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

Optimal. Leaf size=472 \[ \frac {5 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac {5 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {5 a^5 \tan (c+d x) \sec (c+d x)}{16 d}+\frac {5 a^4 b \sec ^7(c+d x)}{7 d}-\frac {25 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac {5 a^3 b^2 \tan (c+d x) \sec ^7(c+d x)}{4 d}-\frac {5 a^3 b^2 \tan (c+d x) \sec ^5(c+d x)}{24 d}-\frac {25 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{96 d}-\frac {25 a^3 b^2 \tan (c+d x) \sec (c+d x)}{64 d}+\frac {10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}+\frac {15 a b^4 \tanh ^{-1}(\sin (c+d x))}{256 d}+\frac {a b^4 \tan ^3(c+d x) \sec ^7(c+d x)}{2 d}-\frac {3 a b^4 \tan (c+d x) \sec ^7(c+d x)}{16 d}+\frac {a b^4 \tan (c+d x) \sec ^5(c+d x)}{32 d}+\frac {5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{128 d}+\frac {15 a b^4 \tan (c+d x) \sec (c+d x)}{256 d}+\frac {b^5 \sec ^{11}(c+d x)}{11 d}-\frac {2 b^5 \sec ^9(c+d x)}{9 d}+\frac {b^5 \sec ^7(c+d x)}{7 d} \]

[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

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Rubi [A]  time = 0.47, antiderivative size = 472, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3090, 3768, 3770, 2606, 30, 2611, 14, 270} \[ \frac {10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac {25 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac {5 a^3 b^2 \tan (c+d x) \sec ^7(c+d x)}{4 d}-\frac {5 a^3 b^2 \tan (c+d x) \sec ^5(c+d x)}{24 d}-\frac {25 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{96 d}-\frac {25 a^3 b^2 \tan (c+d x) \sec (c+d x)}{64 d}+\frac {5 a^4 b \sec ^7(c+d x)}{7 d}+\frac {5 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac {5 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {5 a^5 \tan (c+d x) \sec (c+d x)}{16 d}+\frac {15 a b^4 \tanh ^{-1}(\sin (c+d x))}{256 d}+\frac {a b^4 \tan ^3(c+d x) \sec ^7(c+d x)}{2 d}-\frac {3 a b^4 \tan (c+d x) \sec ^7(c+d x)}{16 d}+\frac {a b^4 \tan (c+d x) \sec ^5(c+d x)}{32 d}+\frac {5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{128 d}+\frac {15 a b^4 \tan (c+d x) \sec (c+d x)}{256 d}+\frac {b^5 \sec ^{11}(c+d x)}{11 d}-\frac {2 b^5 \sec ^9(c+d x)}{9 d}+\frac {b^5 \sec ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[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 270

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 2606

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 2611

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 3090

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 3768

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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=\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 ) \operatorname {Subst}\left (\int x^6 \, dx,x,\sec (c+d x)\right )}{d}+\frac {\left (10 a^2 b^3\right ) \operatorname {Subst}\left (\int x^6 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d}+\frac {b^5 \operatorname {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 ) \operatorname {Subst}\left (\int \left (-x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{d}+\frac {b^5 \operatorname {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 \tanh ^{-1}(\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 \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac {25 a^3 b^2 \tanh ^{-1}(\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 \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac {25 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac {15 a b^4 \tanh ^{-1}(\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*}

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Mathematica [A]  time = 1.79, size = 374, normalized size = 0.79 \[ \frac {13860 a \left (976 a^4+2876 a^2 b^2+1207 b^4\right ) \tan (c+d x) \sec ^9(c+d x)-1774080 a \left (16 a^4-20 a^2 b^2+3 b^4\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec ^{11}(c+d x) \left (6623232 a^5 \sin (4 (c+d x))+2857008 a^5 \sin (6 (c+d x))+591360 a^5 \sin (8 (c+d x))+55440 a^5 \sin (10 (c+d x))[phone] a^4 b+5913600 a^3 b^2 \sin (4 (c+d x))-3571260 a^3 b^2 \sin (6 (c+d x))-739200 a^3 b^2 \sin (8 (c+d x))-69300 a^3 b^2 \sin (10 (c+d x))+1802240 a^2 b^3+3604480 \left (9 a^4 b-4 a^2 b^3-b^5\right ) \cos (2 (c+d x))+1622016 \left (5 a^4 b-10 a^2 b^3+b^5\right ) \cos (4 (c+d x))-6564096 a b^4 \sin (4 (c+d x))+535689 a b^4 \sin (6 (c+d x))+110880 a b^4 \sin (8 (c+d x))+10395 a b^4 \sin (10 (c+d x))+3031040 b^5\right )}{90832896 d} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.67, size = 287, normalized size = 0.61 \[ \frac {3465 \, {\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{11} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3465 \, {\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{11} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 32256 \, b^{5} + 50688 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 78848 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 462 \, {\left (15 \, {\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} + 10 \, {\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 384 \, a b^{4} \cos \left (d x + c\right ) + 8 \, {\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 48 \, {\left (20 \, a^{3} b^{2} - 11 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{354816 \, d \cos \left (d x + c\right )^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 12.98, size = 1096, normalized size = 2.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.28, size = 814, normalized size = 1.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/693*b^5*cos(d*x+c)/d-15/256*a*b^4*sin(d*x+c)/d-5/256*a*b^4*sin(d*x+c)^3/d+25/64*a^3*b^2*sin(d*x+c)/d+5/16*a^
5*sec(d*x+c)*tan(d*x+c)/d+5/24*a^5*sec(d*x+c)^3*tan(d*x+c)/d+1/6*a^5*sec(d*x+c)^5*tan(d*x+c)/d-20/63*a^2*b^3*c
os(d*x+c)/d+5/99/d*b^5*sin(d*x+c)^6/cos(d*x+c)^9+5/7/d*a^4*b/cos(d*x+c)^7+1/11/d*b^5*sin(d*x+c)^6/cos(d*x+c)^1
1+5/4/d*a^3*b^2*sin(d*x+c)^3/cos(d*x+c)^8+10/9/d*a^2*b^3*sin(d*x+c)^4/cos(d*x+c)^9+1/2/d*a*b^4*sin(d*x+c)^5/co
s(d*x+c)^10+1/231/d*b^5*sin(d*x+c)^6/cos(d*x+c)^5-25/64/d*a^3*b^2*ln(sec(d*x+c)+tan(d*x+c))-1/693/d*b^5*sin(d*
x+c)^6/cos(d*x+c)^3+25/64/d*a^3*b^2*sin(d*x+c)^3/cos(d*x+c)^2+10/63/d*a^2*b^3*sin(d*x+c)^4/cos(d*x+c)^3+5/128/
d*a*b^4*sin(d*x+c)^5/cos(d*x+c)^4-10/63/d*cos(d*x+c)*sin(d*x+c)^2*a^2*b^3+25/32/d*a^3*b^2*sin(d*x+c)^3/cos(d*x
+c)^4+10/21/d*a^2*b^3*sin(d*x+c)^4/cos(d*x+c)^5+5/32/d*a*b^4*sin(d*x+c)^5/cos(d*x+c)^6+5/16/d*a^5*ln(sec(d*x+c
)+tan(d*x+c))+15/256/d*a*b^4*ln(sec(d*x+c)+tan(d*x+c))+1/231/d*b^5*sin(d*x+c)^6/cos(d*x+c)+1/231/d*b^5*cos(d*x
+c)*sin(d*x+c)^4+4/693/d*cos(d*x+c)*sin(d*x+c)^2*b^5-10/63/d*a^2*b^3*sin(d*x+c)^4/cos(d*x+c)-5/256/d*a*b^4*sin
(d*x+c)^5/cos(d*x+c)^2+5/231/d*b^5*sin(d*x+c)^6/cos(d*x+c)^7+25/24/d*a^3*b^2*sin(d*x+c)^3/cos(d*x+c)^6+50/63/d
*a^2*b^3*sin(d*x+c)^4/cos(d*x+c)^7+5/16/d*a*b^4*sin(d*x+c)^5/cos(d*x+c)^8

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maxima [A]  time = 0.54, size = 420, normalized size = 0.89 \[ -\frac {693 \, a b^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{9} - 70 \, \sin \left (d x + c\right )^{7} + 128 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{10} - 5 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4620 \, a^{3} b^{2} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{7} - 55 \, \sin \left (d x + c\right )^{5} + 73 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 3696 \, a^{5} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {253440 \, a^{4} b}{\cos \left (d x + c\right )^{7}} + \frac {56320 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} a^{2} b^{3}}{\cos \left (d x + c\right )^{9}} - \frac {512 \, {\left (99 \, \cos \left (d x + c\right )^{4} - 154 \, \cos \left (d x + c\right )^{2} + 63\right )} b^{5}}{\cos \left (d x + c\right )^{11}}}{354816 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 5.92, size = 831, normalized size = 1.76 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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