3.3.7 \(\int \frac {\sin (19 x)+\sin (20 x)}{\cos (19 x)+\cos (20 x)} \, dx\) [207]

Optimal. Leaf size=11 \[ -\frac {2}{39} \log \left (\cos \left (\frac {39 x}{2}\right )\right ) \]

[Out]

-2/39*ln(cos(39/2*x))

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(264\) vs. \(2(11)=22\).
time = 2.75, antiderivative size = 264, normalized size of antiderivative = 24.00, number of steps used = 14, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4486, 2099, 1601, 12, 2125} \begin {gather*} -\frac {1}{780} \log \left (-64 \cos ^6(x)+32 \cos ^5(x)+80 \cos ^4(x)-32 \cos ^3(x)-24 \cos ^2(x)+6 \cos (x)+1\right )-\frac {1}{780} \log (1-2 \cos (x))+\frac {19}{780} \log (\cos (x)+1)-\frac {1}{780} \log \left (4096 \cos ^{12}(x)+2048 \cos ^{11}(x)-12288 \cos ^{10}(x)-6144 \cos ^9(x)+13568 \cos ^8(x)+6784 \cos ^7(x)-6592 \cos ^6(x)-3296 \cos ^5(x)+1264 \cos ^4(x)+632 \cos ^3(x)-48 \cos ^2(x)-24 \cos (x)+1\right )-\frac {1}{20} \log \left (524288 \cos ^{20}(x)+262144 \cos ^{19}(x)-2621440 \cos ^{18}(x)-1245184 \cos ^{17}(x)+5570560 \cos ^{16}(x)+2490368 \cos ^{15}(x)-6553600 \cos ^{14}(x)-2723840 \cos ^{13}(x)+4659200 \cos ^{12}(x)+1770496 \cos ^{11}(x)-2050048 \cos ^{10}(x)-695552 \cos ^9(x)+549120 \cos ^8(x)+160512 \cos ^7(x)-84480 \cos ^6(x)-20064 \cos ^5(x)+6600 \cos ^4(x)+1140 \cos ^3(x)-200 \cos ^2(x)-19 \cos (x)+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sin[19*x] + Sin[20*x])/(Cos[19*x] + Cos[20*x]),x]

[Out]

-1/780*Log[1 - 2*Cos[x]] + (19*Log[1 + Cos[x]])/780 - Log[1 + 6*Cos[x] - 24*Cos[x]^2 - 32*Cos[x]^3 + 80*Cos[x]
^4 + 32*Cos[x]^5 - 64*Cos[x]^6]/780 - Log[1 - 24*Cos[x] - 48*Cos[x]^2 + 632*Cos[x]^3 + 1264*Cos[x]^4 - 3296*Co
s[x]^5 - 6592*Cos[x]^6 + 6784*Cos[x]^7 + 13568*Cos[x]^8 - 6144*Cos[x]^9 - 12288*Cos[x]^10 + 2048*Cos[x]^11 + 4
096*Cos[x]^12]/780 - Log[1 - 19*Cos[x] - 200*Cos[x]^2 + 1140*Cos[x]^3 + 6600*Cos[x]^4 - 20064*Cos[x]^5 - 84480
*Cos[x]^6 + 160512*Cos[x]^7 + 549120*Cos[x]^8 - 695552*Cos[x]^9 - 2050048*Cos[x]^10 + 1770496*Cos[x]^11 + 4659
200*Cos[x]^12 - 2723840*Cos[x]^13 - 6553600*Cos[x]^14 + 2490368*Cos[x]^15 + 5570560*Cos[x]^16 - 1245184*Cos[x]
^17 - 2621440*Cos[x]^18 + 262144*Cos[x]^19 + 524288*Cos[x]^20]/20

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1601

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*(Log[RemoveConte
nt[Qq, x]]/(q*Coeff[Qq, x, q])), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 2125

Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[Coeff[Pm, x, m]*(Log[Qn]/(n*Coef
f[Qn, x, n])), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x
], x]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]

Rule 4486

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /;  !InertTrigFreeQ[u]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \left (\frac {\sin (19 x)}{\cos (19 x)+\cos (20 x)}+\frac {\sin (20 x)}{\cos (19 x)+\cos (20 x)}\right ) \, dx\\ &=\int \frac {\sin (19 x)}{\cos (19 x)+\cos (20 x)} \, dx+\int \frac {\sin (20 x)}{\cos (19 x)+\cos (20 x)} \, dx\\ &=-\text {Subst}\left (\int \frac {4 x \left (-5+330 x^2-6336 x^4+54912 x^6-256256 x^8+698880 x^{10}-1146880 x^{12}+1114112 x^{14}-589824 x^{16}+131072 x^{18}\right )}{1-19 x-200 x^2+1140 x^3+6600 x^4-20064 x^5-84480 x^6+160512 x^7+549120 x^8-695552 x^9-2050048 x^{10}+1770496 x^{11}+4659200 x^{12}-2723840 x^{13}-6553600 x^{14}+2490368 x^{15}+5570560 x^{16}-1245184 x^{17}-2621440 x^{18}+262144 x^{19}+524288 x^{20}} \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \frac {-1+180 x^2-5280 x^4+59136 x^6-329472 x^8+1025024 x^{10}-1863680 x^{12}+1966080 x^{14}-1114112 x^{16}+262144 x^{18}}{1-19 x-200 x^2+1140 x^3+6600 x^4-20064 x^5-84480 x^6+160512 x^7+549120 x^8-695552 x^9-2050048 x^{10}+1770496 x^{11}+4659200 x^{12}-2723840 x^{13}-6553600 x^{14}+2490368 x^{15}+5570560 x^{16}-1245184 x^{17}-2621440 x^{18}+262144 x^{19}+524288 x^{20}} \, dx,x,\cos (x)\right )\\ &=-\left (4 \text {Subst}\left (\int \frac {x \left (-5+330 x^2-6336 x^4+54912 x^6-256256 x^8+698880 x^{10}-1146880 x^{12}+1114112 x^{14}-589824 x^{16}+131072 x^{18}\right )}{1-19 x-200 x^2+1140 x^3+6600 x^4-20064 x^5-84480 x^6+160512 x^7+549120 x^8-695552 x^9-2050048 x^{10}+1770496 x^{11}+4659200 x^{12}-2723840 x^{13}-6553600 x^{14}+2490368 x^{15}+5570560 x^{16}-1245184 x^{17}-2621440 x^{18}+262144 x^{19}+524288 x^{20}} \, dx,x,\cos (x)\right )\right )-\text {Subst}\left (\int \left (-\frac {19}{39 (1+x)}+\frac {2}{39 (-1+2 x)}+\frac {2 \left (-3+24 x+48 x^2-160 x^3-80 x^4+192 x^5\right )}{39 \left (-1-6 x+24 x^2+32 x^3-80 x^4-32 x^5+64 x^6\right )}+\frac {8 \left (-3-12 x+237 x^2+632 x^3-2060 x^4-4944 x^5+5936 x^6+13568 x^7-6912 x^8-15360 x^9+2816 x^{10}+6144 x^{11}\right )}{39 \left (1-24 x-48 x^2+632 x^3+1264 x^4-3296 x^5-6592 x^6+6784 x^7+13568 x^8-6144 x^9-12288 x^{10}+2048 x^{11}+4096 x^{12}\right )}\right ) \, dx,x,\cos (x)\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(131\) vs. \(2(11)=22\).
time = 0.14, size = 131, normalized size = 11.91 \begin {gather*} -\frac {2}{39} \log \left (\cos \left (\frac {x}{2}\right )\right )-\frac {2}{39} \log (1-2 \cos (x)+2 \cos (2 x)-2 \cos (3 x)+2 \cos (4 x)-2 \cos (5 x)+2 \cos (6 x)-2 \cos (7 x)+2 \cos (8 x)-2 \cos (9 x)+2 \cos (10 x)-2 \cos (11 x)+2 \cos (12 x)-2 \cos (13 x)+2 \cos (14 x)-2 \cos (15 x)+2 \cos (16 x)-2 \cos (17 x)+2 \cos (18 x)-2 \cos (19 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sin[19*x] + Sin[20*x])/(Cos[19*x] + Cos[20*x]),x]

[Out]

(-2*Log[Cos[x/2]])/39 - (2*Log[1 - 2*Cos[x] + 2*Cos[2*x] - 2*Cos[3*x] + 2*Cos[4*x] - 2*Cos[5*x] + 2*Cos[6*x] -
2*Cos[7*x] + 2*Cos[8*x] - 2*Cos[9*x] + 2*Cos[10*x] - 2*Cos[11*x] + 2*Cos[12*x] - 2*Cos[13*x] + 2*Cos[14*x] -
2*Cos[15*x] + 2*Cos[16*x] - 2*Cos[17*x] + 2*Cos[18*x] - 2*Cos[19*x]])/39

________________________________________________________________________________________

Maple [C] Result contains complex when optimal does not.
time = 0.43, size = 16, normalized size = 1.45

method	result	size
risch	\(i x -\frac {2 \ln \left ({\mathrm e}^{39 i x}+1\right )}{39}\)	\(16\)
parallelrisch	\(\ln \left (\left (\sec ^{2}\left (10 x \right )\right )^{\frac {1}{39}}\right )+\ln \left (\left (\sec ^{2}\left (\frac {19 x}{2}\right )\right )^{\frac {1}{39}}\right )+\ln \left (\frac {1}{\left (\tan \left (\frac {19 x}{2}\right ) \tan \left (10 x \right )-1\right )^{\frac {2}{39}}}\right )\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sin(20*x)+sin(19*x))/(cos(20*x)+cos(19*x)),x,method=_RETURNVERBOSE)

[Out]

I*x-2/39*ln(exp(39*I*x)+1)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2527 vs. \(2 (7) = 14\).
time = 0.52, size = 2527, normalized size = 229.73 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((sin(20*x)+sin(19*x))/(cos(20*x)+cos(19*x)),x, algorithm="maxima")

[Out]

-1/39*log(2*(cos(23*x) - cos(21*x) - cos(20*x) + cos(18*x) + cos(17*x) - cos(15*x) - cos(14*x) + cos(12*x) - c
os(10*x) - cos(9*x) + cos(7*x) + cos(6*x) - cos(4*x) - cos(3*x) + cos(x) + 1)*cos(24*x) + cos(24*x)^2 - 2*(cos
(21*x) + cos(20*x) - cos(18*x) - cos(17*x) + cos(15*x) + cos(14*x) - cos(12*x) + cos(10*x) + cos(9*x) - cos(7*
x) - cos(6*x) + cos(4*x) + cos(3*x) - cos(x) - 1)*cos(23*x) + cos(23*x)^2 + 2*(cos(20*x) - cos(18*x) - cos(17*
x) + cos(15*x) + cos(14*x) - cos(12*x) + cos(10*x) + cos(9*x) - cos(7*x) - cos(6*x) + cos(4*x) + cos(3*x) - co
s(x) - 1)*cos(21*x) + cos(21*x)^2 - 2*(cos(18*x) + cos(17*x) - cos(15*x) - cos(14*x) + cos(12*x) - cos(10*x) -
cos(9*x) + cos(7*x) + cos(6*x) - cos(4*x) - cos(3*x) + cos(x) + 1)*cos(20*x) + cos(20*x)^2 + 2*(cos(17*x) - c
os(15*x) - cos(14*x) + cos(12*x) - cos(10*x) - cos(9*x) + cos(7*x) + cos(6*x) - cos(4*x) - cos(3*x) + cos(x) +
1)*cos(18*x) + cos(18*x)^2 - 2*(cos(15*x) + cos(14*x) - cos(12*x) + cos(10*x) + cos(9*x) - cos(7*x) - cos(6*x
) + cos(4*x) + cos(3*x) - cos(x) - 1)*cos(17*x) + cos(17*x)^2 + 2*(cos(14*x) - cos(12*x) + cos(10*x) + cos(9*x
) - cos(7*x) - cos(6*x) + cos(4*x) + cos(3*x) - cos(x) - 1)*cos(15*x) + cos(15*x)^2 - 2*(cos(12*x) - cos(10*x)
- cos(9*x) + cos(7*x) + cos(6*x) - cos(4*x) - cos(3*x) + cos(x) + 1)*cos(14*x) + cos(14*x)^2 - 2*(cos(10*x) +
cos(9*x) - cos(7*x) - cos(6*x) + cos(4*x) + cos(3*x) - cos(x) - 1)*cos(12*x) + cos(12*x)^2 + 2*(cos(9*x) - co
s(7*x) - cos(6*x) + cos(4*x) + cos(3*x) - cos(x) - 1)*cos(10*x) + cos(10*x)^2 - 2*(cos(7*x) + cos(6*x) - cos(4
*x) - cos(3*x) + cos(x) + 1)*cos(9*x) + cos(9*x)^2 + 2*(cos(6*x) - cos(4*x) - cos(3*x) + cos(x) + 1)*cos(7*x)
+ cos(7*x)^2 - 2*(cos(4*x) + cos(3*x) - cos(x) - 1)*cos(6*x) + cos(6*x)^2 + 2*(cos(3*x) - cos(x) - 1)*cos(4*x)
+ cos(4*x)^2 - 2*(cos(x) + 1)*cos(3*x) + cos(3*x)^2 + cos(x)^2 + 2*(sin(23*x) - sin(21*x) - sin(20*x) + sin(1
8*x) + sin(17*x) - sin(15*x) - sin(14*x) + sin(12*x) - sin(10*x) - sin(9*x) + sin(7*x) + sin(6*x) - sin(4*x) -
sin(3*x) + sin(x))*sin(24*x) + sin(24*x)^2 - 2*(sin(21*x) + sin(20*x) - sin(18*x) - sin(17*x) + sin(15*x) + s
in(14*x) - sin(12*x) + sin(10*x) + sin(9*x) - sin(7*x) - sin(6*x) + sin(4*x) + sin(3*x) - sin(x))*sin(23*x) +
sin(23*x)^2 + 2*(sin(20*x) - sin(18*x) - sin(17*x) + sin(15*x) + sin(14*x) - sin(12*x) + sin(10*x) + sin(9*x)
- sin(7*x) - sin(6*x) + sin(4*x) + sin(3*x) - sin(x))*sin(21*x) + sin(21*x)^2 - 2*(sin(18*x) + sin(17*x) - sin
(15*x) - sin(14*x) + sin(12*x) - sin(10*x) - sin(9*x) + sin(7*x) + sin(6*x) - sin(4*x) - sin(3*x) + sin(x))*si
n(20*x) + sin(20*x)^2 + 2*(sin(17*x) - sin(15*x) - sin(14*x) + sin(12*x) - sin(10*x) - sin(9*x) + sin(7*x) + s
in(6*x) - sin(4*x) - sin(3*x) + sin(x))*sin(18*x) + sin(18*x)^2 - 2*(sin(15*x) + sin(14*x) - sin(12*x) + sin(1
0*x) + sin(9*x) - sin(7*x) - sin(6*x) + sin(4*x) + sin(3*x) - sin(x))*sin(17*x) + sin(17*x)^2 + 2*(sin(14*x) -
sin(12*x) + sin(10*x) + sin(9*x) - sin(7*x) - sin(6*x) + sin(4*x) + sin(3*x) - sin(x))*sin(15*x) + sin(15*x)^
2 - 2*(sin(12*x) - sin(10*x) - sin(9*x) + sin(7*x) + sin(6*x) - sin(4*x) - sin(3*x) + sin(x))*sin(14*x) + sin(
14*x)^2 - 2*(sin(10*x) + sin(9*x) - sin(7*x) - sin(6*x) + sin(4*x) + sin(3*x) - sin(x))*sin(12*x) + sin(12*x)^
2 + 2*(sin(9*x) - sin(7*x) - sin(6*x) + sin(4*x) + sin(3*x) - sin(x))*sin(10*x) + sin(10*x)^2 - 2*(sin(7*x) +
sin(6*x) - sin(4*x) - sin(3*x) + sin(x))*sin(9*x) + sin(9*x)^2 + 2*(sin(6*x) - sin(4*x) - sin(3*x) + sin(x))*s
in(7*x) + sin(7*x)^2 - 2*(sin(4*x) + sin(3*x) - sin(x))*sin(6*x) + sin(6*x)^2 + 2*(sin(3*x) - sin(x))*sin(4*x)
+ sin(4*x)^2 + sin(3*x)^2 - 2*sin(3*x)*sin(x) + sin(x)^2 + 2*cos(x) + 1) - 1/39*log(-2*(cos(11*x) - cos(10*x)
+ cos(9*x) - cos(8*x) + cos(7*x) - cos(6*x) + cos(5*x) - cos(4*x) + cos(3*x) - cos(2*x) + cos(x) - 1)*cos(12*
x) + cos(12*x)^2 - 2*(cos(10*x) - cos(9*x) + cos(8*x) - cos(7*x) + cos(6*x) - cos(5*x) + cos(4*x) - cos(3*x) +
cos(2*x) - cos(x) + 1)*cos(11*x) + cos(11*x)^2 - 2*(cos(9*x) - cos(8*x) + cos(7*x) - cos(6*x) + cos(5*x) - co
s(4*x) + cos(3*x) - cos(2*x) + cos(x) - 1)*cos(10*x) + cos(10*x)^2 - 2*(cos(8*x) - cos(7*x) + cos(6*x) - cos(5
*x) + cos(4*x) - cos(3*x) + cos(2*x) - cos(x) + 1)*cos(9*x) + cos(9*x)^2 - 2*(cos(7*x) - cos(6*x) + cos(5*x) -
cos(4*x) + cos(3*x) - cos(2*x) + cos(x) - 1)*cos(8*x) + cos(8*x)^2 - 2*(cos(6*x) - cos(5*x) + cos(4*x) - cos(
3*x) + cos(2*x) - cos(x) + 1)*cos(7*x) + cos(7*x)^2 - 2*(cos(5*x) - cos(4*x) + cos(3*x) - cos(2*x) + cos(x) -
1)*cos(6*x) + cos(6*x)^2 - 2*(cos(4*x) - cos(3*x) + cos(2*x) - cos(x) + 1)*cos(5*x) + cos(5*x)^2 - 2*(cos(3*x)
- cos(2*x) + cos(x) - 1)*cos(4*x) + cos(4*x)^2 - 2*(cos(2*x) - cos(x) + 1)*cos(3*x) + cos(3*x)^2 - 2*(cos(x)
- 1)*cos(2*x) + cos(2*x)^2 + cos(x)^2 - 2*(sin(11*x) - sin(10*x) + sin(9*x) - sin(8*x) + sin(7*x) - sin(6*x) +
sin(5*x) - sin(4*x) + sin(3*x) - sin(2*x) + sin(x))*sin(12*x) + sin(12*x)^2 - 2*(sin(10*x) - sin(9*x) + sin(8
*x) - sin(7*x) + sin(6*x) - sin(5*x) + sin(4*x) - sin(3*x) + sin(2*x) - sin(x))*sin(11*x) + sin(11*x)^2 - 2*(s
in(9*x) - sin(8*x) + sin(7*x) - sin(6*x) + sin(...

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (7) = 14\).
time = 0.87, size = 123, normalized size = 11.18 \begin {gather*} -\frac {1}{39} \, \log \left (137438953472 \, \cos \left (x\right )^{39} - 1340029796352 \, \cos \left (x\right )^{37} + 6030134083584 \, \cos \left (x\right )^{35} - 16610786017280 \, \cos \left (x\right )^{33} + 31323196489728 \, \cos \left (x\right )^{31} - 42839077552128 \, \cos \left (x\right )^{29} + 43920872439808 \, \cos \left (x\right )^{27} - 34411219255296 \, \cos \left (x\right )^{25} + 20813237452800 \, \cos \left (x\right )^{23} - 9751387176960 \, \cos \left (x\right )^{21} + 3530674667520 \, \cos \left (x\right )^{19} - 980106117120 \, \cos \left (x\right )^{17} + 205701283840 \, \cos \left (x\right )^{15} - 31950643200 \, \cos \left (x\right )^{13} + 3560214528 \, \cos \left (x\right )^{11} - 271960832 \, \cos \left (x\right )^{9} + 13302432 \, \cos \left (x\right )^{7} - 373464 \, \cos \left (x\right )^{5} + 4940 \, \cos \left (x\right )^{3} - \frac {39}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((sin(20*x)+sin(19*x))/(cos(20*x)+cos(19*x)),x, algorithm="fricas")

[Out]

-1/39*log(137438953472*cos(x)^39 - 1340029796352*cos(x)^37 + 6030134083584*cos(x)^35 - 16610786017280*cos(x)^3
3 + 31323196489728*cos(x)^31 - 42839077552128*cos(x)^29 + 43920872439808*cos(x)^27 - 34411219255296*cos(x)^25
+ 20813237452800*cos(x)^23 - 9751387176960*cos(x)^21 + 3530674667520*cos(x)^19 - 980106117120*cos(x)^17 + 2057
01283840*cos(x)^15 - 31950643200*cos(x)^13 + 3560214528*cos(x)^11 - 271960832*cos(x)^9 + 13302432*cos(x)^7 - 3
73464*cos(x)^5 + 4940*cos(x)^3 - 39/2*cos(x) + 1/2)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((sin(20*x)+sin(19*x))/(cos(20*x)+cos(19*x)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (7) = 14\).
time = 0.46, size = 134, normalized size = 12.18 \begin {gather*} -\frac {1}{39} \, \log \left (\cos \left (x\right ) + 1\right ) - \frac {2}{39} \, \log \left ({\left | 4096 \, \cos \left (x\right )^{12} + 2048 \, \cos \left (x\right )^{11} - 12288 \, \cos \left (x\right )^{10} - 6144 \, \cos \left (x\right )^{9} + 13568 \, \cos \left (x\right )^{8} + 6784 \, \cos \left (x\right )^{7} - 6592 \, \cos \left (x\right )^{6} - 3296 \, \cos \left (x\right )^{5} + 1264 \, \cos \left (x\right )^{4} + 632 \, \cos \left (x\right )^{3} - 48 \, \cos \left (x\right )^{2} - 24 \, \cos \left (x\right ) + 1 \right |}\right ) - \frac {2}{39} \, \log \left ({\left | 64 \, \cos \left (x\right )^{6} - 32 \, \cos \left (x\right )^{5} - 80 \, \cos \left (x\right )^{4} + 32 \, \cos \left (x\right )^{3} + 24 \, \cos \left (x\right )^{2} - 6 \, \cos \left (x\right ) - 1 \right |}\right ) - \frac {2}{39} \, \log \left ({\left | 2 \, \cos \left (x\right ) - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((sin(20*x)+sin(19*x))/(cos(20*x)+cos(19*x)),x, algorithm="giac")

[Out]

-1/39*log(cos(x) + 1) - 2/39*log(abs(4096*cos(x)^12 + 2048*cos(x)^11 - 12288*cos(x)^10 - 6144*cos(x)^9 + 13568
*cos(x)^8 + 6784*cos(x)^7 - 6592*cos(x)^6 - 3296*cos(x)^5 + 1264*cos(x)^4 + 632*cos(x)^3 - 48*cos(x)^2 - 24*co
s(x) + 1)) - 2/39*log(abs(64*cos(x)^6 - 32*cos(x)^5 - 80*cos(x)^4 + 32*cos(x)^3 + 24*cos(x)^2 - 6*cos(x) - 1))
- 2/39*log(abs(2*cos(x) - 1))

________________________________________________________________________________________

Mupad [B]
time = 1.43, size = 15, normalized size = 1.36 \begin {gather*} x\,1{}\mathrm {i}-\frac {2\,\ln \left ({\mathrm {e}}^{x\,39{}\mathrm {i}}+1\right )}{39} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sin(19*x) + sin(20*x))/(cos(19*x) + cos(20*x)),x)

[Out]

x*1i - (2*log(exp(x*39i) + 1))/39

________________________________________________________________________________________

Chatgpt [F] Failed to verify
time = 1.00, size = 7, normalized size = 0.64 \begin {gather*} -\frac {39 \ln \left (\cos \left (\frac {39 x}{2}\right )\right )}{2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int((sin(20*x)+sin(19*x))/(cos(20*x)+cos(19*x)),x)

[Out]

-39/2*ln(cos(39/2*x))

________________________________________________________________________________________