3.4.0 ∫ csc(x) sin(23x) dx [305]

Integrand size = 7, antiderivative size = 86 \[ \int \csc (x) \sin (23 x) \, dx=x+\sin (2 x)+\frac {1}{2} \sin (4 x)+\frac {1}{3} \sin (6 x)+\frac {1}{4} \sin (8 x)+\frac {1}{5} \sin (10 x)+\frac {1}{6} \sin (12 x)+\frac {1}{7} \sin (14 x)+\frac {1}{8} \sin (16 x)+\frac {1}{9} \sin (18 x)+\frac {1}{10} \sin (20 x)+\frac {1}{11} \sin (22 x) \]

x+sin(2*x)+1/2*sin(4*x)+1/3*sin(6*x)+1/4*sin(8*x)+1/5*sin(10*x)+1/6*sin(12*x)+1/7*sin(14*x)+1/8*sin(16*x)+1/9*

Rubi [A] (veriﬁed)

Time = 0.40 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.26, number of steps used = 13, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1828, 1171, 393, 209} \[ \int \csc (x) \sin (23 x) \, dx=x+\frac {2097152}{11} \sin (x) \cos ^{21}(x)-\frac {49545216}{55} \sin (x) \cos ^{19}(x)+\frac {899022848}{495} \sin (x) \cos ^{17}(x)-\frac {1009455104}{495} \sin (x) \cos ^{15}(x)+\frac {322500608}{231} \sin (x) \cos ^{13}(x)-\frac {415607296}{693} \sin (x) \cos ^{11}(x)+\frac {10100480}{63} \sin (x) \cos ^9(x)-\frac {178880}{7} \sin (x) \cos ^7(x)+\frac {6656}{3} \sin (x) \cos ^5(x)-\frac {260}{3} \sin (x) \cos ^3(x)+2 \sin (x) \cos (x) \]

x + 2*Cos[x]*Sin[x] - (260*Cos[x]^3*Sin[x])/3 + (6656*Cos[x]^5*Sin[x])/3 - (178880*Cos[x]^7*Sin[x])/7 + (10100

480*Cos[x]^9*Sin[x])/63 - (415607296*Cos[x]^11*Sin[x])/693 + (322500608*Cos[x]^13*Sin[x])/231 - (1009455104*Co

s[x]^15*Sin[x])/495 + (899022848*Cos[x]^17*Sin[x])/495 - (49545216*Cos[x]^19*Sin[x])/55 + (2097152*Cos[x]^21*S

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p

+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]

/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ

uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,

x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1

)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P

olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*

g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS

um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {23-1771 x^2+33649 x^4-245157 x^6+817190 x^8-1352078 x^{10}+1144066 x^{12}-490314 x^{14}+100947 x^{16}-8855 x^{18}+253 x^{20}-x^{22}}{\left (1+x^2\right )^{12}} \, dx,x,\tan (x)\right ) \\ & = \frac {2097152}{11} \cos ^{21}(x) \sin (x)-\frac {1}{22} \text {Subst}\left (\int \frac {4193798-92235220 x^2+91494942 x^4-86101488 x^6+68123308 x^8-38377592 x^{10}+13208140 x^{12}-2421232 x^{14}+200398 x^{16}-5588 x^{18}+22 x^{20}}{\left (1+x^2\right )^{11}} \, dx,x,\tan (x)\right ) \\ & = -\frac {49545216}{55} \cos ^{19}(x) \sin (x)+\frac {2097152}{11} \cos ^{21}(x) \sin (x)+\frac {1}{440} \text {Subst}\left (\int \frac {312485768-5998654200 x^2+4168755360 x^4-2446725600 x^6+1084259440 x^8-316707600 x^{10}+52544800 x^{12}-4120160 x^{14}+112200 x^{16}-440 x^{18}}{\left (1+x^2\right )^{10}} \, dx,x,\tan (x)\right ) \\ & = \frac {899022848}{495} \cos ^{17}(x) \sin (x)-\frac {49545216}{55} \cos ^{19}(x) \sin (x)+\frac {2097152}{11} \cos ^{21}(x) \sin (x)-\frac {\text {Subst}\left (\int \frac {8759621744-145318060800 x^2+70280464320 x^4-26239403520 x^6+6722733600 x^8-1021996800 x^{10}+76190400 x^{12}-2027520 x^{14}+7920 x^{16}}{\left (1+x^2\right )^9} \, dx,x,\tan (x)\right )}{7920} \\ & = -\frac {1009455104}{495} \cos ^{15}(x) \sin (x)+\frac {899022848}{495} \cos ^{17}(x) \sin (x)-\frac {49545216}{55} \cos ^{19}(x) \sin (x)+\frac {2097152}{11} \cos ^{21}(x) \sin (x)+\frac {\text {Subst}\left (\int \frac {118266558720-1669485185280 x^2+544997756160 x^4-125167299840 x^6+17603562240 x^8-1251613440 x^{10}+32567040 x^{12}-126720 x^{14}}{\left (1+x^2\right )^8} \, dx,x,\tan (x)\right )}{126720} \\ & = \frac {322500608}{231} \cos ^{13}(x) \sin (x)-\frac {1009455104}{495} \cos ^{15}(x) \sin (x)+\frac {899022848}{495} \cos ^{17}(x) \sin (x)-\frac {49545216}{55} \cos ^{19}(x) \sin (x)+\frac {2097152}{11} \cos ^{21}(x) \sin (x)-\frac {\text {Subst}\left (\int \frac {821072847360-9646740956160 x^2+2016772369920 x^4-264430172160 x^6+17980300800 x^8-457712640 x^{10}+1774080 x^{12}}{\left (1+x^2\right )^7} \, dx,x,\tan (x)\right )}{1774080} \\ & = -\frac {415607296}{693} \cos ^{11}(x) \sin (x)+\frac {322500608}{231} \cos ^{13}(x) \sin (x)-\frac {1009455104}{495} \cos ^{15}(x) \sin (x)+\frac {899022848}{495} \cos ^{17}(x) \sin (x)-\frac {49545216}{55} \cos ^{19}(x) \sin (x)+\frac {2097152}{11} \cos ^{21}(x) \sin (x)+\frac {\text {Subst}\left (\int \frac {2914581964800-27595707955200 x^2+3394439516160 x^4-221277450240 x^6+5513840640 x^8-21288960 x^{10}}{\left (1+x^2\right )^6} \, dx,x,\tan (x)\right )}{21288960} \\ & = \frac {10100480}{63} \cos ^9(x) \sin (x)-\frac {415607296}{693} \cos ^{11}(x) \sin (x)+\frac {322500608}{231} \cos ^{13}(x) \sin (x)-\frac {1009455104}{495} \cos ^{15}(x) \sin (x)+\frac {899022848}{495} \cos ^{17}(x) \sin (x)-\frac {49545216}{55} \cos ^{19}(x) \sin (x)+\frac {2097152}{11} \cos ^{21}(x) \sin (x)-\frac {\text {Subst}\left (\int \frac {4985722368000-36212520960000 x^2+2268125798400 x^4-55351296000 x^6+212889600 x^8}{\left (1+x^2\right )^5} \, dx,x,\tan (x)\right )}{212889600} \\ & = -\frac {178880}{7} \cos ^7(x) \sin (x)+\frac {10100480}{63} \cos ^9(x) \sin (x)-\frac {415607296}{693} \cos ^{11}(x) \sin (x)+\frac {322500608}{231} \cos ^{13}(x) \sin (x)-\frac {1009455104}{495} \cos ^{15}(x) \sin (x)+\frac {899022848}{495} \cos ^{17}(x) \sin (x)-\frac {49545216}{55} \cos ^{19}(x) \sin (x)+\frac {2097152}{11} \cos ^{21}(x) \sin (x)+\frac {\text {Subst}\left (\int \frac {3636154368000-18589519872000 x^2+444513484800 x^4-1703116800 x^6}{\left (1+x^2\right )^4} \, dx,x,\tan (x)\right )}{1703116800} \\ & = \frac {6656}{3} \cos ^5(x) \sin (x)-\frac {178880}{7} \cos ^7(x) \sin (x)+\frac {10100480}{63} \cos ^9(x) \sin (x)-\frac {415607296}{693} \cos ^{11}(x) \sin (x)+\frac {322500608}{231} \cos ^{13}(x) \sin (x)-\frac {1009455104}{495} \cos ^{15}(x) \sin (x)+\frac {899022848}{495} \cos ^{17}(x) \sin (x)-\frac {49545216}{55} \cos ^{19}(x) \sin (x)+\frac {2097152}{11} \cos ^{21}(x) \sin (x)-\frac {\text {Subst}\left (\int \frac {854964633600-2677299609600 x^2+10218700800 x^4}{\left (1+x^2\right )^3} \, dx,x,\tan (x)\right )}{10218700800} \\ & = -\frac {260}{3} \cos ^3(x) \sin (x)+\frac {6656}{3} \cos ^5(x) \sin (x)-\frac {178880}{7} \cos ^7(x) \sin (x)+\frac {10100480}{63} \cos ^9(x) \sin (x)-\frac {415607296}{693} \cos ^{11}(x) \sin (x)+\frac {322500608}{231} \cos ^{13}(x) \sin (x)-\frac {1009455104}{495} \cos ^{15}(x) \sin (x)+\frac {899022848}{495} \cos ^{17}(x) \sin (x)-\frac {49545216}{55} \cos ^{19}(x) \sin (x)+\frac {2097152}{11} \cos ^{21}(x) \sin (x)+\frac {\text {Subst}\left (\int \frac {122624409600-40874803200 x^2}{\left (1+x^2\right )^2} \, dx,x,\tan (x)\right )}{40874803200} \\ & = 2 \cos (x) \sin (x)-\frac {260}{3} \cos ^3(x) \sin (x)+\frac {6656}{3} \cos ^5(x) \sin (x)-\frac {178880}{7} \cos ^7(x) \sin (x)+\frac {10100480}{63} \cos ^9(x) \sin (x)-\frac {415607296}{693} \cos ^{11}(x) \sin (x)+\frac {322500608}{231} \cos ^{13}(x) \sin (x)-\frac {1009455104}{495} \cos ^{15}(x) \sin (x)+\frac {899022848}{495} \cos ^{17}(x) \sin (x)-\frac {49545216}{55} \cos ^{19}(x) \sin (x)+\frac {2097152}{11} \cos ^{21}(x) \sin (x)+\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = x+2 \cos (x) \sin (x)-\frac {260}{3} \cos ^3(x) \sin (x)+\frac {6656}{3} \cos ^5(x) \sin (x)-\frac {178880}{7} \cos ^7(x) \sin (x)+\frac {10100480}{63} \cos ^9(x) \sin (x)-\frac {415607296}{693} \cos ^{11}(x) \sin (x)+\frac {322500608}{231} \cos ^{13}(x) \sin (x)-\frac {1009455104}{495} \cos ^{15}(x) \sin (x)+\frac {899022848}{495} \cos ^{17}(x) \sin (x)-\frac {49545216}{55} \cos ^{19}(x) \sin (x)+\frac {2097152}{11} \cos ^{21}(x) \sin (x) \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \csc (x) \sin (23 x) \, dx=x+\sin (2 x)+\frac {1}{2} \sin (4 x)+\frac {1}{3} \sin (6 x)+\frac {1}{4} \sin (8 x)+\frac {1}{5} \sin (10 x)+\frac {1}{6} \sin (12 x)+\frac {1}{7} \sin (14 x)+\frac {1}{8} \sin (16 x)+\frac {1}{9} \sin (18 x)+\frac {1}{10} \sin (20 x)+\frac {1}{11} \sin (22 x) \]

x + Sin[2*x] + Sin[4*x]/2 + Sin[6*x]/3 + Sin[8*x]/4 + Sin[10*x]/5 + Sin[12*x]/6 + Sin[14*x]/7 + Sin[16*x]/8 +

Maple [A] (veriﬁed)

Time = 0.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.78


method	result	size

risch	\(x +\sin \left (2 x \right )+\frac {\sin \left (4 x \right )}{2}+\frac {\sin \left (6 x \right )}{3}+\frac {\sin \left (8 x \right )}{4}+\frac {\sin \left (10 x \right )}{5}+\frac {\sin \left (12 x \right )}{6}+\frac {\sin \left (14 x \right )}{7}+\frac {\sin \left (16 x \right )}{8}+\frac {\sin \left (18 x \right )}{9}+\frac {\sin \left (20 x \right )}{10}+\frac {\sin \left (22 x \right )}{11}\)	\(67\)

x+sin(2*x)+1/2*sin(4*x)+1/3*sin(6*x)+1/4*sin(8*x)+1/5*sin(10*x)+1/6*sin(12*x)+1/7*sin(14*x)+1/8*sin(16*x)+1/9*

Fricas [A] (veriﬁcation not implemented)

Time = 0.36 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.83 \[ \int \csc (x) \sin (23 x) \, dx=\frac {2}{3465} \, {\left (330301440 \, \cos \left (x\right )^{21} - 1560674304 \, \cos \left (x\right )^{19} + 3146579968 \, \cos \left (x\right )^{17} - 3533092864 \, \cos \left (x\right )^{15} + 2418754560 \, \cos \left (x\right )^{13} - 1039018240 \, \cos \left (x\right )^{11} + 277763200 \, \cos \left (x\right )^{9} - 44272800 \, \cos \left (x\right )^{7} + 3843840 \, \cos \left (x\right )^{5} - 150150 \, \cos \left (x\right )^{3} + 3465 \, \cos \left (x\right )\right )} \sin \left (x\right ) + x \]

2/3465*(330301440*cos(x)^21 - 1560674304*cos(x)^19 + 3146579968*cos(x)^17 - 3533092864*cos(x)^15 + 2418754560*

cos(x)^13 - 1039018240*cos(x)^11 + 277763200*cos(x)^9 - 44272800*cos(x)^7 + 3843840*cos(x)^5 - 150150*cos(x)^3

Sympy [A] (veriﬁcation not implemented)

Time = 69.50 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.16 \[ \int \csc (x) \sin (23 x) \, dx=x - \frac {1024 \sin ^{11}{\left (2 x \right )}}{11} + \frac {2560 \sin ^{9}{\left (2 x \right )}}{9} - \frac {2304 \sin ^{7}{\left (2 x \right )}}{7} + \frac {896 \sin ^{5}{\left (2 x \right )}}{5} - \frac {140 \sin ^{3}{\left (2 x \right )}}{3} + 6 \sin {\left (2 x \right )} + \frac {8 \sin ^{5}{\left (4 x \right )}}{5} - \frac {8 \sin ^{3}{\left (4 x \right )}}{3} + \frac {3 \sin {\left (4 x \right )}}{2} + \frac {\sin {\left (8 x \right )}}{4} + \frac {\sin {\left (16 x \right )}}{8} \]

x - 1024*sin(2*x)**11/11 + 2560*sin(2*x)**9/9 - 2304*sin(2*x)**7/7 + 896*sin(2*x)**5/5 - 140*sin(2*x)**3/3 + 6

*sin(2*x) + 8*sin(4*x)**5/5 - 8*sin(4*x)**3/3 + 3*sin(4*x)/2 + sin(8*x)/4 + sin(16*x)/8

Maxima [F(-1)]

Giac [A] (veriﬁcation not implemented)

Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.77 \[ \int \csc (x) \sin (23 x) \, dx=x + \frac {1}{11} \, \sin \left (22 \, x\right ) + \frac {1}{10} \, \sin \left (20 \, x\right ) + \frac {1}{9} \, \sin \left (18 \, x\right ) + \frac {1}{8} \, \sin \left (16 \, x\right ) + \frac {1}{7} \, \sin \left (14 \, x\right ) + \frac {1}{6} \, \sin \left (12 \, x\right ) + \frac {1}{5} \, \sin \left (10 \, x\right ) + \frac {1}{4} \, \sin \left (8 \, x\right ) + \frac {1}{3} \, \sin \left (6 \, x\right ) + \frac {1}{2} \, \sin \left (4 \, x\right ) + \sin \left (2 \, x\right ) \]

x + 1/11*sin(22*x) + 1/10*sin(20*x) + 1/9*sin(18*x) + 1/8*sin(16*x) + 1/7*sin(14*x) + 1/6*sin(12*x) + 1/5*sin(

Mupad [B] (veriﬁcation not implemented)

Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.77 \[ \int \csc (x) \sin (23 x) \, dx=x+\sin \left (2\,x\right )+\frac {\sin \left (4\,x\right )}{2}+\frac {\sin \left (6\,x\right )}{3}+\frac {\sin \left (8\,x\right )}{4}+\frac {\sin \left (10\,x\right )}{5}+\frac {\sin \left (12\,x\right )}{6}+\frac {\sin \left (14\,x\right )}{7}+\frac {\sin \left (16\,x\right )}{8}+\frac {\sin \left (18\,x\right )}{9}+\frac {\sin \left (20\,x\right )}{10}+\frac {\sin \left (22\,x\right )}{11} \]

x + sin(2*x) + sin(4*x)/2 + sin(6*x)/3 + sin(8*x)/4 + sin(10*x)/5 + sin(12*x)/6 + sin(14*x)/7 + sin(16*x)/8 +

\(\int \csc (x) \sin (23 x) \, dx\) [305]

Optimal result