Integrand size = 18, antiderivative size = 61 \[ \int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {128 \sin ^9(a+b x)}{9 b}-\frac {384 \sin ^{11}(a+b x)}{11 b}+\frac {384 \sin ^{13}(a+b x)}{13 b}-\frac {128 \sin ^{15}(a+b x)}{15 b} \] Output:
128/9*sin(b*x+a)^9/b-384/11*sin(b*x+a)^11/b+384/13*sin(b*x+a)^13/b-128/15* sin(b*x+a)^15/b
Time = 0.33 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77 \[ \int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {4 (8330+10755 \cos (2 (a+b x))+3366 \cos (4 (a+b x))+429 \cos (6 (a+b x))) \sin ^9(a+b x)}{6435 b} \] Input:
Integrate[Sin[a + b*x]*Sin[2*a + 2*b*x]^7,x]
Output:
(4*(8330 + 10755*Cos[2*(a + b*x)] + 3366*Cos[4*(a + b*x)] + 429*Cos[6*(a + b*x)])*Sin[a + b*x]^9)/(6435*b)
Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4776, 3042, 3044, 244, 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.
\(\displaystyle \int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x) \sin (2 a+2 b x)^7dx\) |
\(\Big \downarrow \) 4776 |
\(\displaystyle 128 \int \cos ^7(a+b x) \sin ^8(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 128 \int \cos (a+b x)^7 \sin (a+b x)^8dx\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {128 \int \sin ^8(a+b x) \left (1-\sin ^2(a+b x)\right )^3d\sin (a+b x)}{b}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {128 \int \left (-\sin ^{14}(a+b x)+3 \sin ^{12}(a+b x)-3 \sin ^{10}(a+b x)+\sin ^8(a+b x)\right )d\sin (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {128 \left (-\frac {1}{15} \sin ^{15}(a+b x)+\frac {3}{13} \sin ^{13}(a+b x)-\frac {3}{11} \sin ^{11}(a+b x)+\frac {1}{9} \sin ^9(a+b x)\right )}{b}\) |
Input:
Int[Sin[a + b*x]*Sin[2*a + 2*b*x]^7,x]
Output:
(128*(Sin[a + b*x]^9/9 - (3*Sin[a + b*x]^11)/11 + (3*Sin[a + b*x]^13)/13 - Sin[a + b*x]^15/15))/b
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
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 = 25.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.51
method | result | size |
parallelrisch | \(\frac {5005 \sin \left (9 b x +9 a \right )+19305 \sin \left (7 b x +7 a \right )-4095 \sin \left (11 b x +11 a \right )-75075 \sin \left (3 b x +3 a \right )+429 \sin \left (15 b x +15 a \right )-495 \sin \left (13 b x +13 a \right )+225225 \sin \left (b x +a \right )-27027 \sin \left (5 b x +5 a \right )}{823680 b}\) | \(92\) |
default | \(\frac {35 \sin \left (b x +a \right )}{128 b}-\frac {35 \sin \left (3 b x +3 a \right )}{384 b}-\frac {21 \sin \left (5 b x +5 a \right )}{640 b}+\frac {3 \sin \left (7 b x +7 a \right )}{128 b}+\frac {7 \sin \left (9 b x +9 a \right )}{1152 b}-\frac {7 \sin \left (11 b x +11 a \right )}{1408 b}-\frac {\sin \left (13 b x +13 a \right )}{1664 b}+\frac {\sin \left (15 b x +15 a \right )}{1920 b}\) | \(111\) |
risch | \(\frac {35 \sin \left (b x +a \right )}{128 b}-\frac {35 \sin \left (3 b x +3 a \right )}{384 b}-\frac {21 \sin \left (5 b x +5 a \right )}{640 b}+\frac {3 \sin \left (7 b x +7 a \right )}{128 b}+\frac {7 \sin \left (9 b x +9 a \right )}{1152 b}-\frac {7 \sin \left (11 b x +11 a \right )}{1408 b}-\frac {\sin \left (13 b x +13 a \right )}{1664 b}+\frac {\sin \left (15 b x +15 a \right )}{1920 b}\) | \(111\) |
orering | \(\text {Expression too large to display}\) | \(1592\) |
Input:
int(sin(b*x+a)*sin(2*b*x+2*a)^7,x,method=_RETURNVERBOSE)
Output:
1/823680*(5005*sin(9*b*x+9*a)+19305*sin(7*b*x+7*a)-4095*sin(11*b*x+11*a)-7 5075*sin(3*b*x+3*a)+429*sin(15*b*x+15*a)-495*sin(13*b*x+13*a)+225225*sin(b *x+a)-27027*sin(5*b*x+5*a))/b
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.36 \[ \int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {128 \, {\left (429 \, \cos \left (b x + a\right )^{14} - 1518 \, \cos \left (b x + a\right )^{12} + 1854 \, \cos \left (b x + a\right )^{10} - 800 \, \cos \left (b x + a\right )^{8} + 5 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 16\right )} \sin \left (b x + a\right )}{6435 \, b} \] Input:
integrate(sin(b*x+a)*sin(2*b*x+2*a)^7,x, algorithm="fricas")
Output:
128/6435*(429*cos(b*x + a)^14 - 1518*cos(b*x + a)^12 + 1854*cos(b*x + a)^1 0 - 800*cos(b*x + a)^8 + 5*cos(b*x + a)^6 + 6*cos(b*x + a)^4 + 8*cos(b*x + a)^2 + 16)*sin(b*x + a)/b
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (53) = 106\).
Time = 26.39 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.41 \[ \int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx=\begin {cases} - \frac {3838 \sin {\left (a + b x \right )} \sin ^{6}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{6435 b} - \frac {1648 \sin {\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{1287 b} - \frac {768 \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{715 b} - \frac {2048 \sin {\left (a + b x \right )} \cos ^{7}{\left (2 a + 2 b x \right )}}{6435 b} + \frac {1241 \sin ^{7}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{6435 b} + \frac {376 \sin ^{5}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{715 b} + \frac {640 \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{1287 b} + \frac {1024 \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{6}{\left (2 a + 2 b x \right )}}{6435 b} & \text {for}\: b \neq 0 \\x \sin {\left (a \right )} \sin ^{7}{\left (2 a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(sin(b*x+a)*sin(2*b*x+2*a)**7,x)
Output:
Piecewise((-3838*sin(a + b*x)*sin(2*a + 2*b*x)**6*cos(2*a + 2*b*x)/(6435*b ) - 1648*sin(a + b*x)*sin(2*a + 2*b*x)**4*cos(2*a + 2*b*x)**3/(1287*b) - 7 68*sin(a + b*x)*sin(2*a + 2*b*x)**2*cos(2*a + 2*b*x)**5/(715*b) - 2048*sin (a + b*x)*cos(2*a + 2*b*x)**7/(6435*b) + 1241*sin(2*a + 2*b*x)**7*cos(a + b*x)/(6435*b) + 376*sin(2*a + 2*b*x)**5*cos(a + b*x)*cos(2*a + 2*b*x)**2/( 715*b) + 640*sin(2*a + 2*b*x)**3*cos(a + b*x)*cos(2*a + 2*b*x)**4/(1287*b) + 1024*sin(2*a + 2*b*x)*cos(a + b*x)*cos(2*a + 2*b*x)**6/(6435*b), Ne(b, 0)), (x*sin(a)*sin(2*a)**7, True))
Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.49 \[ \int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {429 \, \sin \left (15 \, b x + 15 \, a\right ) - 495 \, \sin \left (13 \, b x + 13 \, a\right ) - 4095 \, \sin \left (11 \, b x + 11 \, a\right ) + 5005 \, \sin \left (9 \, b x + 9 \, a\right ) + 19305 \, \sin \left (7 \, b x + 7 \, a\right ) - 27027 \, \sin \left (5 \, b x + 5 \, a\right ) - 75075 \, \sin \left (3 \, b x + 3 \, a\right ) + 225225 \, \sin \left (b x + a\right )}{823680 \, b} \] Input:
integrate(sin(b*x+a)*sin(2*b*x+2*a)^7,x, algorithm="maxima")
Output:
1/823680*(429*sin(15*b*x + 15*a) - 495*sin(13*b*x + 13*a) - 4095*sin(11*b* x + 11*a) + 5005*sin(9*b*x + 9*a) + 19305*sin(7*b*x + 7*a) - 27027*sin(5*b *x + 5*a) - 75075*sin(3*b*x + 3*a) + 225225*sin(b*x + a))/b
Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx=-\frac {128 \, {\left (429 \, \sin \left (b x + a\right )^{15} - 1485 \, \sin \left (b x + a\right )^{13} + 1755 \, \sin \left (b x + a\right )^{11} - 715 \, \sin \left (b x + a\right )^{9}\right )}}{6435 \, b} \] Input:
integrate(sin(b*x+a)*sin(2*b*x+2*a)^7,x, algorithm="giac")
Output:
-128/6435*(429*sin(b*x + a)^15 - 1485*sin(b*x + a)^13 + 1755*sin(b*x + a)^ 11 - 715*sin(b*x + a)^9)/b
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {-\frac {128\,{\sin \left (a+b\,x\right )}^{15}}{15}+\frac {384\,{\sin \left (a+b\,x\right )}^{13}}{13}-\frac {384\,{\sin \left (a+b\,x\right )}^{11}}{11}+\frac {128\,{\sin \left (a+b\,x\right )}^9}{9}}{b} \] Input:
int(sin(a + b*x)*sin(2*a + 2*b*x)^7,x)
Output:
((128*sin(a + b*x)^9)/9 - (384*sin(a + b*x)^11)/11 + (384*sin(a + b*x)^13) /13 - (128*sin(a + b*x)^15)/15)/b
Time = 0.19 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.97 \[ \int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {-462 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right )^{6} \sin \left (b x +a \right )-560 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right )^{4} \sin \left (b x +a \right )-768 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )-2048 \cos \left (2 b x +2 a \right ) \sin \left (b x +a \right )+33 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{7}+56 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{5}+128 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{3}+1024 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )}{6435 b} \] Input:
int(sin(b*x+a)*sin(2*b*x+2*a)^7,x)
Output:
( - 462*cos(2*a + 2*b*x)*sin(2*a + 2*b*x)**6*sin(a + b*x) - 560*cos(2*a + 2*b*x)*sin(2*a + 2*b*x)**4*sin(a + b*x) - 768*cos(2*a + 2*b*x)*sin(2*a + 2 *b*x)**2*sin(a + b*x) - 2048*cos(2*a + 2*b*x)*sin(a + b*x) + 33*cos(a + b* x)*sin(2*a + 2*b*x)**7 + 56*cos(a + b*x)*sin(2*a + 2*b*x)**5 + 128*cos(a + b*x)*sin(2*a + 2*b*x)**3 + 1024*cos(a + b*x)*sin(2*a + 2*b*x))/(6435*b)