Integrand size = 10, antiderivative size = 66 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^4} \, dx=-\frac {279 \text {arctanh}\left (\frac {\tan (x)}{2}\right )}{2048}+\frac {5 \cos (x) \sin (x)}{24 \left (4-5 \sin ^2(x)\right )^3}-\frac {25 \cos (x) \sin (x)}{128 \left (4-5 \sin ^2(x)\right )^2}+\frac {995 \cos (x) \sin (x)}{3072 \left (4-5 \sin ^2(x)\right )} \] Output:
-279/2048*arctanh(1/2*tan(x))+5/24*cos(x)*sin(x)/(4-5*sin(x)^2)^3-25/128*c os(x)*sin(x)/(4-5*sin(x)^2)^2+995*cos(x)*sin(x)/(12288-15360*sin(x)^2)
Time = 5.16 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^4} \, dx=\frac {837 \log (2 \cos (x)-\sin (x))-837 \log (2 \cos (x)+\sin (x))+\frac {10 (31454 \cos (x)+17075 \cos (3 x)+4975 \cos (5 x)) \sin (x)}{(3+5 \cos (2 x))^3}-\frac {320}{(-2 \cos (x)+\sin (x))^2}+\frac {320}{(2 \cos (x)+\sin (x))^2}}{12288} \] Input:
Integrate[(4 - 5*Sin[x]^2)^(-4),x]
Output:
(837*Log[2*Cos[x] - Sin[x]] - 837*Log[2*Cos[x] + Sin[x]] + (10*(31454*Cos[ x] + 17075*Cos[3*x] + 4975*Cos[5*x])*Sin[x])/(3 + 5*Cos[2*x])^3 - 320/(-2* Cos[x] + Sin[x])^2 + 320/(2*Cos[x] + Sin[x])^2)/12288
Time = 0.45 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {3042, 3663, 3042, 3652, 25, 3042, 3652, 27, 3042, 3660, 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.
| \(\displaystyle \int \frac {1}{\left (4-5 \sin ^2(x)\right )^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (4-5 \sin (x)^2\right )^4}dx\) |
| \(\Big \downarrow \) 3663 |
| \(\displaystyle \frac {1}{24} \int \frac {1-20 \sin ^2(x)}{\left (4-5 \sin ^2(x)\right )^3}dx+\frac {5 \sin (x) \cos (x)}{24 \left (4-5 \sin ^2(x)\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{24} \int \frac {1-20 \sin (x)^2}{\left (4-5 \sin (x)^2\right )^3}dx+\frac {5 \sin (x) \cos (x)}{24 \left (4-5 \sin ^2(x)\right )^3}\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {1}{24} \left (-\frac {1}{16} \int -\frac {150 \sin ^2(x)+79}{\left (4-5 \sin ^2(x)\right )^2}dx-\frac {75 \sin (x) \cos (x)}{16 \left (4-5 \sin ^2(x)\right )^2}\right )+\frac {5 \sin (x) \cos (x)}{24 \left (4-5 \sin ^2(x)\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{16} \int \frac {150 \sin ^2(x)+79}{\left (4-5 \sin ^2(x)\right )^2}dx-\frac {75 \sin (x) \cos (x)}{16 \left (4-5 \sin ^2(x)\right )^2}\right )+\frac {5 \sin (x) \cos (x)}{24 \left (4-5 \sin ^2(x)\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{16} \int \frac {150 \sin (x)^2+79}{\left (4-5 \sin (x)^2\right )^2}dx-\frac {75 \sin (x) \cos (x)}{16 \left (4-5 \sin ^2(x)\right )^2}\right )+\frac {5 \sin (x) \cos (x)}{24 \left (4-5 \sin ^2(x)\right )^3}\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{16} \left (\frac {995 \sin (x) \cos (x)}{8 \left (4-5 \sin ^2(x)\right )}-\frac {1}{8} \int \frac {837}{4-5 \sin ^2(x)}dx\right )-\frac {75 \sin (x) \cos (x)}{16 \left (4-5 \sin ^2(x)\right )^2}\right )+\frac {5 \sin (x) \cos (x)}{24 \left (4-5 \sin ^2(x)\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{16} \left (\frac {995 \sin (x) \cos (x)}{8 \left (4-5 \sin ^2(x)\right )}-\frac {837}{8} \int \frac {1}{4-5 \sin ^2(x)}dx\right )-\frac {75 \sin (x) \cos (x)}{16 \left (4-5 \sin ^2(x)\right )^2}\right )+\frac {5 \sin (x) \cos (x)}{24 \left (4-5 \sin ^2(x)\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{16} \left (\frac {995 \sin (x) \cos (x)}{8 \left (4-5 \sin ^2(x)\right )}-\frac {837}{8} \int \frac {1}{4-5 \sin (x)^2}dx\right )-\frac {75 \sin (x) \cos (x)}{16 \left (4-5 \sin ^2(x)\right )^2}\right )+\frac {5 \sin (x) \cos (x)}{24 \left (4-5 \sin ^2(x)\right )^3}\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{16} \left (\frac {995 \sin (x) \cos (x)}{8 \left (4-5 \sin ^2(x)\right )}-\frac {837}{8} \int \frac {1}{4-\tan ^2(x)}d\tan (x)\right )-\frac {75 \sin (x) \cos (x)}{16 \left (4-5 \sin ^2(x)\right )^2}\right )+\frac {5 \sin (x) \cos (x)}{24 \left (4-5 \sin ^2(x)\right )^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{16} \left (\frac {995 \sin (x) \cos (x)}{8 \left (4-5 \sin ^2(x)\right )}-\frac {837}{16} \text {arctanh}\left (\frac {\tan (x)}{2}\right )\right )-\frac {75 \sin (x) \cos (x)}{16 \left (4-5 \sin ^2(x)\right )^2}\right )+\frac {5 \sin (x) \cos (x)}{24 \left (4-5 \sin ^2(x)\right )^3}\) |
Input:
Int[(4 - 5*Sin[x]^2)^(-4),x]
Output:
(5*Cos[x]*Sin[x])/(24*(4 - 5*Sin[x]^2)^3) + ((-75*Cos[x]*Sin[x])/(16*(4 - 5*Sin[x]^2)^2) + ((-837*ArcTanh[Tan[x]/2])/16 + (995*Cos[x]*Sin[x])/(8*(4 - 5*Sin[x]^2)))/16)/24
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]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b - a*B))*Cos[e + f*x]*Sin[e + f*x ]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(a + b)*(p + 1))), x] - Simp[1/(2* a*(a + b)*(p + 1)) Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[a*B - A*(2*a*( p + 1) + b*(2*p + 3)) + 2*(A*b - a*B)*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[1/(a + (a + b)*ff^2*x^ 2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(p + 1)*(a + b))), x] + Simp[1/(2*a*(p + 1)*(a + b)) Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && LtQ[p, -1]
Time = 0.64 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(-\frac {125}{768 \left (\tan \left (x \right )+2\right )^{3}}+\frac {175}{512 \left (\tan \left (x \right )+2\right )^{2}}-\frac {745}{2048 \left (\tan \left (x \right )+2\right )}-\frac {279 \ln \left (\tan \left (x \right )+2\right )}{4096}-\frac {125}{768 \left (\tan \left (x \right )-2\right )^{3}}-\frac {175}{512 \left (\tan \left (x \right )-2\right )^{2}}-\frac {745}{2048 \left (\tan \left (x \right )-2\right )}+\frac {279 \ln \left (\tan \left (x \right )-2\right )}{4096}\) | \(64\) |
| risch | \(\frac {i \left (20925 \,{\mathrm e}^{10 i x}+62775 \,{\mathrm e}^{8 i x}+111042 \,{\mathrm e}^{6 i x}+119310 \,{\mathrm e}^{4 i x}+68625 \,{\mathrm e}^{2 i x}+24875\right )}{1536 \left (5 \,{\mathrm e}^{4 i x}+6 \,{\mathrm e}^{2 i x}+5\right )^{3}}-\frac {279 \ln \left ({\mathrm e}^{2 i x}+\frac {3}{5}+\frac {4 i}{5}\right )}{4096}+\frac {279 \ln \left ({\mathrm e}^{2 i x}+\frac {3}{5}-\frac {4 i}{5}\right )}{4096}\) | \(84\) |
| norman | \(\frac {-\frac {6545 \tan \left (\frac {x}{2}\right )^{3}}{6144}+\frac {5815 \tan \left (\frac {x}{2}\right )^{5}}{2048}-\frac {5815 \tan \left (\frac {x}{2}\right )^{7}}{2048}+\frac {6545 \tan \left (\frac {x}{2}\right )^{9}}{6144}-\frac {295 \tan \left (\frac {x}{2}\right )^{11}}{2048}+\frac {295 \tan \left (\frac {x}{2}\right )}{2048}}{\left (\tan \left (\frac {x}{2}\right )^{4}-3 \tan \left (\frac {x}{2}\right )^{2}+1\right )^{3}}-\frac {279 \ln \left (\tan \left (\frac {x}{2}\right )^{2}-\tan \left (\frac {x}{2}\right )-1\right )}{4096}+\frac {279 \ln \left (\tan \left (\frac {x}{2}\right )^{2}+\tan \left (\frac {x}{2}\right )-1\right )}{4096}\) | \(100\) |
| parallelrisch | \(\frac {\left (-467046-104625 \cos \left (6 x \right )-376650 \cos \left (4 x \right )-765855 \cos \left (2 x \right )\right ) \ln \left (\tan \left (\frac {x}{2}\right )^{2}-\tan \left (\frac {x}{2}\right )-1\right )+\left (467046+104625 \cos \left (6 x \right )+376650 \cos \left (4 x \right )+765855 \cos \left (2 x \right )\right ) \ln \left (\tan \left (\frac {x}{2}\right )^{2}+\tan \left (\frac {x}{2}\right )-1\right )+226140 \sin \left (2 x \right )+190800 \sin \left (4 x \right )+99500 \sin \left (6 x \right )}{1536000 \cos \left (6 x \right )+11243520 \cos \left (2 x \right )+5529600 \cos \left (4 x \right )+6856704}\) | \(113\) |
Input:
int(1/(4-5*sin(x)^2)^4,x,method=_RETURNVERBOSE)
Output:
-125/768/(tan(x)+2)^3+175/512/(tan(x)+2)^2-745/2048/(tan(x)+2)-279/4096*ln (tan(x)+2)-125/768/(tan(x)-2)^3-175/512/(tan(x)-2)^2-745/2048/(tan(x)-2)+2 79/4096*ln(tan(x)-2)
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (56) = 112\).
Time = 0.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.80 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^4} \, dx=-\frac {837 \, {\left (125 \, \cos \left (x\right )^{6} - 75 \, \cos \left (x\right )^{4} + 15 \, \cos \left (x\right )^{2} - 1\right )} \log \left (\frac {3}{4} \, \cos \left (x\right )^{2} + \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{4}\right ) - 837 \, {\left (125 \, \cos \left (x\right )^{6} - 75 \, \cos \left (x\right )^{4} + 15 \, \cos \left (x\right )^{2} - 1\right )} \log \left (\frac {3}{4} \, \cos \left (x\right )^{2} - \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{4}\right ) - 40 \, {\left (4975 \, \cos \left (x\right )^{5} - 2590 \, \cos \left (x\right )^{3} + 447 \, \cos \left (x\right )\right )} \sin \left (x\right )}{24576 \, {\left (125 \, \cos \left (x\right )^{6} - 75 \, \cos \left (x\right )^{4} + 15 \, \cos \left (x\right )^{2} - 1\right )}} \] Input:
integrate(1/(4-5*sin(x)^2)^4,x, algorithm="fricas")
Output:
-1/24576*(837*(125*cos(x)^6 - 75*cos(x)^4 + 15*cos(x)^2 - 1)*log(3/4*cos(x )^2 + cos(x)*sin(x) + 1/4) - 837*(125*cos(x)^6 - 75*cos(x)^4 + 15*cos(x)^2 - 1)*log(3/4*cos(x)^2 - cos(x)*sin(x) + 1/4) - 40*(4975*cos(x)^5 - 2590*c os(x)^3 + 447*cos(x))*sin(x))/(125*cos(x)^6 - 75*cos(x)^4 + 15*cos(x)^2 - 1)
Leaf count of result is larger than twice the leaf count of optimal. 1421 vs. \(2 (66) = 132\).
Time = 7.85 (sec) , antiderivative size = 1421, normalized size of antiderivative = 21.53 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(4-5*sin(x)**2)**4,x)
Output:
-837*log(tan(x/2)**2 - tan(x/2) - 1)*tan(x/2)**12/(12288*tan(x/2)**12 - 11 0592*tan(x/2)**10 + 368640*tan(x/2)**8 - 552960*tan(x/2)**6 + 368640*tan(x /2)**4 - 110592*tan(x/2)**2 + 12288) + 7533*log(tan(x/2)**2 - tan(x/2) - 1 )*tan(x/2)**10/(12288*tan(x/2)**12 - 110592*tan(x/2)**10 + 368640*tan(x/2) **8 - 552960*tan(x/2)**6 + 368640*tan(x/2)**4 - 110592*tan(x/2)**2 + 12288 ) - 25110*log(tan(x/2)**2 - tan(x/2) - 1)*tan(x/2)**8/(12288*tan(x/2)**12 - 110592*tan(x/2)**10 + 368640*tan(x/2)**8 - 552960*tan(x/2)**6 + 368640*t an(x/2)**4 - 110592*tan(x/2)**2 + 12288) + 37665*log(tan(x/2)**2 - tan(x/2 ) - 1)*tan(x/2)**6/(12288*tan(x/2)**12 - 110592*tan(x/2)**10 + 368640*tan( x/2)**8 - 552960*tan(x/2)**6 + 368640*tan(x/2)**4 - 110592*tan(x/2)**2 + 1 2288) - 25110*log(tan(x/2)**2 - tan(x/2) - 1)*tan(x/2)**4/(12288*tan(x/2)* *12 - 110592*tan(x/2)**10 + 368640*tan(x/2)**8 - 552960*tan(x/2)**6 + 3686 40*tan(x/2)**4 - 110592*tan(x/2)**2 + 12288) + 7533*log(tan(x/2)**2 - tan( x/2) - 1)*tan(x/2)**2/(12288*tan(x/2)**12 - 110592*tan(x/2)**10 + 368640*t an(x/2)**8 - 552960*tan(x/2)**6 + 368640*tan(x/2)**4 - 110592*tan(x/2)**2 + 12288) - 837*log(tan(x/2)**2 - tan(x/2) - 1)/(12288*tan(x/2)**12 - 11059 2*tan(x/2)**10 + 368640*tan(x/2)**8 - 552960*tan(x/2)**6 + 368640*tan(x/2) **4 - 110592*tan(x/2)**2 + 12288) + 837*log(tan(x/2)**2 + tan(x/2) - 1)*ta n(x/2)**12/(12288*tan(x/2)**12 - 110592*tan(x/2)**10 + 368640*tan(x/2)**8 - 552960*tan(x/2)**6 + 368640*tan(x/2)**4 - 110592*tan(x/2)**2 + 12288)...
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^4} \, dx=-\frac {5 \, {\left (447 \, \tan \left (x\right )^{5} - 1696 \, \tan \left (x\right )^{3} + 2832 \, \tan \left (x\right )\right )}}{3072 \, {\left (\tan \left (x\right )^{6} - 12 \, \tan \left (x\right )^{4} + 48 \, \tan \left (x\right )^{2} - 64\right )}} - \frac {279}{4096} \, \log \left (\tan \left (x\right ) + 2\right ) + \frac {279}{4096} \, \log \left (\tan \left (x\right ) - 2\right ) \] Input:
integrate(1/(4-5*sin(x)^2)^4,x, algorithm="maxima")
Output:
-5/3072*(447*tan(x)^5 - 1696*tan(x)^3 + 2832*tan(x))/(tan(x)^6 - 12*tan(x) ^4 + 48*tan(x)^2 - 64) - 279/4096*log(tan(x) + 2) + 279/4096*log(tan(x) - 2)
Time = 0.53 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^4} \, dx=-\frac {5 \, {\left (447 \, \tan \left (x\right )^{5} - 1696 \, \tan \left (x\right )^{3} + 2832 \, \tan \left (x\right )\right )}}{3072 \, {\left (\tan \left (x\right )^{2} - 4\right )}^{3}} - \frac {279}{4096} \, \log \left ({\left | \tan \left (x\right ) + 2 \right |}\right ) + \frac {279}{4096} \, \log \left ({\left | \tan \left (x\right ) - 2 \right |}\right ) \] Input:
integrate(1/(4-5*sin(x)^2)^4,x, algorithm="giac")
Output:
-5/3072*(447*tan(x)^5 - 1696*tan(x)^3 + 2832*tan(x))/(tan(x)^2 - 4)^3 - 27 9/4096*log(abs(tan(x) + 2)) + 279/4096*log(abs(tan(x) - 2))
Time = 36.98 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^4} \, dx=-\frac {279\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (x\right )}{2}\right )}{2048}-\frac {\frac {745\,{\mathrm {tan}\left (x\right )}^5}{1024}-\frac {265\,{\mathrm {tan}\left (x\right )}^3}{96}+\frac {295\,\mathrm {tan}\left (x\right )}{64}}{{\mathrm {tan}\left (x\right )}^6-12\,{\mathrm {tan}\left (x\right )}^4+48\,{\mathrm {tan}\left (x\right )}^2-64} \] Input:
int(1/(5*sin(x)^2 - 4)^4,x)
Output:
- (279*atanh(tan(x)/2))/2048 - ((295*tan(x))/64 - (265*tan(x)^3)/96 + (745 *tan(x)^5)/1024)/(48*tan(x)^2 - 12*tan(x)^4 + tan(x)^6 - 64)
Time = 0.18 (sec) , antiderivative size = 318, normalized size of antiderivative = 4.82 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^4} \, dx=\frac {-99500 \cos \left (x \right ) \sin \left (x \right )^{5}+147200 \cos \left (x \right ) \sin \left (x \right )^{3}-56640 \cos \left (x \right ) \sin \left (x \right )-104625 \,\mathrm {log}\left (-\sqrt {5}+2 \tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{6}+251100 \,\mathrm {log}\left (-\sqrt {5}+2 \tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{4}-200880 \,\mathrm {log}\left (-\sqrt {5}+2 \tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{2}+53568 \,\mathrm {log}\left (-\sqrt {5}+2 \tan \left (\frac {x}{2}\right )-1\right )+104625 \,\mathrm {log}\left (-\sqrt {5}+2 \tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{6}-251100 \,\mathrm {log}\left (-\sqrt {5}+2 \tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{4}+200880 \,\mathrm {log}\left (-\sqrt {5}+2 \tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{2}-53568 \,\mathrm {log}\left (-\sqrt {5}+2 \tan \left (\frac {x}{2}\right )+1\right )-104625 \,\mathrm {log}\left (\sqrt {5}+2 \tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{6}+251100 \,\mathrm {log}\left (\sqrt {5}+2 \tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{4}-200880 \,\mathrm {log}\left (\sqrt {5}+2 \tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{2}+53568 \,\mathrm {log}\left (\sqrt {5}+2 \tan \left (\frac {x}{2}\right )-1\right )+104625 \,\mathrm {log}\left (\sqrt {5}+2 \tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{6}-251100 \,\mathrm {log}\left (\sqrt {5}+2 \tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{4}+200880 \,\mathrm {log}\left (\sqrt {5}+2 \tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{2}-53568 \,\mathrm {log}\left (\sqrt {5}+2 \tan \left (\frac {x}{2}\right )+1\right )}{1536000 \sin \left (x \right )^{6}-3686400 \sin \left (x \right )^{4}+2949120 \sin \left (x \right )^{2}-786432} \] Input:
int(1/(4-5*sin(x)^2)^4,x)
Output:
( - 99500*cos(x)*sin(x)**5 + 147200*cos(x)*sin(x)**3 - 56640*cos(x)*sin(x) - 104625*log( - sqrt(5) + 2*tan(x/2) - 1)*sin(x)**6 + 251100*log( - sqrt( 5) + 2*tan(x/2) - 1)*sin(x)**4 - 200880*log( - sqrt(5) + 2*tan(x/2) - 1)*s in(x)**2 + 53568*log( - sqrt(5) + 2*tan(x/2) - 1) + 104625*log( - sqrt(5) + 2*tan(x/2) + 1)*sin(x)**6 - 251100*log( - sqrt(5) + 2*tan(x/2) + 1)*sin( x)**4 + 200880*log( - sqrt(5) + 2*tan(x/2) + 1)*sin(x)**2 - 53568*log( - s qrt(5) + 2*tan(x/2) + 1) - 104625*log(sqrt(5) + 2*tan(x/2) - 1)*sin(x)**6 + 251100*log(sqrt(5) + 2*tan(x/2) - 1)*sin(x)**4 - 200880*log(sqrt(5) + 2* tan(x/2) - 1)*sin(x)**2 + 53568*log(sqrt(5) + 2*tan(x/2) - 1) + 104625*log (sqrt(5) + 2*tan(x/2) + 1)*sin(x)**6 - 251100*log(sqrt(5) + 2*tan(x/2) + 1 )*sin(x)**4 + 200880*log(sqrt(5) + 2*tan(x/2) + 1)*sin(x)**2 - 53568*log(s qrt(5) + 2*tan(x/2) + 1))/(12288*(125*sin(x)**6 - 300*sin(x)**4 + 240*sin( x)**2 - 64))