+\frac {16\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {190\,a^4\,b}{3}-\frac {200\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )+\frac {10\,a^4\,b}{3}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^5+\frac {5\,a^3\,b^2}{2}-\frac {5\,a\,b^4}{8}\right )+\frac {16\,b^5}{105}-\frac {8\,a^2\,b^3}{3}+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\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)^8,x)

[Out]

(atanh(tan(c/2 + (d*x)/2))*((5*a*b^4)/8 + a^5 - (5*a^3*b^2)/2))/d - (tan(c/2 + (d*x)/2)^3*((25*a*b^4)/6 - 4*a^
5 + 10*a^3*b^2) - tan(c/2 + (d*x)/2)^10*(40*a^4*b - 40*a^2*b^3) - tan(c/2 + (d*x)/2)^13*(a^5 - (5*a*b^4)/8 + (
5*a^3*b^2)/2) - tan(c/2 + (d*x)/2)^11*((25*a*b^4)/6 - 4*a^5 + 10*a^3*b^2) + tan(c/2 + (d*x)/2)^5*((485*a*b^4)/
24 + 5*a^5 - (55*a^3*b^2)/2) - tan(c/2 + (d*x)/2)^9*((485*a*b^4)/24 + 5*a^5 - (55*a^3*b^2)/2) + tan(c/2 + (d*x
)/2)^4*(30*a^4*b + (16*b^5)/5 - 16*a^2*b^3) - tan(c/2 + (d*x)/2)^2*((40*a^4*b)/3 + (16*b^5)/15 - (56*a^2*b^3)/
3) + tan(c/2 + (d*x)/2)^6*((16*b^5)/3 - (160*a^4*b)/3 + (80*a^2*b^3)/3) + tan(c/2 + (d*x)/2)^8*((190*a^4*b)/3
+ (32*b^5)/3 - (200*a^2*b^3)/3) + (10*a^4*b)/3 + tan(c/2 + (d*x)/2)*(a^5 - (5*a*b^4)/8 + (5*a^3*b^2)/2) + (16*
b^5)/105 - (8*a^2*b^3)/3 + 10*a^4*b*tan(c/2 + (d*x)/2)^12)/(d*(7*tan(c/2 + (d*x)/2)^2 - 21*tan(c/2 + (d*x)/2)^
4 + 35*tan(c/2 + (d*x)/2)^6 - 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 - 7*tan(c/2 + (d*x)/2)^12 + t
an(c/2 + (d*x)/2)^14 - 1))