\(\int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx\) [319]

Optimal result

Integrand size = 47, antiderivative size = 64 \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=7 x+4 \sin (2 x)-\frac {1}{2} \sin (4 x)-\frac {4}{3} \sin (6 x)-\sin (8 x)-\frac {2}{5} \sin (10 x)+\frac {1}{6} \sin (12 x)+\frac {2}{7} \sin (14 x)+\frac {1}{8} \sin (16 x) \]

[Out]

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

Rubi [A] (verified)

Time = 2.20 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {1828, 1171, 393, 209} \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=7 x+4096 \sin (x) \cos ^{15}(x)-\frac {83968}{7} \sin (x) \cos ^{13}(x)+\frac {279040}{21} \sin (x) \cos ^{11}(x)-\frac {744704}{105} \sin (x) \cos ^9(x)+\frac {67936}{35} \sin (x) \cos ^7(x)-\frac {4112}{15} \sin (x) \cos ^5(x)+\frac {76}{3} \sin (x) \cos ^3(x)+6 \sin (x) \cos (x) \]

[In]

Int[Csc[x]^2*Csc[6*x]^2*Csc[10*x]^2*Csc[15*x]^2*Sin[2*x]^2*Sin[3*x]^2*Sin[5*x]^2*Sin[30*x]^2,x]

[Out]

7*x + 6*Cos[x]*Sin[x] + (76*Cos[x]^3*Sin[x])/3 - (4112*Cos[x]^5*Sin[x])/15 + (67936*Cos[x]^7*Sin[x])/35 - (744
704*Cos[x]^9*Sin[x])/105 + (279040*Cos[x]^11*Sin[x])/21 - (83968*Cos[x]^13*Sin[x])/7 + 4096*Cos[x]^15*Sin[x]

Rule 209

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] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 393

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])

Rule 1171

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] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 1828

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 {\left (1-92 x^2+134 x^4-28 x^6+x^8\right )^2}{\left (1+x^2\right )^9} \, dx,x,\tan (x)\right ) \\ & = 4096 \cos ^{15}(x) \sin (x)-\frac {1}{16} \text {Subst}\left (\int \frac {65520-1045616 x^2+905904 x^4-510512 x^6+140752 x^8-17744 x^{10}+912 x^{12}-16 x^{14}}{\left (1+x^2\right )^8} \, dx,x,\tan (x)\right ) \\ & = -\frac {83968}{7} \cos ^{13}(x) \sin (x)+4096 \cos ^{15}(x) \sin (x)+\frac {1}{224} \text {Subst}\left (\int \frac {1769696-22061760 x^2+9379104 x^4-2231936 x^6+261408 x^8-12992 x^{10}+224 x^{12}}{\left (1+x^2\right )^7} \, dx,x,\tan (x)\right ) \\ & = \frac {279040}{21} \cos ^{11}(x) \sin (x)-\frac {83968}{7} \cos ^{13}(x) \sin (x)+4096 \cos ^{15}(x) \sin (x)-\frac {\text {Subst}\left (\int \frac {14480768-142627968 x^2+30078720 x^4-3295488 x^6+158592 x^8-2688 x^{10}}{\left (1+x^2\right )^6} \, dx,x,\tan (x)\right )}{2688} \\ & = -\frac {744704}{105} \cos ^9(x) \sin (x)+\frac {279040}{21} \cos ^{11}(x) \sin (x)-\frac {83968}{7} \cos ^{13}(x) \sin (x)+4096 \cos ^{15}(x) \sin (x)+\frac {\text {Subst}\left (\int \frac {45836544-335354880 x^2+34567680 x^4-1612800 x^6+26880 x^8}{\left (1+x^2\right )^5} \, dx,x,\tan (x)\right )}{26880} \\ & = \frac {67936}{35} \cos ^7(x) \sin (x)-\frac {744704}{105} \cos ^9(x) \sin (x)+\frac {279040}{21} \cos ^{11}(x) \sin (x)-\frac {83968}{7} \cos ^{13}(x) \sin (x)+4096 \cos ^{15}(x) \sin (x)-\frac {\text {Subst}\left (\int \frac {50706432-289658880 x^2+13117440 x^4-215040 x^6}{\left (1+x^2\right )^4} \, dx,x,\tan (x)\right )}{215040} \\ & = -\frac {4112}{15} \cos ^5(x) \sin (x)+\frac {67936}{35} \cos ^7(x) \sin (x)-\frac {744704}{105} \cos ^9(x) \sin (x)+\frac {279040}{21} \cos ^{11}(x) \sin (x)-\frac {83968}{7} \cos ^{13}(x) \sin (x)+4096 \cos ^{15}(x) \sin (x)+\frac {\text {Subst}\left (\int \frac {49459200-79994880 x^2+1290240 x^4}{\left (1+x^2\right )^3} \, dx,x,\tan (x)\right )}{1290240} \\ & = \frac {76}{3} \cos ^3(x) \sin (x)-\frac {4112}{15} \cos ^5(x) \sin (x)+\frac {67936}{35} \cos ^7(x) \sin (x)-\frac {744704}{105} \cos ^9(x) \sin (x)+\frac {279040}{21} \cos ^{11}(x) \sin (x)-\frac {83968}{7} \cos ^{13}(x) \sin (x)+4096 \cos ^{15}(x) \sin (x)-\frac {\text {Subst}\left (\int \frac {-67092480-5160960 x^2}{\left (1+x^2\right )^2} \, dx,x,\tan (x)\right )}{5160960} \\ & = 6 \cos (x) \sin (x)+\frac {76}{3} \cos ^3(x) \sin (x)-\frac {4112}{15} \cos ^5(x) \sin (x)+\frac {67936}{35} \cos ^7(x) \sin (x)-\frac {744704}{105} \cos ^9(x) \sin (x)+\frac {279040}{21} \cos ^{11}(x) \sin (x)-\frac {83968}{7} \cos ^{13}(x) \sin (x)+4096 \cos ^{15}(x) \sin (x)+7 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = 7 x+6 \cos (x) \sin (x)+\frac {76}{3} \cos ^3(x) \sin (x)-\frac {4112}{15} \cos ^5(x) \sin (x)+\frac {67936}{35} \cos ^7(x) \sin (x)-\frac {744704}{105} \cos ^9(x) \sin (x)+\frac {279040}{21} \cos ^{11}(x) \sin (x)-\frac {83968}{7} \cos ^{13}(x) \sin (x)+4096 \cos ^{15}(x) \sin (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00 \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=7 x+4 \sin (2 x)-\frac {1}{2} \sin (4 x)-\frac {4}{3} \sin (6 x)-\sin (8 x)-\frac {2}{5} \sin (10 x)+\frac {1}{6} \sin (12 x)+\frac {2}{7} \sin (14 x)+\frac {1}{8} \sin (16 x) \]

[In]

Integrate[Csc[x]^2*Csc[6*x]^2*Csc[10*x]^2*Csc[15*x]^2*Sin[2*x]^2*Sin[3*x]^2*Sin[5*x]^2*Sin[30*x]^2,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(227\) vs. \(2(52)=104\).

Time = 0.74 (sec) , antiderivative size = 228, normalized size of antiderivative = 3.56

\[4096 \left (\cos \left (x \right )^{15}+\frac {15 \cos \left (x \right )^{13}}{14}+\frac {65 \cos \left (x \right )^{11}}{56}+\frac {143 \cos \left (x \right )^{9}}{112}+\frac {1287 \cos \left (x \right )^{7}}{896}+\frac {429 \cos \left (x \right )^{5}}{256}+\frac {2145 \cos \left (x \right )^{3}}{1024}+\frac {6435 \cos \left (x \right )}{2048}\right ) \sin \left (x \right )+7 x -16384 \left (\cos \left (x \right )^{13}+\frac {13 \cos \left (x \right )^{11}}{12}+\frac {143 \cos \left (x \right )^{9}}{120}+\frac {429 \cos \left (x \right )^{7}}{320}+\frac {1001 \cos \left (x \right )^{5}}{640}+\frac {1001 \cos \left (x \right )^{3}}{512}+\frac {3003 \cos \left (x \right )}{1024}\right ) \sin \left (x \right )+\frac {78848 \left (\cos \left (x \right )^{11}+\frac {11 \cos \left (x \right )^{9}}{10}+\frac {99 \cos \left (x \right )^{7}}{80}+\frac {231 \cos \left (x \right )^{5}}{160}+\frac {231 \cos \left (x \right )^{3}}{128}+\frac {693 \cos \left (x \right )}{256}\right ) \sin \left (x \right )}{3}-\frac {108544 \left (\cos \left (x \right )^{9}+\frac {9 \cos \left (x \right )^{7}}{8}+\frac {21 \cos \left (x \right )^{5}}{16}+\frac {105 \cos \left (x \right )^{3}}{64}+\frac {315 \cos \left (x \right )}{128}\right ) \sin \left (x \right )}{5}+9920 \left (\cos \left (x \right )^{7}+\frac {7 \cos \left (x \right )^{5}}{6}+\frac {35 \cos \left (x \right )^{3}}{24}+\frac {35 \cos \left (x \right )}{16}\right ) \sin \left (x \right )-\frac {7616 \left (\cos \left (x \right )^{5}+\frac {5 \cos \left (x \right )^{3}}{4}+\frac {15 \cos \left (x \right )}{8}\right ) \sin \left (x \right )}{3}+368 \left (\cos \left (x \right )^{3}+\frac {3 \cos \left (x \right )}{2}\right ) \sin \left (x \right )-32 \cos \left (x \right ) \sin \left (x \right )\]

[In]

int(sin(2*x)^2*sin(3*x)^2*sin(5*x)^2*sin(30*x)^2/sin(x)^2/sin(6*x)^2/sin(10*x)^2/sin(15*x)^2,x)

[Out]

4096*(cos(x)^15+15/14*cos(x)^13+65/56*cos(x)^11+143/112*cos(x)^9+1287/896*cos(x)^7+429/256*cos(x)^5+2145/1024*
cos(x)^3+6435/2048*cos(x))*sin(x)+7*x-16384*(cos(x)^13+13/12*cos(x)^11+143/120*cos(x)^9+429/320*cos(x)^7+1001/
640*cos(x)^5+1001/512*cos(x)^3+3003/1024*cos(x))*sin(x)+78848/3*(cos(x)^11+11/10*cos(x)^9+99/80*cos(x)^7+231/1
60*cos(x)^5+231/128*cos(x)^3+693/256*cos(x))*sin(x)-108544/5*(cos(x)^9+9/8*cos(x)^7+21/16*cos(x)^5+105/64*cos(
x)^3+315/128*cos(x))*sin(x)+9920*(cos(x)^7+7/6*cos(x)^5+35/24*cos(x)^3+35/16*cos(x))*sin(x)-7616/3*(cos(x)^5+5
/4*cos(x)^3+15/8*cos(x))*sin(x)+368*(cos(x)^3+3/2*cos(x))*sin(x)-32*cos(x)*sin(x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=\frac {2}{105} \, {\left (215040 \, \cos \left (x\right )^{15} - 629760 \, \cos \left (x\right )^{13} + 697600 \, \cos \left (x\right )^{11} - 372352 \, \cos \left (x\right )^{9} + 101904 \, \cos \left (x\right )^{7} - 14392 \, \cos \left (x\right )^{5} + 1330 \, \cos \left (x\right )^{3} + 315 \, \cos \left (x\right )\right )} \sin \left (x\right ) + 7 \, x \]

[In]

integrate(sin(2*x)^2*sin(3*x)^2*sin(5*x)^2*sin(30*x)^2/sin(x)^2/sin(6*x)^2/sin(10*x)^2/sin(15*x)^2,x, algorith
m="fricas")

[Out]

2/105*(215040*cos(x)^15 - 629760*cos(x)^13 + 697600*cos(x)^11 - 372352*cos(x)^9 + 101904*cos(x)^7 - 14392*cos(
x)^5 + 1330*cos(x)^3 + 315*cos(x))*sin(x) + 7*x

Sympy [F(-1)]

Timed out. \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=\text {Timed out} \]

[In]

integrate(sin(2*x)**2*sin(3*x)**2*sin(5*x)**2*sin(30*x)**2/sin(x)**2/sin(6*x)**2/sin(10*x)**2/sin(15*x)**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.81 \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=7 \, x + \frac {1}{8} \, \sin \left (16 \, x\right ) + \frac {2}{7} \, \sin \left (14 \, x\right ) + \frac {1}{6} \, \sin \left (12 \, x\right ) - \frac {2}{5} \, \sin \left (10 \, x\right ) - \sin \left (8 \, x\right ) - \frac {4}{3} \, \sin \left (6 \, x\right ) - \frac {1}{2} \, \sin \left (4 \, x\right ) + 4 \, \sin \left (2 \, x\right ) \]

[In]

integrate(sin(2*x)^2*sin(3*x)^2*sin(5*x)^2*sin(30*x)^2/sin(x)^2/sin(6*x)^2/sin(10*x)^2/sin(15*x)^2,x, algorith
m="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 37.52 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=7 \, x + \frac {2 \, {\left (315 \, \tan \left (x\right )^{15} + 3535 \, \tan \left (x\right )^{13} + 203 \, \tan \left (x\right )^{11} + 60919 \, \tan \left (x\right )^{9} - 71031 \, \tan \left (x\right )^{7} + 74613 \, \tan \left (x\right )^{5} - 5775 \, \tan \left (x\right )^{3} - 315 \, \tan \left (x\right )\right )}}{105 \, {\left (\tan \left (x\right )^{2} + 1\right )}^{8}} \]

[In]

integrate(sin(2*x)^2*sin(3*x)^2*sin(5*x)^2*sin(30*x)^2/sin(x)^2/sin(6*x)^2/sin(10*x)^2/sin(15*x)^2,x, algorith
m="giac")

[Out]

7*x + 2/105*(315*tan(x)^15 + 3535*tan(x)^13 + 203*tan(x)^11 + 60919*tan(x)^9 - 71031*tan(x)^7 + 74613*tan(x)^5
- 5775*tan(x)^3 - 315*tan(x))/(tan(x)^2 + 1)^8

Mupad [B] (verification not implemented)

Time = 19.38 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.03 \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=4096\,\sin \left (x\right )\,{\cos \left (x\right )}^{15}-\frac {83968\,\sin \left (x\right )\,{\cos \left (x\right )}^{13}}{7}+\frac {279040\,\sin \left (x\right )\,{\cos \left (x\right )}^{11}}{21}-\frac {744704\,\sin \left (x\right )\,{\cos \left (x\right )}^9}{105}+\frac {67936\,\sin \left (x\right )\,{\cos \left (x\right )}^7}{35}-\frac {4112\,\sin \left (x\right )\,{\cos \left (x\right )}^5}{15}+\frac {76\,\sin \left (x\right )\,{\cos \left (x\right )}^3}{3}+6\,\sin \left (x\right )\,\cos \left (x\right )+7\,x \]

[In]

int((sin(2*x)^2*sin(3*x)^2*sin(5*x)^2*sin(30*x)^2)/(sin(6*x)^2*sin(10*x)^2*sin(15*x)^2*sin(x)^2),x)

[Out]

7*x + 6*cos(x)*sin(x) + (76*cos(x)^3*sin(x))/3 - (4112*cos(x)^5*sin(x))/15 + (67936*cos(x)^7*sin(x))/35 - (744
704*cos(x)^9*sin(x))/105 + (279040*cos(x)^11*sin(x))/21 - (83968*cos(x)^13*sin(x))/7 + 4096*cos(x)^15*sin(x)