Integrand size = 28, antiderivative size = 87 \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {2} \cos (x)}{\sqrt {\cos (2 x)}}\right )}{\sqrt {2}}-\frac {11 \cos (x)}{20 \cos ^{\frac {3}{2}}(2 x)}-\frac {2 \cos ^3(x)}{3 \cos ^{\frac {3}{2}}(2 x)}+\frac {63 \cos (x)}{20 \sqrt {\cos (2 x)}}+\frac {3 \cos (x) \sin ^2(x)}{10 \cos ^{\frac {5}{2}}(2 x)} \]
-11/20*cos(x)/cos(2*x)^(3/2)-2/3*cos(x)^3/cos(2*x)^(3/2)+3/10*cos(x)*sin(x )^2/cos(2*x)^(5/2)-1/2*arctanh(cos(x)*2^(1/2)/cos(2*x)^(1/2))*2^(1/2)+63/2 0*cos(x)/cos(2*x)^(1/2)
Time = 0.43 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=\frac {250 \cos (x)+45 \cos (3 x)+169 \cos (5 x)-120 \sqrt {2} \cos ^{\frac {5}{2}}(2 x) \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right )}{240 \cos ^{\frac {5}{2}}(2 x)} \]
(250*Cos[x] + 45*Cos[3*x] + 169*Cos[5*x] - 120*Sqrt[2]*Cos[2*x]^(5/2)*Log[ Sqrt[2]*Cos[x] + Sqrt[Cos[2*x]]])/(240*Cos[2*x]^(5/2))
Time = 0.54 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.80, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4877, 25, 3042, 4866, 292, 292, 208, 4879, 27, 357, 252, 224, 219}
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.
| \(\displaystyle \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\sin (4 x) \cos (x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^2 \left (3 \sin (x)^3-\sin (4 x) \cos (x)\right )}{\cos (2 x)^{7/2}}dx\) |
| \(\Big \downarrow \) 4877 |
| \(\displaystyle 3 \int \frac {\sin ^5(x)}{\cos ^{\frac {7}{2}}(2 x)}dx+\int -\frac {\cos (x) \sin ^2(x) \sin (4 x)}{\cos ^{\frac {7}{2}}(2 x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \int \frac {\sin ^5(x)}{\cos ^{\frac {7}{2}}(2 x)}dx-\int \frac {\cos (x) \sin ^2(x) \sin (4 x)}{\cos ^{\frac {7}{2}}(2 x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \int \frac {\sin (x)^5}{\cos (2 x)^{7/2}}dx-\int \frac {\cos (x) \sin (x)^2 \sin (4 x)}{\cos (2 x)^{7/2}}dx\) |
| \(\Big \downarrow \) 4866 |
| \(\displaystyle -3 \int \frac {\left (1-\cos ^2(x)\right )^2}{\left (2 \cos ^2(x)-1\right )^{7/2}}d\cos (x)-\int \frac {\cos (x) \sin (x)^2 \sin (4 x)}{\cos (2 x)^{7/2}}dx\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle -3 \left (-\frac {4}{5} \int \frac {1-\cos ^2(x)}{\left (2 \cos ^2(x)-1\right )^{5/2}}d\cos (x)-\frac {\cos (x) \left (1-\cos ^2(x)\right )^2}{5 \left (2 \cos ^2(x)-1\right )^{5/2}}\right )-\int \frac {\cos (x) \sin (x)^2 \sin (4 x)}{\cos (2 x)^{7/2}}dx\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle -3 \left (-\frac {4}{5} \left (-\frac {2}{3} \int \frac {1}{\left (2 \cos ^2(x)-1\right )^{3/2}}d\cos (x)-\frac {\cos (x) \left (1-\cos ^2(x)\right )}{3 \left (2 \cos ^2(x)-1\right )^{3/2}}\right )-\frac {\cos (x) \left (1-\cos ^2(x)\right )^2}{5 \left (2 \cos ^2(x)-1\right )^{5/2}}\right )-\int \frac {\cos (x) \sin (x)^2 \sin (4 x)}{\cos (2 x)^{7/2}}dx\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\int \frac {\cos (x) \sin (x)^2 \sin (4 x)}{\cos (2 x)^{7/2}}dx-3 \left (-\frac {\cos (x) \left (1-\cos ^2(x)\right )^2}{5 \left (2 \cos ^2(x)-1\right )^{5/2}}-\frac {4}{5} \left (\frac {2 \cos (x)}{3 \sqrt {2 \cos ^2(x)-1}}-\frac {\cos (x) \left (1-\cos ^2(x)\right )}{3 \left (2 \cos ^2(x)-1\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 4879 |
| \(\displaystyle \int \frac {4 \cos ^2(x) \left (1-\cos ^2(x)\right )}{\left (2 \cos ^2(x)-1\right )^{5/2}}d\cos (x)-3 \left (-\frac {\cos (x) \left (1-\cos ^2(x)\right )^2}{5 \left (2 \cos ^2(x)-1\right )^{5/2}}-\frac {4}{5} \left (\frac {2 \cos (x)}{3 \sqrt {2 \cos ^2(x)-1}}-\frac {\cos (x) \left (1-\cos ^2(x)\right )}{3 \left (2 \cos ^2(x)-1\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {\cos ^2(x) \left (1-\cos ^2(x)\right )}{\left (2 \cos ^2(x)-1\right )^{5/2}}d\cos (x)-3 \left (-\frac {\cos (x) \left (1-\cos ^2(x)\right )^2}{5 \left (2 \cos ^2(x)-1\right )^{5/2}}-\frac {4}{5} \left (\frac {2 \cos (x)}{3 \sqrt {2 \cos ^2(x)-1}}-\frac {\cos (x) \left (1-\cos ^2(x)\right )}{3 \left (2 \cos ^2(x)-1\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 357 |
| \(\displaystyle 4 \left (-\frac {1}{2} \int \frac {\cos ^2(x)}{\left (2 \cos ^2(x)-1\right )^{3/2}}d\cos (x)-\frac {\cos ^3(x)}{6 \left (2 \cos ^2(x)-1\right )^{3/2}}\right )-3 \left (-\frac {\cos (x) \left (1-\cos ^2(x)\right )^2}{5 \left (2 \cos ^2(x)-1\right )^{5/2}}-\frac {4}{5} \left (\frac {2 \cos (x)}{3 \sqrt {2 \cos ^2(x)-1}}-\frac {\cos (x) \left (1-\cos ^2(x)\right )}{3 \left (2 \cos ^2(x)-1\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle 4 \left (\frac {1}{2} \left (\frac {\cos (x)}{2 \sqrt {2 \cos ^2(x)-1}}-\frac {1}{2} \int \frac {1}{\sqrt {2 \cos ^2(x)-1}}d\cos (x)\right )-\frac {\cos ^3(x)}{6 \left (2 \cos ^2(x)-1\right )^{3/2}}\right )-3 \left (-\frac {\cos (x) \left (1-\cos ^2(x)\right )^2}{5 \left (2 \cos ^2(x)-1\right )^{5/2}}-\frac {4}{5} \left (\frac {2 \cos (x)}{3 \sqrt {2 \cos ^2(x)-1}}-\frac {\cos (x) \left (1-\cos ^2(x)\right )}{3 \left (2 \cos ^2(x)-1\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle 4 \left (\frac {1}{2} \left (\frac {\cos (x)}{2 \sqrt {2 \cos ^2(x)-1}}-\frac {1}{2} \int \frac {1}{1-\frac {2 \cos ^2(x)}{2 \cos ^2(x)-1}}d\frac {\cos (x)}{\sqrt {2 \cos ^2(x)-1}}\right )-\frac {\cos ^3(x)}{6 \left (2 \cos ^2(x)-1\right )^{3/2}}\right )-3 \left (-\frac {\cos (x) \left (1-\cos ^2(x)\right )^2}{5 \left (2 \cos ^2(x)-1\right )^{5/2}}-\frac {4}{5} \left (\frac {2 \cos (x)}{3 \sqrt {2 \cos ^2(x)-1}}-\frac {\cos (x) \left (1-\cos ^2(x)\right )}{3 \left (2 \cos ^2(x)-1\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 4 \left (\frac {1}{2} \left (\frac {\cos (x)}{2 \sqrt {2 \cos ^2(x)-1}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \cos (x)}{\sqrt {2 \cos ^2(x)-1}}\right )}{2 \sqrt {2}}\right )-\frac {\cos ^3(x)}{6 \left (2 \cos ^2(x)-1\right )^{3/2}}\right )-3 \left (-\frac {\cos (x) \left (1-\cos ^2(x)\right )^2}{5 \left (2 \cos ^2(x)-1\right )^{5/2}}-\frac {4}{5} \left (\frac {2 \cos (x)}{3 \sqrt {2 \cos ^2(x)-1}}-\frac {\cos (x) \left (1-\cos ^2(x)\right )}{3 \left (2 \cos ^2(x)-1\right )^{3/2}}\right )\right )\) |
4*(-1/6*Cos[x]^3/(-1 + 2*Cos[x]^2)^(3/2) + (-1/2*ArcTanh[(Sqrt[2]*Cos[x])/ Sqrt[-1 + 2*Cos[x]^2]]/Sqrt[2] + Cos[x]/(2*Sqrt[-1 + 2*Cos[x]^2]))/2) - 3* (-1/5*(Cos[x]*(1 - Cos[x]^2)^2)/(-1 + 2*Cos[x]^2)^(5/2) - (4*(-1/3*(Cos[x] *(1 - Cos[x]^2))/(-1 + 2*Cos[x]^2)^(3/2) + (2*Cos[x])/(3*Sqrt[-1 + 2*Cos[x ]^2])))/5)
3.5.33.3.1 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)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Si mp[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] - Simp[c*(q/( a*(p + 1))) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1), x], x] /; FreeQ[ {a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0] && Gt Q[q, 0] && NeQ[p, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_ Symbol] :> Simp[(b*c - a*d)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*b*e*(m + 1))), x] + Simp[d/b Int[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]
Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = Free Factors[Cos[c*(a + b*x)], x]}, Simp[-d/(b*c) Subst[Int[SubstFor[(1 - d^2* x^2)^((n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && Intege rQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sin] || EqQ[F, sin])
Int[(u_)*((v_) + (d_.)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_.)), x_Symbol] : > With[{e = FreeFactors[Cos[c*(a + b*x)], x]}, Int[ActivateTrig[u*v], x] + Simp[d Int[ActivateTrig[u]*Sin[c*(a + b*x)]^n, x], x] /; FunctionOfQ[Cos[ c*(a + b*x)]/e, u, x]] /; FreeQ[{a, b, c, d}, x] && !FreeQ[v, x] && Intege rQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sin] || EqQ[F, sin])
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa ctors[Cos[v], x]}, -d/Coefficient[v, x, 1] Subst[Int[SubstFor[1, Cos[v]/d , u/Sin[v], x], x], x, Cos[v]/d]], x] /; !FalseQ[v] && FunctionOfQ[Nonfree Factors[Cos[v], x], u/Sin[v], x]]
Leaf count of result is larger than twice the leaf count of optimal. \(179\) vs. \(2(65)=130\).
Time = 2.61 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.07
| method | result | size |
| default | \(-\frac {120 \sqrt {2}\, \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \left (\sin ^{6}\left (x \right )\right )+338 \sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \cos \left (x \right ) \left (\sin ^{4}\left (x \right )\right )-180 \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \sqrt {2}\, \left (\sin ^{4}\left (x \right )\right )-276 \sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \left (\sin ^{2}\left (x \right )\right ) \cos \left (x \right )+90 \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \sqrt {2}\, \left (\sin ^{2}\left (x \right )\right )+58 \sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \cos \left (x \right )-15 \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \sqrt {2}}{30 \left (8 \left (\sin ^{6}\left (x \right )\right )-12 \left (\sin ^{4}\left (x \right )\right )+6 \left (\sin ^{2}\left (x \right )\right )-1\right )}\) | \(180\) |
| parts | \(-\frac {\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \cos \left (x \right ) \left (43 \left (\sin ^{4}\left (x \right )\right )-36 \left (\sin ^{2}\left (x \right )\right )+8\right )}{5 \left (8 \left (\sin ^{6}\left (x \right )\right )-12 \left (\sin ^{4}\left (x \right )\right )+6 \left (\sin ^{2}\left (x \right )\right )-1\right )}-\frac {12 \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \sqrt {2}\, \left (\sin ^{4}\left (x \right )\right )+8 \sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \left (\sin ^{2}\left (x \right )\right ) \cos \left (x \right )-12 \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \sqrt {2}\, \left (\sin ^{2}\left (x \right )\right )-2 \sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \cos \left (x \right )+3 \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \sqrt {2}}{6 \left (4 \left (\sin ^{4}\left (x \right )\right )-4 \left (\sin ^{2}\left (x \right )\right )+1\right )}\) | \(180\) |
-1/30/(8*sin(x)^6-12*sin(x)^4+6*sin(x)^2-1)*(120*2^(1/2)*ln(cos(x)*2^(1/2) +(1-2*sin(x)^2)^(1/2))*sin(x)^6+338*(1-2*sin(x)^2)^(1/2)*cos(x)*sin(x)^4-1 80*ln(cos(x)*2^(1/2)+(1-2*sin(x)^2)^(1/2))*2^(1/2)*sin(x)^4-276*(1-2*sin(x )^2)^(1/2)*sin(x)^2*cos(x)+90*ln(cos(x)*2^(1/2)+(1-2*sin(x)^2)^(1/2))*2^(1 /2)*sin(x)^2+58*(1-2*sin(x)^2)^(1/2)*cos(x)-15*ln(cos(x)*2^(1/2)+(1-2*sin( x)^2)^(1/2))*2^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (65) = 130\).
Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.87 \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=\frac {15 \, {\left (8 \, \sqrt {2} \cos \left (x\right )^{6} - 12 \, \sqrt {2} \cos \left (x\right )^{4} + 6 \, \sqrt {2} \cos \left (x\right )^{2} - \sqrt {2}\right )} \log \left (2048 \, \cos \left (x\right )^{8} - 2048 \, \cos \left (x\right )^{6} + 640 \, \cos \left (x\right )^{4} - 64 \, \cos \left (x\right )^{2} - 8 \, {\left (128 \, \sqrt {2} \cos \left (x\right )^{7} - 96 \, \sqrt {2} \cos \left (x\right )^{5} + 20 \, \sqrt {2} \cos \left (x\right )^{3} - \sqrt {2} \cos \left (x\right )\right )} \sqrt {2 \, \cos \left (x\right )^{2} - 1} + 1\right ) + 16 \, {\left (169 \, \cos \left (x\right )^{5} - 200 \, \cos \left (x\right )^{3} + 60 \, \cos \left (x\right )\right )} \sqrt {2 \, \cos \left (x\right )^{2} - 1}}{240 \, {\left (8 \, \cos \left (x\right )^{6} - 12 \, \cos \left (x\right )^{4} + 6 \, \cos \left (x\right )^{2} - 1\right )}} \]
1/240*(15*(8*sqrt(2)*cos(x)^6 - 12*sqrt(2)*cos(x)^4 + 6*sqrt(2)*cos(x)^2 - sqrt(2))*log(2048*cos(x)^8 - 2048*cos(x)^6 + 640*cos(x)^4 - 64*cos(x)^2 - 8*(128*sqrt(2)*cos(x)^7 - 96*sqrt(2)*cos(x)^5 + 20*sqrt(2)*cos(x)^3 - sqr t(2)*cos(x))*sqrt(2*cos(x)^2 - 1) + 1) + 16*(169*cos(x)^5 - 200*cos(x)^3 + 60*cos(x))*sqrt(2*cos(x)^2 - 1))/(8*cos(x)^6 - 12*cos(x)^4 + 6*cos(x)^2 - 1)
Timed out. \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1359 vs. \(2 (65) = 130\).
Time = 0.41 (sec) , antiderivative size = 1359, normalized size of antiderivative = 15.62 \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=\text {Too large to display} \]
1/48*(4*(4*sqrt(2)*sin(4*x)*sin(5/2*arctan2(sin(4*x), cos(4*x))) + 4*(sqrt (2)*cos(4*x) + sqrt(2))*cos(5/2*arctan2(sin(4*x), cos(4*x))) + 3*sqrt(2)*c os(8*x) + 7*sqrt(2)*cos(4*x) + 4*sqrt(2))*cos(5/2*arctan2(sin(4*x), cos(4* x) + 1)) + 12*sqrt(2)*sqrt(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)*cos(3 /2*arctan2(sin(4*x), cos(4*x) + 1)) - 12*(sqrt(2)*cos(4*x)^2 + sqrt(2)*sin (4*x)^2 + 2*sqrt(2)*cos(4*x) + sqrt(2))*cos(1/2*arctan2(sin(4*x), cos(4*x) + 1)) - 4*(4*sqrt(2)*cos(5/2*arctan2(sin(4*x), cos(4*x)))*sin(4*x) - 4*(s qrt(2)*cos(4*x) + sqrt(2))*sin(5/2*arctan2(sin(4*x), cos(4*x))) - 3*sqrt(2 )*sin(8*x) - 7*sqrt(2)*sin(4*x))*sin(5/2*arctan2(sin(4*x), cos(4*x) + 1)) - 3*(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*((sqrt(2)*cos(4*x)^2 + sqrt(2)*sin(4*x)^2 + 2*sqrt(2)*cos(4*x) + sqrt(2))*log(sqrt(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)*cos(1/2*arctan2(sin(4*x), cos(4*x) + 1))^2 + sqrt(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)*sin(1/2*arctan2(sin(4*x), cos(4*x) + 1))^2 + 2*(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*cos( 1/2*arctan2(sin(4*x), cos(4*x) + 1)) + 1) - (sqrt(2)*cos(4*x)^2 + sqrt(2)* sin(4*x)^2 + 2*sqrt(2)*cos(4*x) + sqrt(2))*log(sqrt(cos(4*x)^2 + sin(4*x)^ 2 + 2*cos(4*x) + 1)*cos(1/2*arctan2(sin(4*x), cos(4*x) + 1))^2 + sqrt(cos( 4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)*sin(1/2*arctan2(sin(4*x), cos(4*x) + 1))^2 - 2*(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*cos(1/2*arctan 2(sin(4*x), cos(4*x) + 1)) + 1) + (sqrt(2)*cos(4*x)^2 + sqrt(2)*sin(4*x...
Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.63 \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left ({\left | -\sqrt {2} \cos \left (x\right ) + \sqrt {2 \, \cos \left (x\right )^{2} - 1} \right |}\right ) + \frac {{\left ({\left (169 \, \cos \left (x\right )^{2} - 200\right )} \cos \left (x\right )^{2} + 60\right )} \cos \left (x\right )}{15 \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}^{\frac {5}{2}}} \]
1/2*sqrt(2)*log(abs(-sqrt(2)*cos(x) + sqrt(2*cos(x)^2 - 1))) + 1/15*((169* cos(x)^2 - 200)*cos(x)^2 + 60)*cos(x)/(2*cos(x)^2 - 1)^(5/2)
Timed out. \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=\int \frac {{\sin \left (x\right )}^2\,\left (3\,{\sin \left (x\right )}^3-\sin \left (4\,x\right )\,\cos \left (x\right )\right )}{{\cos \left (2\,x\right )}^{7/2}} \,d x \]
\[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=-\left (\int \frac {\sqrt {\cos \left (2 x \right )}\, \cos \left (x \right ) \sin \left (4 x \right )}{\cos \left (2 x \right )^{4} \csc \left (x \right )^{2}}d x \right )+3 \left (\int \frac {\sqrt {\cos \left (2 x \right )}\, \sin \left (x \right )^{3}}{\cos \left (2 x \right )^{4} \csc \left (x \right )^{2}}d x \right ) \]