Integrand size = 20, antiderivative size = 60 \[ \int \csc ^5(2 a+2 b x) \sin ^2(a+b x) \, dx=-\frac {\cot ^2(a+b x)}{64 b}+\frac {3 \log (\tan (a+b x))}{32 b}+\frac {3 \tan ^2(a+b x)}{64 b}+\frac {\tan ^4(a+b x)}{128 b} \] Output:
-1/64*cot(b*x+a)^2/b+3/32*ln(tan(b*x+a))/b+3/64*tan(b*x+a)^2/b+1/128*tan(b *x+a)^4/b
Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.93 \[ \int \csc ^5(2 a+2 b x) \sin ^2(a+b x) \, dx=-\frac {2 \csc ^2(a+b x)+12 \log (\cos (a+b x))-12 \log (\sin (a+b x))-4 \sec ^2(a+b x)-\sec ^4(a+b x)}{128 b} \] Input:
Integrate[Csc[2*a + 2*b*x]^5*Sin[a + b*x]^2,x]
Output:
-1/128*(2*Csc[a + b*x]^2 + 12*Log[Cos[a + b*x]] - 12*Log[Sin[a + b*x]] - 4 *Sec[a + b*x]^2 - Sec[a + b*x]^4)/b
Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3042, 4776, 3042, 3100, 243, 49, 2009}
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.
\(\displaystyle \int \sin ^2(a+b x) \csc ^5(2 a+2 b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (a+b x)^2}{\sin (2 a+2 b x)^5}dx\) |
\(\Big \downarrow \) 4776 |
\(\displaystyle \frac {1}{32} \int \csc ^3(a+b x) \sec ^5(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{32} \int \csc (a+b x)^3 \sec (a+b x)^5dx\) |
\(\Big \downarrow \) 3100 |
\(\displaystyle \frac {\int \cot ^3(a+b x) \left (\tan ^2(a+b x)+1\right )^3d\tan (a+b x)}{32 b}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\int \cot ^2(a+b x) \left (\tan ^2(a+b x)+1\right )^3d\tan ^2(a+b x)}{64 b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (\cot ^2(a+b x)+3 \cot (a+b x)+\tan ^2(a+b x)+3\right )d\tan ^2(a+b x)}{64 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{2} \tan ^4(a+b x)+3 \tan ^2(a+b x)-\cot (a+b x)+3 \log \left (\tan ^2(a+b x)\right )}{64 b}\) |
Input:
Int[Csc[2*a + 2*b*x]^5*Sin[a + b*x]^2,x]
Output:
(-Cot[a + b*x] + 3*Log[Tan[a + b*x]^2] + 3*Tan[a + b*x]^2 + Tan[a + b*x]^4 /2)/(64*b)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[1/f Subst[Int[(1 + x^2)^((m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]] , x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
Int[((f_.)*sin[(a_.) + (b_.)*(x_)])^(n_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_ Symbol] :> Simp[2^p/f^p Int[Cos[a + b*x]^p*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, f, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && I ntegerQ[p]
Time = 1.70 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {\frac {1}{4 \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )^{4}}+\frac {3}{4 \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )^{2}}-\frac {3}{2 \sin \left (b x +a \right )^{2}}+3 \ln \left (\tan \left (b x +a \right )\right )}{32 b}\) | \(62\) |
parallelrisch | \(-\frac {\sec \left (b x +a \right )^{4} \csc \left (b x +a \right )^{4} \left (-64 \cos \left (4 b x +4 a \right )+16 \cos \left (8 b x +8 a \right )+66 \cos \left (2 b x +2 a \right )-18 \cos \left (6 b x +6 a \right )+36 \ln \left (\tan \left (b x +a \right )\right ) \cos \left (4 b x +4 a \right )-9 \ln \left (\tan \left (b x +a \right )\right ) \cos \left (8 b x +8 a \right )-27 \ln \left (\tan \left (b x +a \right )\right )\right )}{12288 b}\) | \(112\) |
risch | \(\frac {3 \,{\mathrm e}^{10 i \left (b x +a \right )}+6 \,{\mathrm e}^{8 i \left (b x +a \right )}-2 \,{\mathrm e}^{6 i \left (b x +a \right )}+6 \,{\mathrm e}^{4 i \left (b x +a \right )}+3 \,{\mathrm e}^{2 i \left (b x +a \right )}}{16 b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{4} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{32 b}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{32 b}\) | \(123\) |
Input:
int(csc(2*b*x+2*a)^5*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/32/b*(1/4/sin(b*x+a)^2/cos(b*x+a)^4+3/4/sin(b*x+a)^2/cos(b*x+a)^2-3/2/si n(b*x+a)^2+3*ln(tan(b*x+a)))
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (52) = 104\).
Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.87 \[ \int \csc ^5(2 a+2 b x) \sin ^2(a+b x) \, dx=\frac {6 \, \cos \left (b x + a\right )^{4} - 3 \, \cos \left (b x + a\right )^{2} - 6 \, {\left (\cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} \log \left (\cos \left (b x + a\right )^{2}\right ) + 6 \, {\left (\cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) - 1}{128 \, {\left (b \cos \left (b x + a\right )^{6} - b \cos \left (b x + a\right )^{4}\right )}} \] Input:
integrate(csc(2*b*x+2*a)^5*sin(b*x+a)^2,x, algorithm="fricas")
Output:
1/128*(6*cos(b*x + a)^4 - 3*cos(b*x + a)^2 - 6*(cos(b*x + a)^6 - cos(b*x + a)^4)*log(cos(b*x + a)^2) + 6*(cos(b*x + a)^6 - cos(b*x + a)^4)*log(-1/4* cos(b*x + a)^2 + 1/4) - 1)/(b*cos(b*x + a)^6 - b*cos(b*x + a)^4)
Timed out. \[ \int \csc ^5(2 a+2 b x) \sin ^2(a+b x) \, dx=\text {Timed out} \] Input:
integrate(csc(2*b*x+2*a)**5*sin(b*x+a)**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 3164 vs. \(2 (52) = 104\).
Time = 0.22 (sec) , antiderivative size = 3164, normalized size of antiderivative = 52.73 \[ \int \csc ^5(2 a+2 b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(2*b*x+2*a)^5*sin(b*x+a)^2,x, algorithm="maxima")
Output:
1/64*(4*(3*cos(10*b*x + 10*a) + 6*cos(8*b*x + 8*a) - 2*cos(6*b*x + 6*a) + 6*cos(4*b*x + 4*a) + 3*cos(2*b*x + 2*a))*cos(12*b*x + 12*a) + 4*(9*cos(8*b *x + 8*a) - 16*cos(6*b*x + 6*a) + 9*cos(4*b*x + 4*a) + 12*cos(2*b*x + 2*a) + 3)*cos(10*b*x + 10*a) + 24*cos(10*b*x + 10*a)^2 - 4*(22*cos(6*b*x + 6*a ) + 12*cos(4*b*x + 4*a) - 9*cos(2*b*x + 2*a) - 6)*cos(8*b*x + 8*a) - 24*co s(8*b*x + 8*a)^2 - 8*(11*cos(4*b*x + 4*a) + 8*cos(2*b*x + 2*a) + 1)*cos(6* b*x + 6*a) + 32*cos(6*b*x + 6*a)^2 + 12*(3*cos(2*b*x + 2*a) + 2)*cos(4*b*x + 4*a) - 24*cos(4*b*x + 4*a)^2 + 24*cos(2*b*x + 2*a)^2 - 3*(2*(2*cos(10*b *x + 10*a) - cos(8*b*x + 8*a) - 4*cos(6*b*x + 6*a) - cos(4*b*x + 4*a) + 2* cos(2*b*x + 2*a) + 1)*cos(12*b*x + 12*a) + cos(12*b*x + 12*a)^2 - 4*(cos(8 *b*x + 8*a) + 4*cos(6*b*x + 6*a) + cos(4*b*x + 4*a) - 2*cos(2*b*x + 2*a) - 1)*cos(10*b*x + 10*a) + 4*cos(10*b*x + 10*a)^2 + 2*(4*cos(6*b*x + 6*a) + cos(4*b*x + 4*a) - 2*cos(2*b*x + 2*a) - 1)*cos(8*b*x + 8*a) + cos(8*b*x + 8*a)^2 + 8*(cos(4*b*x + 4*a) - 2*cos(2*b*x + 2*a) - 1)*cos(6*b*x + 6*a) + 16*cos(6*b*x + 6*a)^2 - 2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos( 4*b*x + 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + 2*(2*sin(10*b*x + 10*a) - sin(8*b* x + 8*a) - 4*sin(6*b*x + 6*a) - sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*sin (12*b*x + 12*a) + sin(12*b*x + 12*a)^2 - 4*(sin(8*b*x + 8*a) + 4*sin(6*b*x + 6*a) + sin(4*b*x + 4*a) - 2*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) + 4*si n(10*b*x + 10*a)^2 + 2*(4*sin(6*b*x + 6*a) + sin(4*b*x + 4*a) - 2*sin(2...
Time = 0.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.23 \[ \int \csc ^5(2 a+2 b x) \sin ^2(a+b x) \, dx=-\frac {\frac {6 \, \sin \left (b x + a\right )^{4} - 9 \, \sin \left (b x + a\right )^{2} + 2}{{\left (\sin \left (b x + a\right )^{2} - 1\right )}^{2} \sin \left (b x + a\right )^{2}} + 6 \, \log \left (-\sin \left (b x + a\right )^{2} + 1\right ) - 12 \, \log \left ({\left | \sin \left (b x + a\right ) \right |}\right )}{128 \, b} \] Input:
integrate(csc(2*b*x+2*a)^5*sin(b*x+a)^2,x, algorithm="giac")
Output:
-1/128*((6*sin(b*x + a)^4 - 9*sin(b*x + a)^2 + 2)/((sin(b*x + a)^2 - 1)^2* sin(b*x + a)^2) + 6*log(-sin(b*x + a)^2 + 1) - 12*log(abs(sin(b*x + a))))/ b
Time = 0.17 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.23 \[ \int \csc ^5(2 a+2 b x) \sin ^2(a+b x) \, dx=\frac {3\,\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{64\,b}-\frac {3\,\ln \left (\cos \left (a+b\,x\right )\right )}{32\,b}+\frac {-\frac {3\,{\cos \left (a+b\,x\right )}^4}{64}+\frac {3\,{\cos \left (a+b\,x\right )}^2}{128}+\frac {1}{128}}{b\,\left ({\cos \left (a+b\,x\right )}^4-{\cos \left (a+b\,x\right )}^6\right )} \] Input:
int(sin(a + b*x)^2/sin(2*a + 2*b*x)^5,x)
Output:
(3*log(sin(a + b*x)^2))/(64*b) - (3*log(cos(a + b*x)))/(32*b) + ((3*cos(a + b*x)^2)/128 - (3*cos(a + b*x)^4)/64 + 1/128)/(b*(cos(a + b*x)^4 - cos(a + b*x)^6))
Time = 0.18 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.60 \[ \int \csc ^5(2 a+2 b x) \sin ^2(a+b x) \, dx=\frac {-16 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )^{2}-\cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right )^{2}-12 \cos \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{2}-16 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{3} \sin \left (b x +a \right )-4 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )+9 \,\mathrm {log}\left (\tan \left (b x +a \right )\right ) \sin \left (2 b x +2 a \right )^{4}}{96 \sin \left (2 b x +2 a \right )^{4} b} \] Input:
int(csc(2*b*x+2*a)^5*sin(b*x+a)^2,x)
Output:
( - 16*cos(2*a + 2*b*x)*sin(2*a + 2*b*x)**2*sin(a + b*x)**2 - cos(2*a + 2* b*x)*sin(2*a + 2*b*x)**2 - 12*cos(2*a + 2*b*x)*sin(a + b*x)**2 - 16*cos(a + b*x)*sin(2*a + 2*b*x)**3*sin(a + b*x) - 4*cos(a + b*x)*sin(2*a + 2*b*x)* sin(a + b*x) + 9*log(tan(a + b*x))*sin(2*a + 2*b*x)**4)/(96*sin(2*a + 2*b* x)**4*b)