Integrand size = 47, antiderivative size = 64 \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=7 x+4 \sin (2 x)-\frac {1}{2} \sin (4 x)-\frac {4}{3} \sin (6 x)-\sin (8 x)-\frac {2}{5} \sin (10 x)+\frac {1}{6} \sin (12 x)+\frac {2}{7} \sin (14 x)+\frac {1}{8} \sin (16 x) \]
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)
Time = 0.48 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00 \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=7 x+4 \sin (2 x)-\frac {1}{2} \sin (4 x)-\frac {4}{3} \sin (6 x)-\sin (8 x)-\frac {2}{5} \sin (10 x)+\frac {1}{6} \sin (12 x)+\frac {2}{7} \sin (14 x)+\frac {1}{8} \sin (16 x) \]
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]
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
Leaf count is larger than twice the leaf count of optimal. \(136\) vs. \(2(64)=128\).
Time = 2.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.12, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.383, Rules used = {3042, 4889, 2345, 27, 2345, 27, 2345, 27, 2345, 27, 2345, 27, 2345, 27, 1471, 27, 298, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (2 x)^2 \sin (3 x)^2 \sin (5 x)^2 \sin (30 x)^2 \csc (x)^2 \csc (6 x)^2 \csc (10 x)^2 \csc (15 x)^2dx\) |
| \(\Big \downarrow \) 4889 |
| \(\displaystyle \int \frac {\left (\tan ^8(x)-28 \tan ^6(x)+134 \tan ^4(x)-92 \tan ^2(x)+1\right )^2}{\left (\tan ^2(x)+1\right )^9}d\tan (x)\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}-\frac {1}{16} \int \frac {16 \left (-\tan ^{14}(x)+57 \tan ^{12}(x)-1109 \tan ^{10}(x)+8797 \tan ^8(x)-31907 \tan ^6(x)+56619 \tan ^4(x)-65351 \tan ^2(x)+4095\right )}{\left (\tan ^2(x)+1\right )^8}d\tan (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}-\int \frac {-\tan ^{14}(x)+57 \tan ^{12}(x)-1109 \tan ^{10}(x)+8797 \tan ^8(x)-31907 \tan ^6(x)+56619 \tan ^4(x)-65351 \tan ^2(x)+4095}{\left (\tan ^2(x)+1\right )^8}d\tan (x)\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {1}{14} \int \frac {2 \left (7 \tan ^{12}(x)-406 \tan ^{10}(x)+8169 \tan ^8(x)-69748 \tan ^6(x)+293097 \tan ^4(x)-689430 \tan ^2(x)+55303\right )}{\left (\tan ^2(x)+1\right )^7}d\tan (x)-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {7 \tan ^{12}(x)-406 \tan ^{10}(x)+8169 \tan ^8(x)-69748 \tan ^6(x)+293097 \tan ^4(x)-689430 \tan ^2(x)+55303}{\left (\tan ^2(x)+1\right )^7}d\tan (x)-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {1}{7} \left (\frac {279040 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}-\frac {1}{12} \int \frac {4 \left (-21 \tan ^{10}(x)+1239 \tan ^8(x)-25746 \tan ^6(x)+234990 \tan ^4(x)-1114281 \tan ^2(x)+113131\right )}{\left (\tan ^2(x)+1\right )^6}d\tan (x)\right )-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {279040 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}-\frac {1}{3} \int \frac {-21 \tan ^{10}(x)+1239 \tan ^8(x)-25746 \tan ^6(x)+234990 \tan ^4(x)-1114281 \tan ^2(x)+113131}{\left (\tan ^2(x)+1\right )^6}d\tan (x)\right )-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{10} \int \frac {6 \left (35 \tan ^8(x)-2100 \tan ^6(x)+45010 \tan ^4(x)-436660 \tan ^2(x)+59683\right )}{\left (\tan ^2(x)+1\right )^5}d\tan (x)-\frac {744704 \tan (x)}{5 \left (\tan ^2(x)+1\right )^5}\right )+\frac {279040 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {3}{5} \int \frac {35 \tan ^8(x)-2100 \tan ^6(x)+45010 \tan ^4(x)-436660 \tan ^2(x)+59683}{\left (\tan ^2(x)+1\right )^5}d\tan (x)-\frac {744704 \tan (x)}{5 \left (\tan ^2(x)+1\right )^5}\right )+\frac {279040 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {3}{5} \left (\frac {67936 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-\frac {1}{8} \int \frac {56 \left (-5 \tan ^6(x)+305 \tan ^4(x)-6735 \tan ^2(x)+1179\right )}{\left (\tan ^2(x)+1\right )^4}d\tan (x)\right )-\frac {744704 \tan (x)}{5 \left (\tan ^2(x)+1\right )^5}\right )+\frac {279040 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {3}{5} \left (\frac {67936 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \int \frac {-5 \tan ^6(x)+305 \tan ^4(x)-6735 \tan ^2(x)+1179}{\left (\tan ^2(x)+1\right )^4}d\tan (x)\right )-\frac {744704 \tan (x)}{5 \left (\tan ^2(x)+1\right )^5}\right )+\frac {279040 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {3}{5} \left (\frac {67936 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \left (\frac {4112 \tan (x)}{3 \left (\tan ^2(x)+1\right )^3}-\frac {1}{6} \int \frac {10 \left (3 \tan ^4(x)-186 \tan ^2(x)+115\right )}{\left (\tan ^2(x)+1\right )^3}d\tan (x)\right )\right )-\frac {744704 \tan (x)}{5 \left (\tan ^2(x)+1\right )^5}\right )+\frac {279040 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {3}{5} \left (\frac {67936 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \left (\frac {4112 \tan (x)}{3 \left (\tan ^2(x)+1\right )^3}-\frac {5}{3} \int \frac {3 \tan ^4(x)-186 \tan ^2(x)+115}{\left (\tan ^2(x)+1\right )^3}d\tan (x)\right )\right )-\frac {744704 \tan (x)}{5 \left (\tan ^2(x)+1\right )^5}\right )+\frac {279040 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {3}{5} \left (\frac {67936 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \left (\frac {4112 \tan (x)}{3 \left (\tan ^2(x)+1\right )^3}-\frac {5}{3} \left (\frac {76 \tan (x)}{\left (\tan ^2(x)+1\right )^2}-\frac {1}{4} \int -\frac {12 \left (\tan ^2(x)+13\right )}{\left (\tan ^2(x)+1\right )^2}d\tan (x)\right )\right )\right )-\frac {744704 \tan (x)}{5 \left (\tan ^2(x)+1\right )^5}\right )+\frac {279040 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {3}{5} \left (\frac {67936 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \left (\frac {4112 \tan (x)}{3 \left (\tan ^2(x)+1\right )^3}-\frac {5}{3} \left (3 \int \frac {\tan ^2(x)+13}{\left (\tan ^2(x)+1\right )^2}d\tan (x)+\frac {76 \tan (x)}{\left (\tan ^2(x)+1\right )^2}\right )\right )\right )-\frac {744704 \tan (x)}{5 \left (\tan ^2(x)+1\right )^5}\right )+\frac {279040 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {3}{5} \left (\frac {67936 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \left (\frac {4112 \tan (x)}{3 \left (\tan ^2(x)+1\right )^3}-\frac {5}{3} \left (3 \left (7 \int \frac {1}{\tan ^2(x)+1}d\tan (x)+\frac {6 \tan (x)}{\tan ^2(x)+1}\right )+\frac {76 \tan (x)}{\left (\tan ^2(x)+1\right )^2}\right )\right )\right )-\frac {744704 \tan (x)}{5 \left (\tan ^2(x)+1\right )^5}\right )+\frac {279040 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {3}{5} \left (\frac {67936 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \left (\frac {4112 \tan (x)}{3 \left (\tan ^2(x)+1\right )^3}-\frac {5}{3} \left (3 \left (7 \arctan (\tan (x))+\frac {6 \tan (x)}{\tan ^2(x)+1}\right )+\frac {76 \tan (x)}{\left (\tan ^2(x)+1\right )^2}\right )\right )\right )-\frac {744704 \tan (x)}{5 \left (\tan ^2(x)+1\right )^5}\right )+\frac {279040 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )-\frac {83968 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}+\frac {4096 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\) |
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]
(4096*Tan[x])/(1 + Tan[x]^2)^8 - (83968*Tan[x])/(7*(1 + Tan[x]^2)^7) + ((2 79040*Tan[x])/(3*(1 + Tan[x]^2)^6) + ((-744704*Tan[x])/(5*(1 + Tan[x]^2)^5 ) + (3*((67936*Tan[x])/(1 + Tan[x]^2)^4 - 7*((4112*Tan[x])/(3*(1 + Tan[x]^ 2)^3) - (5*((76*Tan[x])/(1 + Tan[x]^2)^2 + 3*(7*ArcTan[Tan[x]] + (6*Tan[x] )/(1 + Tan[x]^2))))/3)))/5)/3)/7
3.4.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(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] + Simp[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]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[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] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors [Tan[v], x]}, Simp[d/Coefficient[v, x, 1] Subst[Int[SubstFor[1/(1 + d^2*x ^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /; !FalseQ[v] && FunctionOfQ[N onfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] && !MatchQ[ u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && I ntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
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
\[4096 \left (\cos \left (x \right )^{15}+\frac {15 \cos \left (x \right )^{13}}{14}+\frac {65 \cos \left (x \right )^{11}}{56}+\frac {143 \cos \left (x \right )^{9}}{112}+\frac {1287 \cos \left (x \right )^{7}}{896}+\frac {429 \cos \left (x \right )^{5}}{256}+\frac {2145 \cos \left (x \right )^{3}}{1024}+\frac {6435 \cos \left (x \right )}{2048}\right ) \sin \left (x \right )+7 x -16384 \left (\cos \left (x \right )^{13}+\frac {13 \cos \left (x \right )^{11}}{12}+\frac {143 \cos \left (x \right )^{9}}{120}+\frac {429 \cos \left (x \right )^{7}}{320}+\frac {1001 \cos \left (x \right )^{5}}{640}+\frac {1001 \cos \left (x \right )^{3}}{512}+\frac {3003 \cos \left (x \right )}{1024}\right ) \sin \left (x \right )+\frac {78848 \left (\cos \left (x \right )^{11}+\frac {11 \cos \left (x \right )^{9}}{10}+\frac {99 \cos \left (x \right )^{7}}{80}+\frac {231 \cos \left (x \right )^{5}}{160}+\frac {231 \cos \left (x \right )^{3}}{128}+\frac {693 \cos \left (x \right )}{256}\right ) \sin \left (x \right )}{3}-\frac {108544 \left (\cos \left (x \right )^{9}+\frac {9 \cos \left (x \right )^{7}}{8}+\frac {21 \cos \left (x \right )^{5}}{16}+\frac {105 \cos \left (x \right )^{3}}{64}+\frac {315 \cos \left (x \right )}{128}\right ) \sin \left (x \right )}{5}+9920 \left (\cos \left (x \right )^{7}+\frac {7 \cos \left (x \right )^{5}}{6}+\frac {35 \cos \left (x \right )^{3}}{24}+\frac {35 \cos \left (x \right )}{16}\right ) \sin \left (x \right )-\frac {7616 \left (\cos \left (x \right )^{5}+\frac {5 \cos \left (x \right )^{3}}{4}+\frac {15 \cos \left (x \right )}{8}\right ) \sin \left (x \right )}{3}+368 \left (\cos \left (x \right )^{3}+\frac {3 \cos \left (x \right )}{2}\right ) \sin \left (x \right )-32 \cos \left (x \right ) \sin \left (x \right )\]
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(1 0*x)^2/sin(15*x)^2,x)
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+1001/64 0*cos(x)^5+1001/512*cos(x)^3+3003/1024*cos(x))*sin(x)+78848/3*(cos(x)^11+1 1/10*cos(x)^9+99/80*cos(x)^7+231/160*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*c os(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)
Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=\frac {2}{105} \, {\left (215040 \, \cos \left (x\right )^{15} - 629760 \, \cos \left (x\right )^{13} + 697600 \, \cos \left (x\right )^{11} - 372352 \, \cos \left (x\right )^{9} + 101904 \, \cos \left (x\right )^{7} - 14392 \, \cos \left (x\right )^{5} + 1330 \, \cos \left (x\right )^{3} + 315 \, \cos \left (x\right )\right )} \sin \left (x\right ) + 7 \, x \]
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, algorithm="fricas")
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
Timed out. \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=\text {Timed out} \]
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)
Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.81 \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=7 \, x + \frac {1}{8} \, \sin \left (16 \, x\right ) + \frac {2}{7} \, \sin \left (14 \, x\right ) + \frac {1}{6} \, \sin \left (12 \, x\right ) - \frac {2}{5} \, \sin \left (10 \, x\right ) - \sin \left (8 \, x\right ) - \frac {4}{3} \, \sin \left (6 \, x\right ) - \frac {1}{2} \, \sin \left (4 \, x\right ) + 4 \, \sin \left (2 \, x\right ) \]
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, algorithm="maxima")
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)
Time = 37.52 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=7 \, x + \frac {2 \, {\left (315 \, \tan \left (x\right )^{15} + 3535 \, \tan \left (x\right )^{13} + 203 \, \tan \left (x\right )^{11} + 60919 \, \tan \left (x\right )^{9} - 71031 \, \tan \left (x\right )^{7} + 74613 \, \tan \left (x\right )^{5} - 5775 \, \tan \left (x\right )^{3} - 315 \, \tan \left (x\right )\right )}}{105 \, {\left (\tan \left (x\right )^{2} + 1\right )}^{8}} \]
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, algorithm="giac")
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
Time = 19.38 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.03 \[ \int \csc ^2(x) \csc ^2(6 x) \csc ^2(10 x) \csc ^2(15 x) \sin ^2(2 x) \sin ^2(3 x) \sin ^2(5 x) \sin ^2(30 x) \, dx=4096\,\sin \left (x\right )\,{\cos \left (x\right )}^{15}-\frac {83968\,\sin \left (x\right )\,{\cos \left (x\right )}^{13}}{7}+\frac {279040\,\sin \left (x\right )\,{\cos \left (x\right )}^{11}}{21}-\frac {744704\,\sin \left (x\right )\,{\cos \left (x\right )}^9}{105}+\frac {67936\,\sin \left (x\right )\,{\cos \left (x\right )}^7}{35}-\frac {4112\,\sin \left (x\right )\,{\cos \left (x\right )}^5}{15}+\frac {76\,\sin \left (x\right )\,{\cos \left (x\right )}^3}{3}+6\,\sin \left (x\right )\,\cos \left (x\right )+7\,x \]
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)