Integrand size = 17, antiderivative size = 36 \[ \int \cos (x) \sin ^3(x) \left (a+b \sin ^2(x)\right )^3 \, dx=-\frac {a \left (a+b \sin ^2(x)\right )^4}{8 b^2}+\frac {\left (a+b \sin ^2(x)\right )^5}{10 b^2} \] Output:
-1/8*a*(a+b*sin(x)^2)^4/b^2+1/10*(a+b*sin(x)^2)^5/b^2
Leaf count is larger than twice the leaf count of optimal. \(128\) vs. \(2(36)=72\).
Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.56 \[ \int \cos (x) \sin ^3(x) \left (a+b \sin ^2(x)\right )^3 \, dx=\frac {-20 \left (64 a^3+24 a b^2+7 b^3\right ) \cos (2 x)+20 \left (16 a^3+18 a b^2+5 b^3\right ) \cos (4 x)+b \left (-10 b (16 a+5 b) \cos (6 x)+15 b (2 a+b) \cos (8 x)-2 b^2 \cos (10 x)+3840 a^2 \sin ^4(x)+2560 a b \sin ^6(x)+640 b^2 \sin ^8(x)-1280 a^2 \sin ^3(x) \sin (3 x)\right )}{10240} \] Input:
Integrate[Cos[x]*Sin[x]^3*(a + b*Sin[x]^2)^3,x]
Output:
(-20*(64*a^3 + 24*a*b^2 + 7*b^3)*Cos[2*x] + 20*(16*a^3 + 18*a*b^2 + 5*b^3) *Cos[4*x] + b*(-10*b*(16*a + 5*b)*Cos[6*x] + 15*b*(2*a + b)*Cos[8*x] - 2*b ^2*Cos[10*x] + 3840*a^2*Sin[x]^4 + 2560*a*b*Sin[x]^6 + 640*b^2*Sin[x]^8 - 1280*a^2*Sin[x]^3*Sin[3*x]))/10240
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3042, 3677, 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.
\(\displaystyle \int \sin ^3(x) \cos (x) \left (a+b \sin ^2(x)\right )^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (x)^3 \cos (x) \left (a+b \sin (x)^2\right )^3dx\) |
\(\Big \downarrow \) 3677 |
\(\displaystyle \int \sin ^3(x) \left (a+b \sin ^2(x)\right )^3d\sin (x)\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \sin ^2(x) \left (b \sin ^2(x)+a\right )^3d\sin ^2(x)\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (\frac {\left (b \sin ^2(x)+a\right )^4}{b}-\frac {a \left (b \sin ^2(x)+a\right )^3}{b}\right )d\sin ^2(x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (a+b \sin ^2(x)\right )^5}{5 b^2}-\frac {a \left (a+b \sin ^2(x)\right )^4}{4 b^2}\right )\) |
Input:
Int[Cos[x]*Sin[x]^3*(a + b*Sin[x]^2)^3,x]
Output:
(-1/4*(a*(a + b*Sin[x]^2)^4)/b^2 + (a + b*Sin[x]^2)^5/(5*b^2))/2
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[cos[(e_.) + (f_.)*(x_)]^(m_.)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a _) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeFa ctors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[(d*ff*x)^n*(1 - ff^2*x^2)^(( m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(m - 1)/2]
Time = 63.88 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(\frac {\sin \left (x \right )^{10} b^{3}}{10}+\frac {3 \sin \left (x \right )^{8} a \,b^{2}}{8}+\frac {\sin \left (x \right )^{6} a^{2} b}{2}+\frac {a^{3} \sin \left (x \right )^{4}}{4}\) | \(40\) |
default | \(\frac {\sin \left (x \right )^{10} b^{3}}{10}+\frac {3 \sin \left (x \right )^{8} a \,b^{2}}{8}+\frac {\sin \left (x \right )^{6} a^{2} b}{2}+\frac {a^{3} \sin \left (x \right )^{4}}{4}\) | \(40\) |
parallelrisch | \(\frac {\left (-64 a^{3}-120 a^{2} b -84 a \,b^{2}-21 b^{3}\right ) \cos \left (2 x \right )}{512}+\frac {\left (8 a^{3}+24 a^{2} b +21 a \,b^{2}+6 b^{3}\right ) \cos \left (4 x \right )}{256}-\frac {\left (a +\frac {3 b}{4}\right )^{2} b \cos \left (6 x \right )}{64}+\frac {\left (3 a \,b^{2}+2 b^{3}\right ) \cos \left (8 x \right )}{1024}-\frac {b^{3} \cos \left (10 x \right )}{5120}+\frac {3 a^{3}}{32}+\frac {5 a^{2} b}{32}+\frac {105 a \,b^{2}}{1024}+\frac {63 b^{3}}{2560}\) | \(123\) |
risch | \(-\frac {b^{3} \cos \left (10 x \right )}{5120}+\frac {3 \cos \left (8 x \right ) a \,b^{2}}{1024}+\frac {\cos \left (8 x \right ) b^{3}}{512}-\frac {\cos \left (6 x \right ) a^{2} b}{64}-\frac {3 \cos \left (6 x \right ) a \,b^{2}}{128}-\frac {9 \cos \left (6 x \right ) b^{3}}{1024}+\frac {\cos \left (4 x \right ) a^{3}}{32}+\frac {3 \cos \left (4 x \right ) a^{2} b}{32}+\frac {21 \cos \left (4 x \right ) a \,b^{2}}{256}+\frac {3 \cos \left (4 x \right ) b^{3}}{128}-\frac {\cos \left (2 x \right ) a^{3}}{8}-\frac {15 \cos \left (2 x \right ) a^{2} b}{64}-\frac {21 \cos \left (2 x \right ) a \,b^{2}}{128}-\frac {21 \cos \left (2 x \right ) b^{3}}{512}\) | \(135\) |
orering | \(-\frac {21 \cos \left (x \right )^{8} \sin \left (x \right )^{2} b^{3}}{1024}+\frac {5 \sin \left (x \right )^{4} \left (a +b \sin \left (x \right )^{2}\right )^{3}}{32}-\frac {3 \cos \left (x \right )^{2} \sin \left (x \right )^{2} \left (a +b \sin \left (x \right )^{2}\right )^{3}}{16}+\frac {3 \cos \left (x \right )^{2} \sin \left (x \right )^{4} \left (a +b \sin \left (x \right )^{2}\right )^{2} b}{32}-\frac {3 \cos \left (x \right )^{4} \sin \left (x \right )^{6} b^{3}}{512}+\frac {3 \cos \left (x \right )^{2} \sin \left (x \right )^{8} b^{3}}{512}-\frac {63 \cos \left (x \right )^{10} b^{3}}{2560}+\frac {\cos \left (x \right )^{6} \sin \left (x \right )^{4} b^{3}}{128}-\frac {49 \sin \left (x \right )^{10} b^{3}}{5120}-\frac {\sin \left (x \right )^{6} \left (a +b \sin \left (x \right )^{2}\right )^{2} b}{8}-\frac {3 \cos \left (x \right )^{4} \left (a +b \sin \left (x \right )^{2}\right )^{3}}{32}+\frac {55 \sin \left (x \right )^{8} \left (a +b \sin \left (x \right )^{2}\right ) b^{2}}{1024}-\frac {9 \sin \left (x \right )^{6} \left (a +b \sin \left (x \right )^{2}\right ) b^{2} \cos \left (x \right )^{2}}{256}-\frac {3 \cos \left (x \right )^{4} \sin \left (x \right )^{2} \left (a +b \sin \left (x \right )^{2}\right )^{2} b}{16}+\frac {21 \cos \left (x \right )^{4} \sin \left (x \right )^{4} \left (a +b \sin \left (x \right )^{2}\right ) b^{2}}{512}-\frac {5 \cos \left (x \right )^{6} \left (a +b \sin \left (x \right )^{2}\right )^{2} b}{32}-\frac {25 \cos \left (x \right )^{6} \left (a +b \sin \left (x \right )^{2}\right ) b^{2} \sin \left (x \right )^{2}}{256}-\frac {105 \cos \left (x \right )^{8} \left (a +b \sin \left (x \right )^{2}\right ) b^{2}}{1024}\) | \(297\) |
Input:
int(cos(x)*sin(x)^3*(a+b*sin(x)^2)^3,x,method=_RETURNVERBOSE)
Output:
1/10*sin(x)^10*b^3+3/8*sin(x)^8*a*b^2+1/2*sin(x)^6*a^2*b+1/4*a^3*sin(x)^4
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (32) = 64\).
Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.86 \[ \int \cos (x) \sin ^3(x) \left (a+b \sin ^2(x)\right )^3 \, dx=-\frac {1}{10} \, b^{3} \cos \left (x\right )^{10} + \frac {1}{8} \, {\left (3 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (x\right )^{8} - \frac {1}{2} \, {\left (a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} \cos \left (x\right )^{6} + \frac {1}{4} \, {\left (a^{3} + 6 \, a^{2} b + 9 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (x\right )^{4} - \frac {1}{2} \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{2} \] Input:
integrate(cos(x)*sin(x)^3*(a+b*sin(x)^2)^3,x, algorithm="fricas")
Output:
-1/10*b^3*cos(x)^10 + 1/8*(3*a*b^2 + 4*b^3)*cos(x)^8 - 1/2*(a^2*b + 3*a*b^ 2 + 2*b^3)*cos(x)^6 + 1/4*(a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cos(x)^4 - 1/2 *(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(x)^2
Time = 1.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int \cos (x) \sin ^3(x) \left (a+b \sin ^2(x)\right )^3 \, dx=\frac {a^{3} \sin ^{4}{\left (x \right )}}{4} + \frac {a^{2} b \sin ^{6}{\left (x \right )}}{2} + \frac {3 a b^{2} \sin ^{8}{\left (x \right )}}{8} + \frac {b^{3} \sin ^{10}{\left (x \right )}}{10} \] Input:
integrate(cos(x)*sin(x)**3*(a+b*sin(x)**2)**3,x)
Output:
a**3*sin(x)**4/4 + a**2*b*sin(x)**6/2 + 3*a*b**2*sin(x)**8/8 + b**3*sin(x) **10/10
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int \cos (x) \sin ^3(x) \left (a+b \sin ^2(x)\right )^3 \, dx=\frac {1}{10} \, b^{3} \sin \left (x\right )^{10} + \frac {3}{8} \, a b^{2} \sin \left (x\right )^{8} + \frac {1}{2} \, a^{2} b \sin \left (x\right )^{6} + \frac {1}{4} \, a^{3} \sin \left (x\right )^{4} \] Input:
integrate(cos(x)*sin(x)^3*(a+b*sin(x)^2)^3,x, algorithm="maxima")
Output:
1/10*b^3*sin(x)^10 + 3/8*a*b^2*sin(x)^8 + 1/2*a^2*b*sin(x)^6 + 1/4*a^3*sin (x)^4
Time = 0.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int \cos (x) \sin ^3(x) \left (a+b \sin ^2(x)\right )^3 \, dx=\frac {1}{10} \, b^{3} \sin \left (x\right )^{10} + \frac {3}{8} \, a b^{2} \sin \left (x\right )^{8} + \frac {1}{2} \, a^{2} b \sin \left (x\right )^{6} + \frac {1}{4} \, a^{3} \sin \left (x\right )^{4} \] Input:
integrate(cos(x)*sin(x)^3*(a+b*sin(x)^2)^3,x, algorithm="giac")
Output:
1/10*b^3*sin(x)^10 + 3/8*a*b^2*sin(x)^8 + 1/2*a^2*b*sin(x)^6 + 1/4*a^3*sin (x)^4
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.03 \[ \int \cos (x) \sin ^3(x) \left (a+b \sin ^2(x)\right )^3 \, dx=\frac {b^2\,{\cos \left (x\right )}^8\,\left (3\,a+4\,b\right )}{8}-\frac {b^3\,{\cos \left (x\right )}^{10}}{10}-\frac {{\cos \left (x\right )}^2\,{\left (a+b\right )}^3}{2}-\frac {b\,{\cos \left (x\right )}^6\,\left (a^2+3\,a\,b+2\,b^2\right )}{2}+\frac {{\cos \left (x\right )}^4\,{\left (a+b\right )}^2\,\left (a+4\,b\right )}{4} \] Input:
int(cos(x)*sin(x)^3*(a + b*sin(x)^2)^3,x)
Output:
(b^2*cos(x)^8*(3*a + 4*b))/8 - (b^3*cos(x)^10)/10 - (cos(x)^2*(a + b)^3)/2 - (b*cos(x)^6*(3*a*b + a^2 + 2*b^2))/2 + (cos(x)^4*(a + b)^2*(a + 4*b))/4
Time = 0.17 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int \cos (x) \sin ^3(x) \left (a+b \sin ^2(x)\right )^3 \, dx=\frac {\sin \left (x \right )^{4} \left (4 \sin \left (x \right )^{6} b^{3}+15 \sin \left (x \right )^{4} a \,b^{2}+20 \sin \left (x \right )^{2} a^{2} b +10 a^{3}\right )}{40} \] Input:
int(cos(x)*sin(x)^3*(a+b*sin(x)^2)^3,x)
Output:
(sin(x)**4*(4*sin(x)**6*b**3 + 15*sin(x)**4*a*b**2 + 20*sin(x)**2*a**2*b + 10*a**3))/40