Integrand size = 20, antiderivative size = 49 \[ \int \sin ^2(a+b x) \sin ^2(2 a+2 b x) \, dx=\frac {x}{4}-\frac {\cos (2 a+2 b x) \sin (2 a+2 b x)}{8 b}-\frac {\sin ^3(2 a+2 b x)}{12 b} \] Output:
1/4*x-1/8*cos(2*b*x+2*a)*sin(2*b*x+2*a)/b-1/12*sin(2*b*x+2*a)^3/b
Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82 \[ \int \sin ^2(a+b x) \sin ^2(2 a+2 b x) \, dx=\frac {12 b x-3 \sin (2 (a+b x))-3 \sin (4 (a+b x))+\sin (6 (a+b x))}{48 b} \] Input:
Integrate[Sin[a + b*x]^2*Sin[2*a + 2*b*x]^2,x]
Output:
(12*b*x - 3*Sin[2*(a + b*x)] - 3*Sin[4*(a + b*x)] + Sin[6*(a + b*x)])/(48* b)
Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3042, 4774, 3042, 3044, 15, 3115, 24}
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.
\(\displaystyle \int \sin ^2(a+b x) \sin ^2(2 a+2 b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^2 \sin (2 a+2 b x)^2dx\) |
\(\Big \downarrow \) 4774 |
\(\displaystyle \frac {1}{2} \int \sin ^2(2 a+2 b x)dx-\frac {1}{2} \int \cos (2 a+2 b x) \sin ^2(2 a+2 b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \sin (2 a+2 b x)^2dx-\frac {1}{2} \int \cos (2 a+2 b x) \sin (2 a+2 b x)^2dx\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {1}{2} \int \sin (2 a+2 b x)^2dx-\frac {\int \sin ^2(2 a+2 b x)d\sin (2 a+2 b x)}{4 b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {1}{2} \int \sin (2 a+2 b x)^2dx-\frac {\sin ^3(2 a+2 b x)}{12 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{2} \left (\frac {\int 1dx}{2}-\frac {\sin (2 a+2 b x) \cos (2 a+2 b x)}{4 b}\right )-\frac {\sin ^3(2 a+2 b x)}{12 b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{2} \left (\frac {x}{2}-\frac {\sin (2 a+2 b x) \cos (2 a+2 b x)}{4 b}\right )-\frac {\sin ^3(2 a+2 b x)}{12 b}\) |
Input:
Int[Sin[a + b*x]^2*Sin[2*a + 2*b*x]^2,x]
Output:
-1/12*Sin[2*a + 2*b*x]^3/b + (x/2 - (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(4 *b))/2
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[sin[(a_.) + (b_.)*(x_)]^2*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbo l] :> Simp[1/2 Int[(g*Sin[c + d*x])^p, x], x] - Simp[1/2 Int[Cos[c + d* x]*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d , 0] && EqQ[d/b, 2] && IGtQ[p/2, 0]
Time = 2.36 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86
method | result | size |
parallelrisch | \(\frac {12 b x -3 \sin \left (2 b x +2 a \right )-3 \sin \left (4 b x +4 a \right )+\sin \left (6 b x +6 a \right )}{48 b}\) | \(42\) |
default | \(\frac {x}{4}-\frac {\sin \left (2 b x +2 a \right )}{16 b}-\frac {\sin \left (4 b x +4 a \right )}{16 b}+\frac {\sin \left (6 b x +6 a \right )}{48 b}\) | \(47\) |
risch | \(\frac {x}{4}-\frac {\sin \left (2 b x +2 a \right )}{16 b}-\frac {\sin \left (4 b x +4 a \right )}{16 b}+\frac {\sin \left (6 b x +6 a \right )}{48 b}\) | \(47\) |
orering | \(x \sin \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right )^{2}-\frac {49 \left (2 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{2} b \sin \left (b x +a \right )+4 b \sin \left (b x +a \right )^{2} \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right )\right )}{144 b^{2}}+\frac {49 x \left (-10 b^{2} \sin \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right )^{2}+16 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right ) b^{2} \sin \left (b x +a \right ) \cos \left (2 b x +2 a \right )+2 \cos \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right )^{2} b^{2}+8 b^{2} \sin \left (b x +a \right )^{2} \cos \left (2 b x +2 a \right )^{2}\right )}{144 b^{2}}-\frac {7 \left (-56 b^{3} \sin \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{2} \cos \left (b x +a \right )-88 b^{3} \sin \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right ) \cos \left (2 b x +2 a \right )+48 \cos \left (b x +a \right ) b^{3} \cos \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )+24 \cos \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right ) b^{3} \cos \left (2 b x +2 a \right )\right )}{288 b^{4}}+\frac {7 x \left (-104 b^{4} \cos \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right )^{2}-640 b^{4} \sin \left (b x +a \right ) \sin \left (2 b x +2 a \right ) \cos \left (b x +a \right ) \cos \left (2 b x +2 a \right )+232 b^{4} \sin \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right )^{2}-224 b^{4} \sin \left (b x +a \right )^{2} \cos \left (2 b x +2 a \right )^{2}+96 \cos \left (b x +a \right )^{2} b^{4} \cos \left (2 b x +2 a \right )^{2}\right )}{288 b^{4}}-\frac {1952 b^{5} \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )-1440 b^{5} \cos \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right ) \cos \left (2 b x +2 a \right )-1920 b^{5} \sin \left (b x +a \right ) \cos \left (2 b x +2 a \right )^{2} \cos \left (b x +a \right )+2464 b^{5} \sin \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right ) \cos \left (2 b x +2 a \right )}{2304 b^{6}}+\frac {x \left (-6880 b^{6} \sin \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right )^{2}+23296 b^{6} \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right ) \cos \left (2 b x +2 a \right )+4832 b^{6} \cos \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right )^{2}-4800 b^{6} \cos \left (b x +a \right )^{2} \cos \left (2 b x +2 a \right )^{2}+6848 b^{6} \sin \left (b x +a \right )^{2} \cos \left (2 b x +2 a \right )^{2}\right )}{2304 b^{6}}\) | \(722\) |
Input:
int(sin(b*x+a)^2*sin(2*b*x+2*a)^2,x,method=_RETURNVERBOSE)
Output:
1/48*(12*b*x-3*sin(2*b*x+2*a)-3*sin(4*b*x+4*a)+sin(6*b*x+6*a))/b
Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int \sin ^2(a+b x) \sin ^2(2 a+2 b x) \, dx=\frac {3 \, b x + {\left (8 \, \cos \left (b x + a\right )^{5} - 14 \, \cos \left (b x + a\right )^{3} + 3 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{12 \, b} \] Input:
integrate(sin(b*x+a)^2*sin(2*b*x+2*a)^2,x, algorithm="fricas")
Output:
1/12*(3*b*x + (8*cos(b*x + a)^5 - 14*cos(b*x + a)^3 + 3*cos(b*x + a))*sin( b*x + a))/b
Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (41) = 82\).
Time = 0.85 (sec) , antiderivative size = 231, normalized size of antiderivative = 4.71 \[ \int \sin ^2(a+b x) \sin ^2(2 a+2 b x) \, dx=\begin {cases} \frac {x \sin ^{2}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )}}{4} + \frac {x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{4} + \frac {x \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {x \cos ^{2}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{4} - \frac {7 \sin ^{2}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{24 b} - \frac {\sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{6 b} - \frac {\sin {\left (a + b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{3 b} + \frac {\sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{24 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\left (a \right )} \sin ^{2}{\left (2 a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(sin(b*x+a)**2*sin(2*b*x+2*a)**2,x)
Output:
Piecewise((x*sin(a + b*x)**2*sin(2*a + 2*b*x)**2/4 + x*sin(a + b*x)**2*cos (2*a + 2*b*x)**2/4 + x*sin(2*a + 2*b*x)**2*cos(a + b*x)**2/4 + x*cos(a + b *x)**2*cos(2*a + 2*b*x)**2/4 - 7*sin(a + b*x)**2*sin(2*a + 2*b*x)*cos(2*a + 2*b*x)/(24*b) - sin(a + b*x)*sin(2*a + 2*b*x)**2*cos(a + b*x)/(6*b) - si n(a + b*x)*cos(a + b*x)*cos(2*a + 2*b*x)**2/(3*b) + sin(2*a + 2*b*x)*cos(a + b*x)**2*cos(2*a + 2*b*x)/(24*b), Ne(b, 0)), (x*sin(a)**2*sin(2*a)**2, T rue))
Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int \sin ^2(a+b x) \sin ^2(2 a+2 b x) \, dx=\frac {12 \, b x + \sin \left (6 \, b x + 6 \, a\right ) - 3 \, \sin \left (4 \, b x + 4 \, a\right ) - 3 \, \sin \left (2 \, b x + 2 \, a\right )}{48 \, b} \] Input:
integrate(sin(b*x+a)^2*sin(2*b*x+2*a)^2,x, algorithm="maxima")
Output:
1/48*(12*b*x + sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - 3*sin(2*b*x + 2*a)) /b
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int \sin ^2(a+b x) \sin ^2(2 a+2 b x) \, dx=\frac {12 \, b x + 12 \, a + \sin \left (6 \, b x + 6 \, a\right ) - 3 \, \sin \left (4 \, b x + 4 \, a\right ) - 3 \, \sin \left (2 \, b x + 2 \, a\right )}{48 \, b} \] Input:
integrate(sin(b*x+a)^2*sin(2*b*x+2*a)^2,x, algorithm="giac")
Output:
1/48*(12*b*x + 12*a + sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - 3*sin(2*b*x + 2*a))/b
Time = 18.59 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int \sin ^2(a+b x) \sin ^2(2 a+2 b x) \, dx=\frac {x}{4}-\frac {\frac {\sin \left (2\,a+2\,b\,x\right )}{16}+\frac {\sin \left (4\,a+4\,b\,x\right )}{16}-\frac {\sin \left (6\,a+6\,b\,x\right )}{48}}{b} \] Input:
int(sin(a + b*x)^2*sin(2*a + 2*b*x)^2,x)
Output:
x/4 - (sin(2*a + 2*b*x)/16 + sin(4*a + 4*b*x)/16 - sin(6*a + 6*b*x)/48)/b
Time = 0.17 (sec) , antiderivative size = 149, normalized size of antiderivative = 3.04 \[ \int \sin ^2(a+b x) \sin ^2(2 a+2 b x) \, dx=\frac {-8 \cos \left (2 b x +2 a \right ) \cos \left (b x +a \right ) \sin \left (b x +a \right )-8 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{2}+\cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right )+4 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )-8 \cos \left (b x +a \right ) \sin \left (b x +a \right )-8 \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{2}+4 \sin \left (2 b x +2 a \right )+6 b x}{24 b} \] Input:
int(sin(b*x+a)^2*sin(2*b*x+2*a)^2,x)
Output:
( - 8*cos(2*a + 2*b*x)*cos(a + b*x)*sin(a + b*x) - 8*cos(2*a + 2*b*x)*sin( 2*a + 2*b*x)*sin(a + b*x)**2 + cos(2*a + 2*b*x)*sin(2*a + 2*b*x) + 4*cos(a + b*x)*sin(2*a + 2*b*x)**2*sin(a + b*x) - 8*cos(a + b*x)*sin(a + b*x) - 8 *sin(2*a + 2*b*x)*sin(a + b*x)**2 + 4*sin(2*a + 2*b*x) + 6*b*x)/(24*b)