Integrand size = 20, antiderivative size = 31 \[ \int \sin ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=\frac {8 \sin ^7(a+b x)}{7 b}-\frac {8 \sin ^9(a+b x)}{9 b} \] Output:
8/7*sin(b*x+a)^7/b-8/9*sin(b*x+a)^9/b
Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \sin ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=\frac {4 (11+7 \cos (2 (a+b x))) \sin ^7(a+b x)}{63 b} \] Input:
Integrate[Sin[a + b*x]^3*Sin[2*a + 2*b*x]^3,x]
Output:
(4*(11 + 7*Cos[2*(a + b*x)])*Sin[a + b*x]^7)/(63*b)
Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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.
\(\displaystyle \int \sin ^3(a+b x) \sin ^3(2 a+2 b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^3 \sin (2 a+2 b x)^3dx\) |
\(\Big \downarrow \) 4776 |
\(\displaystyle 8 \int \cos ^3(a+b x) \sin ^6(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 8 \int \cos (a+b x)^3 \sin (a+b x)^6dx\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {8 \int \sin ^6(a+b x) \left (1-\sin ^2(a+b x)\right )d\sin (a+b x)}{b}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {8 \int \left (\sin ^6(a+b x)-\sin ^8(a+b x)\right )d\sin (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {8 \left (\frac {1}{7} \sin ^7(a+b x)-\frac {1}{9} \sin ^9(a+b x)\right )}{b}\) |
Input:
Int[Sin[a + b*x]^3*Sin[2*a + 2*b*x]^3,x]
Output:
(8*(Sin[a + b*x]^7/7 - Sin[a + b*x]^9/9))/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 = 5.54 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77
method | result | size |
default | \(\frac {3 \sin \left (b x +a \right )}{16 b}-\frac {\sin \left (3 b x +3 a \right )}{12 b}+\frac {3 \sin \left (7 b x +7 a \right )}{224 b}-\frac {\sin \left (9 b x +9 a \right )}{288 b}\) | \(55\) |
risch | \(\frac {3 \sin \left (b x +a \right )}{16 b}-\frac {\sin \left (3 b x +3 a \right )}{12 b}+\frac {3 \sin \left (7 b x +7 a \right )}{224 b}-\frac {\sin \left (9 b x +9 a \right )}{288 b}\) | \(55\) |
parallelrisch | \(\frac {16 \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{6} \tan \left (b x +a \right )^{3}+\left (-96 \tan \left (b x +a \right )^{4}+96 \tan \left (b x +a \right )^{2}\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{5}+\left (192 \tan \left (b x +a \right )^{5}-720 \tan \left (b x +a \right )^{3}+192 \tan \left (b x +a \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{4}+\left (-128 \tan \left (b x +a \right )^{6}+1920 \tan \left (b x +a \right )^{4}-1920 \tan \left (b x +a \right )^{2}+128\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{3}+\left (-2112 \tan \left (b x +a \right )^{5}-3120 \tan \left (b x +a \right )^{3}-2112 \tan \left (b x +a \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}+\left (768 \tan \left (b x +a \right )^{6}+672 \tan \left (b x +a \right )^{4}-672 \tan \left (b x +a \right )^{2}-768\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )+384 \tan \left (b x +a \right )^{5}+752 \tan \left (b x +a \right )^{3}+384 \tan \left (b x +a \right )}{315 b \left (1+\tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}\right )^{3} \left (\tan \left (b x +a \right )^{2}+1\right )^{3}}\) | \(284\) |
orering | \(-\frac {4540 \left (3 \sin \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right )^{3} b \cos \left (b x +a \right )+6 \sin \left (b x +a \right )^{3} \sin \left (2 b x +2 a \right )^{2} b \cos \left (2 b x +2 a \right )\right )}{3969 b^{2}}-\frac {754 \left (-129 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right )^{3} b^{3}+108 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{2} b^{3} \cos \left (2 b x +2 a \right )+6 \cos \left (b x +a \right )^{3} b^{3} \sin \left (2 b x +2 a \right )^{3}+216 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right ) b^{3} \cos \left (2 b x +2 a \right )^{2}-222 b^{3} \sin \left (b x +a \right )^{3} \sin \left (2 b x +2 a \right )^{2} \cos \left (2 b x +2 a \right )+48 \cos \left (2 b x +2 a \right )^{3} \sin \left (b x +a \right )^{3} b^{3}\right )}{5103 b^{4}}-\frac {20 \left (7743 \cos \left (b x +a \right ) b^{5} \sin \left (2 b x +2 a \right )^{3} \sin \left (b x +a \right )^{2}+11526 \sin \left (b x +a \right )^{3} \sin \left (2 b x +2 a \right )^{2} b^{5} \cos \left (2 b x +2 a \right )-11880 \cos \left (b x +a \right )^{2} b^{5} \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right ) \cos \left (2 b x +2 a \right )-780 \cos \left (b x +a \right )^{3} b^{5} \sin \left (2 b x +2 a \right )^{3}-19440 \cos \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )^{2} b^{5} \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )+1440 \cos \left (2 b x +2 a \right )^{2} b^{5} \cos \left (b x +a \right )^{3} \sin \left (2 b x +2 a \right )+2880 \cos \left (2 b x +2 a \right )^{3} b^{5} \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )-3360 \cos \left (2 b x +2 a \right )^{3} \sin \left (b x +a \right )^{3} b^{5}\right )}{5103 b^{6}}-\frac {-540669 b^{7} \sin \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right )^{3} \cos \left (b x +a \right )-710142 b^{7} \sin \left (b x +a \right )^{3} \sin \left (2 b x +2 a \right )^{2} \cos \left (2 b x +2 a \right )+1531656 \cos \left (b x +a \right ) b^{7} \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{2} \cos \left (2 b x +2 a \right )^{2}+1113588 \cos \left (b x +a \right )^{2} b^{7} \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right ) \cos \left (2 b x +2 a \right )+86226 \cos \left (b x +a \right )^{3} b^{7} \sin \left (2 b x +2 a \right )^{3}+226128 \sin \left (b x +a \right )^{3} b^{7} \cos \left (2 b x +2 a \right )^{3}-342720 \cos \left (b x +a \right )^{2} b^{7} \cos \left (2 b x +2 a \right )^{3} \sin \left (b x +a \right )-231840 \cos \left (b x +a \right )^{3} b^{7} \sin \left (2 b x +2 a \right ) \cos \left (2 b x +2 a \right )^{2}}{35721 b^{8}}\) | \(778\) |
Input:
int(sin(b*x+a)^3*sin(2*b*x+2*a)^3,x,method=_RETURNVERBOSE)
Output:
3/16*sin(b*x+a)/b-1/12*sin(3*b*x+3*a)/b+3/224/b*sin(7*b*x+7*a)-1/288/b*sin (9*b*x+9*a)
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \sin ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=-\frac {8 \, {\left (7 \, \cos \left (b x + a\right )^{8} - 19 \, \cos \left (b x + a\right )^{6} + 15 \, \cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2} - 2\right )} \sin \left (b x + a\right )}{63 \, b} \] Input:
integrate(sin(b*x+a)^3*sin(2*b*x+2*a)^3,x, algorithm="fricas")
Output:
-8/63*(7*cos(b*x + a)^8 - 19*cos(b*x + a)^6 + 15*cos(b*x + a)^4 - cos(b*x + a)^2 - 2)*sin(b*x + a)/b
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (26) = 52\).
Time = 4.92 (sec) , antiderivative size = 284, normalized size of antiderivative = 9.16 \[ \int \sin ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=\begin {cases} - \frac {46 \sin ^{3}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{105 b} - \frac {16 \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{63 b} - \frac {13 \sin ^{2}{\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{105 b} - \frac {8 \sin ^{2}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{35 b} - \frac {4 \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{7 b} - \frac {64 \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{105 b} + \frac {94 \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )}}{315 b} + \frac {32 \sin {\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{105 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\left (a \right )} \sin ^{3}{\left (2 a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(sin(b*x+a)**3*sin(2*b*x+2*a)**3,x)
Output:
Piecewise((-46*sin(a + b*x)**3*sin(2*a + 2*b*x)**2*cos(2*a + 2*b*x)/(105*b ) - 16*sin(a + b*x)**3*cos(2*a + 2*b*x)**3/(63*b) - 13*sin(a + b*x)**2*sin (2*a + 2*b*x)**3*cos(a + b*x)/(105*b) - 8*sin(a + b*x)**2*sin(2*a + 2*b*x) *cos(a + b*x)*cos(2*a + 2*b*x)**2/(35*b) - 4*sin(a + b*x)*sin(2*a + 2*b*x) **2*cos(a + b*x)**2*cos(2*a + 2*b*x)/(7*b) - 64*sin(a + b*x)*cos(a + b*x)* *2*cos(2*a + 2*b*x)**3/(105*b) + 94*sin(2*a + 2*b*x)**3*cos(a + b*x)**3/(3 15*b) + 32*sin(2*a + 2*b*x)*cos(a + b*x)**3*cos(2*a + 2*b*x)**2/(105*b), N e(b, 0)), (x*sin(a)**3*sin(2*a)**3, True))
Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \sin ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=-\frac {7 \, \sin \left (9 \, b x + 9 \, a\right ) - 27 \, \sin \left (7 \, b x + 7 \, a\right ) + 168 \, \sin \left (3 \, b x + 3 \, a\right ) - 378 \, \sin \left (b x + a\right )}{2016 \, b} \] Input:
integrate(sin(b*x+a)^3*sin(2*b*x+2*a)^3,x, algorithm="maxima")
Output:
-1/2016*(7*sin(9*b*x + 9*a) - 27*sin(7*b*x + 7*a) + 168*sin(3*b*x + 3*a) - 378*sin(b*x + a))/b
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \sin ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=-\frac {8 \, {\left (7 \, \sin \left (b x + a\right )^{9} - 9 \, \sin \left (b x + a\right )^{7}\right )}}{63 \, b} \] Input:
integrate(sin(b*x+a)^3*sin(2*b*x+2*a)^3,x, algorithm="giac")
Output:
-8/63*(7*sin(b*x + a)^9 - 9*sin(b*x + a)^7)/b
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \sin ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=\frac {8\,\left (9\,{\sin \left (a+b\,x\right )}^7-7\,{\sin \left (a+b\,x\right )}^9\right )}{63\,b} \] Input:
int(sin(a + b*x)^3*sin(2*a + 2*b*x)^3,x)
Output:
(8*(9*sin(a + b*x)^7 - 7*sin(a + b*x)^9))/(63*b)
Time = 0.18 (sec) , antiderivative size = 241, normalized size of antiderivative = 7.77 \[ \int \sin ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=\frac {-144 \cos \left (2 b x +2 a \right ) \cos \left (b x +a \right ) \sin \left (b x +a \right )-70 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )^{3}+12 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )+112 \cos \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{3}-192 \cos \left (2 b x +2 a \right ) \sin \left (b x +a \right )+35 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{3} \sin \left (b x +a \right )^{2}-2 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{3}-168 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{2}+96 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )-144 \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{2}+72 \sin \left (2 b x +2 a \right )}{315 b} \] Input:
int(sin(b*x+a)^3*sin(2*b*x+2*a)^3,x)
Output:
( - 144*cos(2*a + 2*b*x)*cos(a + b*x)*sin(a + b*x) - 70*cos(2*a + 2*b*x)*s in(2*a + 2*b*x)**2*sin(a + b*x)**3 + 12*cos(2*a + 2*b*x)*sin(2*a + 2*b*x)* *2*sin(a + b*x) + 112*cos(2*a + 2*b*x)*sin(a + b*x)**3 - 192*cos(2*a + 2*b *x)*sin(a + b*x) + 35*cos(a + b*x)*sin(2*a + 2*b*x)**3*sin(a + b*x)**2 - 2 *cos(a + b*x)*sin(2*a + 2*b*x)**3 - 168*cos(a + b*x)*sin(2*a + 2*b*x)*sin( a + b*x)**2 + 96*cos(a + b*x)*sin(2*a + 2*b*x) - 144*sin(2*a + 2*b*x)*sin( a + b*x)**2 + 72*sin(2*a + 2*b*x))/(315*b)