Integrand size = 10, antiderivative size = 60 \[ \int \left (4-3 \sin ^2(x)\right )^4 \, dx=\frac {10643 x}{128}+\frac {14573}{128} \cos (x) \sin (x)-\frac {2193}{64} \cos (x) \sin ^3(x)+\frac {35}{16} \cos (x) \sin (x) \left (4-3 \sin ^2(x)\right )^2+\frac {3}{8} \cos (x) \sin (x) \left (4-3 \sin ^2(x)\right )^3 \] Output:
10643/128*x+14573/128*cos(x)*sin(x)-2193/64*cos(x)*sin(x)^3+35/16*cos(x)*s in(x)*(4-3*sin(x)^2)^2+3/8*cos(x)*sin(x)*(4-3*sin(x)^2)^3
Time = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.63 \[ \int \left (4-3 \sin ^2(x)\right )^4 \, dx=\frac {10643 x}{128}+\frac {1905}{32} \sin (2 x)+\frac {1431}{128} \sin (4 x)+\frac {45}{32} \sin (6 x)+\frac {81 \sin (8 x)}{1024} \] Input:
Integrate[(4 - 3*Sin[x]^2)^4,x]
Output:
(10643*x)/128 + (1905*Sin[2*x])/32 + (1431*Sin[4*x])/128 + (45*Sin[6*x])/3 2 + (81*Sin[8*x])/1024
Time = 0.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3042, 3659, 3042, 3649, 27, 3042, 3648}
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.
\(\displaystyle \int \left (4-3 \sin ^2(x)\right )^4 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (4-3 \sin (x)^2\right )^4dx\) |
\(\Big \downarrow \) 3659 |
\(\displaystyle \frac {1}{8} \int \left (116-105 \sin ^2(x)\right ) \left (4-3 \sin ^2(x)\right )^2dx+\frac {3}{8} \sin (x) \left (4-3 \sin ^2(x)\right )^3 \cos (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \int \left (116-105 \sin (x)^2\right ) \left (4-3 \sin (x)^2\right )^2dx+\frac {3}{8} \sin (x) \left (4-3 \sin ^2(x)\right )^3 \cos (x)\) |
\(\Big \downarrow \) 3649 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int 3 \left (788-731 \sin ^2(x)\right ) \left (4-3 \sin ^2(x)\right )dx+\frac {35}{2} \sin (x) \left (4-3 \sin ^2(x)\right )^2 \cos (x)\right )+\frac {3}{8} \sin (x) \left (4-3 \sin ^2(x)\right )^3 \cos (x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \int \left (788-731 \sin ^2(x)\right ) \left (4-3 \sin ^2(x)\right )dx+\frac {35}{2} \sin (x) \left (4-3 \sin ^2(x)\right )^2 \cos (x)\right )+\frac {3}{8} \sin (x) \left (4-3 \sin ^2(x)\right )^3 \cos (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \int \left (788-731 \sin (x)^2\right ) \left (4-3 \sin (x)^2\right )dx+\frac {35}{2} \sin (x) \left (4-3 \sin ^2(x)\right )^2 \cos (x)\right )+\frac {3}{8} \sin (x) \left (4-3 \sin ^2(x)\right )^3 \cos (x)\) |
\(\Big \downarrow \) 3648 |
\(\displaystyle \frac {3}{8} \sin (x) \left (4-3 \sin ^2(x)\right )^3 \cos (x)+\frac {1}{8} \left (\frac {1}{2} \left (\frac {10643 x}{8}-\frac {2193}{4} \sin ^3(x) \cos (x)+\frac {14573}{8} \sin (x) \cos (x)\right )+\frac {35}{2} \sin (x) \left (4-3 \sin ^2(x)\right )^2 \cos (x)\right )\) |
Input:
Int[(4 - 3*Sin[x]^2)^4,x]
Output:
(3*Cos[x]*Sin[x]*(4 - 3*Sin[x]^2)^3)/8 + ((35*Cos[x]*Sin[x]*(4 - 3*Sin[x]^ 2)^2)/2 + ((10643*x)/8 + (14573*Cos[x]*Sin[x])/8 - (2193*Cos[x]*Sin[x]^3)/ 4)/2)/8
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_.)*sin[(e_.) + (f_.)*(x_)]^2)*((A_.) + (B_.)*sin[(e_.) + (f_ .)*(x_)]^2), x_Symbol] :> Simp[(4*A*(2*a + b) + B*(4*a + 3*b))*(x/8), x] + (-Simp[b*B*Cos[e + f*x]*(Sin[e + f*x]^3/(4*f)), x] - Simp[(4*A*b + B*(4*a + 3*b))*Cos[e + f*x]*(Sin[e + f*x]/(8*f)), x]) /; FreeQ[{a, b, e, f, A, B}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-B)*Cos[e + f*x]*Sin[e + f*x]*((a + b* Sin[e + f*x]^2)^p/(2*f*(p + 1))), x] + Simp[1/(2*(p + 1)) Int[(a + b*Sin[ e + f*x]^2)^(p - 1)*Simp[a*B + 2*a*A*(p + 1) + (2*A*b*(p + 1) + B*(b + 2*a* p + 2*b*p))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && G tQ[p, 0]
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*f*p)), x] + Sim p[1/(2*p) Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(2*a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[ a + b, 0] && GtQ[p, 1]
Time = 11.95 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.48
method | result | size |
risch | \(\frac {10643 x}{128}+\frac {81 \sin \left (8 x \right )}{1024}+\frac {45 \sin \left (6 x \right )}{32}+\frac {1431 \sin \left (4 x \right )}{128}+\frac {1905 \sin \left (2 x \right )}{32}\) | \(29\) |
parallelrisch | \(\frac {10643 x}{128}+\frac {81 \sin \left (8 x \right )}{1024}+\frac {45 \sin \left (6 x \right )}{32}+\frac {1431 \sin \left (4 x \right )}{128}+\frac {1905 \sin \left (2 x \right )}{32}\) | \(29\) |
default | \(-\frac {81 \left (\sin \left (x \right )^{7}+\frac {7 \sin \left (x \right )^{5}}{6}+\frac {35 \sin \left (x \right )^{3}}{24}+\frac {35 \sin \left (x \right )}{16}\right ) \cos \left (x \right )}{8}+\frac {10643 x}{128}+72 \left (\sin \left (x \right )^{5}+\frac {5 \sin \left (x \right )^{3}}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )-216 \left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )+384 \cos \left (x \right ) \sin \left (x \right )\) | \(68\) |
parts | \(-\frac {81 \left (\sin \left (x \right )^{7}+\frac {7 \sin \left (x \right )^{5}}{6}+\frac {35 \sin \left (x \right )^{3}}{24}+\frac {35 \sin \left (x \right )}{16}\right ) \cos \left (x \right )}{8}+\frac {10643 x}{128}+72 \left (\sin \left (x \right )^{5}+\frac {5 \sin \left (x \right )^{3}}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )-216 \left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )+384 \cos \left (x \right ) \sin \left (x \right )\) | \(68\) |
norman | \(\frac {\frac {10643 x}{128}+\frac {38553 \tan \left (\frac {x}{2}\right )^{3}}{64}+\frac {106173 \tan \left (\frac {x}{2}\right )^{5}}{64}+\frac {6801 \tan \left (\frac {x}{2}\right )^{7}}{64}-\frac {6801 \tan \left (\frac {x}{2}\right )^{9}}{64}-\frac {106173 \tan \left (\frac {x}{2}\right )^{11}}{64}-\frac {38553 \tan \left (\frac {x}{2}\right )^{13}}{64}-\frac {22125 \tan \left (\frac {x}{2}\right )^{15}}{64}+\frac {10643 x \tan \left (\frac {x}{2}\right )^{2}}{16}+\frac {74501 x \tan \left (\frac {x}{2}\right )^{4}}{32}+\frac {74501 x \tan \left (\frac {x}{2}\right )^{6}}{16}+\frac {372505 x \tan \left (\frac {x}{2}\right )^{8}}{64}+\frac {74501 x \tan \left (\frac {x}{2}\right )^{10}}{16}+\frac {74501 x \tan \left (\frac {x}{2}\right )^{12}}{32}+\frac {10643 x \tan \left (\frac {x}{2}\right )^{14}}{16}+\frac {10643 x \tan \left (\frac {x}{2}\right )^{16}}{128}+\frac {22125 \tan \left (\frac {x}{2}\right )}{64}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{8}}\) | \(150\) |
Input:
int((4-3*sin(x)^2)^4,x,method=_RETURNVERBOSE)
Output:
10643/128*x+81/1024*sin(8*x)+45/32*sin(6*x)+1431/128*sin(4*x)+1905/32*sin( 2*x)
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.52 \[ \int \left (4-3 \sin ^2(x)\right )^4 \, dx=\frac {3}{128} \, {\left (432 \, \cos \left (x\right )^{7} + 1272 \, \cos \left (x\right )^{5} + 2166 \, \cos \left (x\right )^{3} + 3505 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {10643}{128} \, x \] Input:
integrate((4-3*sin(x)^2)^4,x, algorithm="fricas")
Output:
3/128*(432*cos(x)^7 + 1272*cos(x)^5 + 2166*cos(x)^3 + 3505*cos(x))*sin(x) + 10643/128*x
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (66) = 132\).
Time = 0.47 (sec) , antiderivative size = 272, normalized size of antiderivative = 4.53 \[ \int \left (4-3 \sin ^2(x)\right )^4 \, dx=\frac {2835 x \sin ^{8}{\left (x \right )}}{128} + \frac {2835 x \sin ^{6}{\left (x \right )} \cos ^{2}{\left (x \right )}}{32} - 135 x \sin ^{6}{\left (x \right )} + \frac {8505 x \sin ^{4}{\left (x \right )} \cos ^{4}{\left (x \right )}}{64} - 405 x \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )} + 324 x \sin ^{4}{\left (x \right )} + \frac {2835 x \sin ^{2}{\left (x \right )} \cos ^{6}{\left (x \right )}}{32} - 405 x \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )} + 648 x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )} - 384 x \sin ^{2}{\left (x \right )} + \frac {2835 x \cos ^{8}{\left (x \right )}}{128} - 135 x \cos ^{6}{\left (x \right )} + 324 x \cos ^{4}{\left (x \right )} - 384 x \cos ^{2}{\left (x \right )} + 256 x - \frac {7533 \sin ^{7}{\left (x \right )} \cos {\left (x \right )}}{128} - \frac {13797 \sin ^{5}{\left (x \right )} \cos ^{3}{\left (x \right )}}{128} + 297 \sin ^{5}{\left (x \right )} \cos {\left (x \right )} - \frac {10395 \sin ^{3}{\left (x \right )} \cos ^{5}{\left (x \right )}}{128} + 360 \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )} - 540 \sin ^{3}{\left (x \right )} \cos {\left (x \right )} - \frac {2835 \sin {\left (x \right )} \cos ^{7}{\left (x \right )}}{128} + 135 \sin {\left (x \right )} \cos ^{5}{\left (x \right )} - 324 \sin {\left (x \right )} \cos ^{3}{\left (x \right )} + 384 \sin {\left (x \right )} \cos {\left (x \right )} \] Input:
integrate((4-3*sin(x)**2)**4,x)
Output:
2835*x*sin(x)**8/128 + 2835*x*sin(x)**6*cos(x)**2/32 - 135*x*sin(x)**6 + 8 505*x*sin(x)**4*cos(x)**4/64 - 405*x*sin(x)**4*cos(x)**2 + 324*x*sin(x)**4 + 2835*x*sin(x)**2*cos(x)**6/32 - 405*x*sin(x)**2*cos(x)**4 + 648*x*sin(x )**2*cos(x)**2 - 384*x*sin(x)**2 + 2835*x*cos(x)**8/128 - 135*x*cos(x)**6 + 324*x*cos(x)**4 - 384*x*cos(x)**2 + 256*x - 7533*sin(x)**7*cos(x)/128 - 13797*sin(x)**5*cos(x)**3/128 + 297*sin(x)**5*cos(x) - 10395*sin(x)**3*cos (x)**5/128 + 360*sin(x)**3*cos(x)**3 - 540*sin(x)**3*cos(x) - 2835*sin(x)* cos(x)**7/128 + 135*sin(x)*cos(x)**5 - 324*sin(x)*cos(x)**3 + 384*sin(x)*c os(x)
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.50 \[ \int \left (4-3 \sin ^2(x)\right )^4 \, dx=-\frac {45}{8} \, \sin \left (2 \, x\right )^{3} + \frac {10643}{128} \, x + \frac {81}{1024} \, \sin \left (8 \, x\right ) + \frac {1431}{128} \, \sin \left (4 \, x\right ) + \frac {255}{4} \, \sin \left (2 \, x\right ) \] Input:
integrate((4-3*sin(x)^2)^4,x, algorithm="maxima")
Output:
-45/8*sin(2*x)^3 + 10643/128*x + 81/1024*sin(8*x) + 1431/128*sin(4*x) + 25 5/4*sin(2*x)
Time = 0.45 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.47 \[ \int \left (4-3 \sin ^2(x)\right )^4 \, dx=\frac {10643}{128} \, x + \frac {81}{1024} \, \sin \left (8 \, x\right ) + \frac {45}{32} \, \sin \left (6 \, x\right ) + \frac {1431}{128} \, \sin \left (4 \, x\right ) + \frac {1905}{32} \, \sin \left (2 \, x\right ) \] Input:
integrate((4-3*sin(x)^2)^4,x, algorithm="giac")
Output:
10643/128*x + 81/1024*sin(8*x) + 45/32*sin(6*x) + 1431/128*sin(4*x) + 1905 /32*sin(2*x)
Time = 35.90 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.60 \[ \int \left (4-3 \sin ^2(x)\right )^4 \, dx=\frac {10643\,x}{128}+\frac {\frac {10515\,{\mathrm {tan}\left (x\right )}^7}{128}+\frac {38043\,{\mathrm {tan}\left (x\right )}^5}{128}+\frac {48357\,{\mathrm {tan}\left (x\right )}^3}{128}+\frac {22125\,\mathrm {tan}\left (x\right )}{128}}{{\left ({\mathrm {tan}\left (x\right )}^2+1\right )}^4} \] Input:
int((3*sin(x)^2 - 4)^4,x)
Output:
(10643*x)/128 + ((22125*tan(x))/128 + (48357*tan(x)^3)/128 + (38043*tan(x) ^5)/128 + (10515*tan(x)^7)/128)/(tan(x)^2 + 1)^4
Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.57 \[ \int \left (4-3 \sin ^2(x)\right )^4 \, dx=-\frac {81 \cos \left (x \right ) \sin \left (x \right )^{7}}{8}+\frac {963 \cos \left (x \right ) \sin \left (x \right )^{5}}{16}-\frac {9009 \cos \left (x \right ) \sin \left (x \right )^{3}}{64}+\frac {22125 \cos \left (x \right ) \sin \left (x \right )}{128}+\frac {10643 x}{128} \] Input:
int((4-3*sin(x)^2)^4,x)
Output:
( - 1296*cos(x)*sin(x)**7 + 7704*cos(x)*sin(x)**5 - 18018*cos(x)*sin(x)**3 + 22125*cos(x)*sin(x) + 10643*x)/128