Integrand size = 10, antiderivative size = 87 \[ \int \left (a+b \cos ^2(x)\right )^3 \, dx=\frac {1}{16} (2 a+b) \left (8 a^2+8 a b+5 b^2\right ) x+\frac {1}{48} b \left (64 a^2+54 a b+15 b^2\right ) \cos (x) \sin (x)+\frac {5}{24} b^2 (2 a+b) \cos ^3(x) \sin (x)+\frac {1}{6} b \cos (x) \left (a+b \cos ^2(x)\right )^2 \sin (x) \] Output:
1/16*(2*a+b)*(8*a^2+8*a*b+5*b^2)*x+1/48*b*(64*a^2+54*a*b+15*b^2)*cos(x)*si n(x)+5/24*b^2*(2*a+b)*cos(x)^3*sin(x)+1/6*b*cos(x)*(a+b*cos(x)^2)^2*sin(x)
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \left (a+b \cos ^2(x)\right )^3 \, dx=\frac {1}{192} \left (12 (2 a+b) \left (8 a^2+8 a b+5 b^2\right ) x+9 b (4 a+(2-i) b) (4 a+(2+i) b) \sin (2 x)+9 b^2 (2 a+b) \sin (4 x)+b^3 \sin (6 x)\right ) \] Input:
Integrate[(a + b*Cos[x]^2)^3,x]
Output:
(12*(2*a + b)*(8*a^2 + 8*a*b + 5*b^2)*x + 9*b*(4*a + (2 - I)*b)*(4*a + (2 + I)*b)*Sin[2*x] + 9*b^2*(2*a + b)*Sin[4*x] + b^3*Sin[6*x])/192
Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3659, 3042, 3648}
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.
\(\displaystyle \int \left (a+b \cos ^2(x)\right )^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a+b \sin \left (x+\frac {\pi }{2}\right )^2\right )^3dx\) |
\(\Big \downarrow \) 3659 |
\(\displaystyle \frac {1}{6} \int \left (b \cos ^2(x)+a\right ) \left (5 b (2 a+b) \cos ^2(x)+a (6 a+b)\right )dx+\frac {1}{6} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^2\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \int \left (b \sin \left (x+\frac {\pi }{2}\right )^2+a\right ) \left (5 b (2 a+b) \sin \left (x+\frac {\pi }{2}\right )^2+a (6 a+b)\right )dx+\frac {1}{6} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^2\) |
\(\Big \downarrow \) 3648 |
\(\displaystyle \frac {1}{6} \left (\frac {3}{8} x (2 a+b) \left (8 a^2+8 a b+5 b^2\right )+\frac {1}{8} b \left (64 a^2+54 a b+15 b^2\right ) \sin (x) \cos (x)+\frac {5}{4} b^2 (2 a+b) \sin (x) \cos ^3(x)\right )+\frac {1}{6} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^2\) |
Input:
Int[(a + b*Cos[x]^2)^3,x]
Output:
(b*Cos[x]*(a + b*Cos[x]^2)^2*Sin[x])/6 + ((3*(2*a + b)*(8*a^2 + 8*a*b + 5* b^2)*x)/8 + (b*(64*a^2 + 54*a*b + 15*b^2)*Cos[x]*Sin[x])/8 + (5*b^2*(2*a + b)*Cos[x]^3*Sin[x])/4)/6
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)*((A_.) + (B_.)*sin[(e_.) + (f_ .)*(x_)]^2), x_Symbol] :> Simp[(4*A*(2*a + b) + B*(4*a + 3*b))*(x/8), x] + (-Simp[b*B*Cos[e + f*x]*(Sin[e + f*x]^3/(4*f)), x] - Simp[(4*A*b + B*(4*a + 3*b))*Cos[e + f*x]*(Sin[e + f*x]/(8*f)), x]) /; FreeQ[{a, b, e, f, A, B}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p - 1)/(2*f*p)), x] + Sim p[1/(2*p) Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(2*a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[ a + b, 0] && GtQ[p, 1]
Time = 1.94 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80
method | result | size |
parallelrisch | \(\frac {3 \left (16 a^{2} b +16 a \,b^{2}+5 b^{3}\right ) \sin \left (2 x \right )}{64}+\frac {3 \left (2 a \,b^{2}+b^{3}\right ) \sin \left (4 x \right )}{64}+\frac {b^{3} \sin \left (6 x \right )}{192}+x \left (a +\frac {b}{2}\right ) \left (a^{2}+b a +\frac {5}{8} b^{2}\right )\) | \(70\) |
default | \(b^{3} \left (\frac {\left (\cos \left (x \right )^{5}+\frac {5 \cos \left (x \right )^{3}}{4}+\frac {15 \cos \left (x \right )}{8}\right ) \sin \left (x \right )}{6}+\frac {5 x}{16}\right )+3 a \,b^{2} \left (\frac {\left (\cos \left (x \right )^{3}+\frac {3 \cos \left (x \right )}{2}\right ) \sin \left (x \right )}{4}+\frac {3 x}{8}\right )+3 a^{2} b \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+a^{3} x\) | \(73\) |
parts | \(b^{3} \left (\frac {\left (\cos \left (x \right )^{5}+\frac {5 \cos \left (x \right )^{3}}{4}+\frac {15 \cos \left (x \right )}{8}\right ) \sin \left (x \right )}{6}+\frac {5 x}{16}\right )+3 a \,b^{2} \left (\frac {\left (\cos \left (x \right )^{3}+\frac {3 \cos \left (x \right )}{2}\right ) \sin \left (x \right )}{4}+\frac {3 x}{8}\right )+3 a^{2} b \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+a^{3} x\) | \(73\) |
risch | \(a^{3} x +\frac {3 a^{2} b x}{2}+\frac {9 a \,b^{2} x}{8}+\frac {5 b^{3} x}{16}+\frac {b^{3} \sin \left (6 x \right )}{192}+\frac {3 \sin \left (4 x \right ) a \,b^{2}}{32}+\frac {3 \sin \left (4 x \right ) b^{3}}{64}+\frac {3 \sin \left (2 x \right ) a^{2} b}{4}+\frac {3 \sin \left (2 x \right ) a \,b^{2}}{4}+\frac {15 \sin \left (2 x \right ) b^{3}}{64}\) | \(84\) |
norman | \(\frac {\left (-9 a^{2} b -\frac {21}{4} a \,b^{2}+\frac {5}{24} b^{3}\right ) \tan \left (\frac {x}{2}\right )^{9}+\left (-6 a^{2} b -\frac {3}{2} a \,b^{2}-\frac {15}{4} b^{3}\right ) \tan \left (\frac {x}{2}\right )^{7}+\left (-3 a^{2} b -\frac {15}{4} a \,b^{2}-\frac {11}{8} b^{3}\right ) \tan \left (\frac {x}{2}\right )^{11}+\left (3 a^{2} b +\frac {15}{4} a \,b^{2}+\frac {11}{8} b^{3}\right ) \tan \left (\frac {x}{2}\right )+\left (6 a^{2} b +\frac {3}{2} a \,b^{2}+\frac {15}{4} b^{3}\right ) \tan \left (\frac {x}{2}\right )^{5}+\left (9 a^{2} b +\frac {21}{4} a \,b^{2}-\frac {5}{24} b^{3}\right ) \tan \left (\frac {x}{2}\right )^{3}+\left (a^{3}+\frac {3}{2} a^{2} b +\frac {9}{8} a \,b^{2}+\frac {5}{16} b^{3}\right ) x +\left (a^{3}+\frac {3}{2} a^{2} b +\frac {9}{8} a \,b^{2}+\frac {5}{16} b^{3}\right ) x \tan \left (\frac {x}{2}\right )^{12}+\left (6 a^{3}+9 a^{2} b +\frac {27}{4} a \,b^{2}+\frac {15}{8} b^{3}\right ) x \tan \left (\frac {x}{2}\right )^{2}+\left (6 a^{3}+9 a^{2} b +\frac {27}{4} a \,b^{2}+\frac {15}{8} b^{3}\right ) x \tan \left (\frac {x}{2}\right )^{10}+\left (15 a^{3}+\frac {45}{2} a^{2} b +\frac {135}{8} a \,b^{2}+\frac {75}{16} b^{3}\right ) x \tan \left (\frac {x}{2}\right )^{4}+\left (15 a^{3}+\frac {45}{2} a^{2} b +\frac {135}{8} a \,b^{2}+\frac {75}{16} b^{3}\right ) x \tan \left (\frac {x}{2}\right )^{8}+\left (20 a^{3}+30 a^{2} b +\frac {45}{2} a \,b^{2}+\frac {25}{4} b^{3}\right ) x \tan \left (\frac {x}{2}\right )^{6}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{6}}\) | \(368\) |
orering | \(\text {Expression too large to display}\) | \(884\) |
Input:
int((a+b*cos(x)^2)^3,x,method=_RETURNVERBOSE)
Output:
3/64*(16*a^2*b+16*a*b^2+5*b^3)*sin(2*x)+3/64*(2*a*b^2+b^3)*sin(4*x)+1/192* b^3*sin(6*x)+x*(a+1/2*b)*(a^2+b*a+5/8*b^2)
Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93 \[ \int \left (a+b \cos ^2(x)\right )^3 \, dx=\frac {1}{16} \, {\left (16 \, a^{3} + 24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} x + \frac {1}{48} \, {\left (8 \, b^{3} \cos \left (x\right )^{5} + 2 \, {\left (18 \, a b^{2} + 5 \, b^{3}\right )} \cos \left (x\right )^{3} + 3 \, {\left (24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right ) \] Input:
integrate((a+b*cos(x)^2)^3,x, algorithm="fricas")
Output:
1/16*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*x + 1/48*(8*b^3*cos(x)^5 + 2*( 18*a*b^2 + 5*b^3)*cos(x)^3 + 3*(24*a^2*b + 18*a*b^2 + 5*b^3)*cos(x))*sin(x )
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (88) = 176\).
Time = 0.25 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.83 \[ \int \left (a+b \cos ^2(x)\right )^3 \, dx=a^{3} x + \frac {3 a^{2} b x \sin ^{2}{\left (x \right )}}{2} + \frac {3 a^{2} b x \cos ^{2}{\left (x \right )}}{2} + \frac {3 a^{2} b \sin {\left (x \right )} \cos {\left (x \right )}}{2} + \frac {9 a b^{2} x \sin ^{4}{\left (x \right )}}{8} + \frac {9 a b^{2} x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{4} + \frac {9 a b^{2} x \cos ^{4}{\left (x \right )}}{8} + \frac {9 a b^{2} \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{8} + \frac {15 a b^{2} \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{8} + \frac {5 b^{3} x \sin ^{6}{\left (x \right )}}{16} + \frac {15 b^{3} x \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{16} + \frac {15 b^{3} x \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{16} + \frac {5 b^{3} x \cos ^{6}{\left (x \right )}}{16} + \frac {5 b^{3} \sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{16} + \frac {5 b^{3} \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{6} + \frac {11 b^{3} \sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{16} \] Input:
integrate((a+b*cos(x)**2)**3,x)
Output:
a**3*x + 3*a**2*b*x*sin(x)**2/2 + 3*a**2*b*x*cos(x)**2/2 + 3*a**2*b*sin(x) *cos(x)/2 + 9*a*b**2*x*sin(x)**4/8 + 9*a*b**2*x*sin(x)**2*cos(x)**2/4 + 9* a*b**2*x*cos(x)**4/8 + 9*a*b**2*sin(x)**3*cos(x)/8 + 15*a*b**2*sin(x)*cos( x)**3/8 + 5*b**3*x*sin(x)**6/16 + 15*b**3*x*sin(x)**4*cos(x)**2/16 + 15*b* *3*x*sin(x)**2*cos(x)**4/16 + 5*b**3*x*cos(x)**6/16 + 5*b**3*sin(x)**5*cos (x)/16 + 5*b**3*sin(x)**3*cos(x)**3/6 + 11*b**3*sin(x)*cos(x)**5/16
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.79 \[ \int \left (a+b \cos ^2(x)\right )^3 \, dx=-\frac {1}{192} \, {\left (4 \, \sin \left (2 \, x\right )^{3} - 60 \, x - 9 \, \sin \left (4 \, x\right ) - 48 \, \sin \left (2 \, x\right )\right )} b^{3} + \frac {3}{32} \, a b^{2} {\left (12 \, x + \sin \left (4 \, x\right ) + 8 \, \sin \left (2 \, x\right )\right )} + \frac {3}{4} \, a^{2} b {\left (2 \, x + \sin \left (2 \, x\right )\right )} + a^{3} x \] Input:
integrate((a+b*cos(x)^2)^3,x, algorithm="maxima")
Output:
-1/192*(4*sin(2*x)^3 - 60*x - 9*sin(4*x) - 48*sin(2*x))*b^3 + 3/32*a*b^2*( 12*x + sin(4*x) + 8*sin(2*x)) + 3/4*a^2*b*(2*x + sin(2*x)) + a^3*x
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int \left (a+b \cos ^2(x)\right )^3 \, dx=\frac {1}{192} \, b^{3} \sin \left (6 \, x\right ) + \frac {1}{16} \, {\left (16 \, a^{3} + 24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} x + \frac {3}{64} \, {\left (2 \, a b^{2} + b^{3}\right )} \sin \left (4 \, x\right ) + \frac {3}{64} \, {\left (16 \, a^{2} b + 16 \, a b^{2} + 5 \, b^{3}\right )} \sin \left (2 \, x\right ) \] Input:
integrate((a+b*cos(x)^2)^3,x, algorithm="giac")
Output:
1/192*b^3*sin(6*x) + 1/16*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*x + 3/64* (2*a*b^2 + b^3)*sin(4*x) + 3/64*(16*a^2*b + 16*a*b^2 + 5*b^3)*sin(2*x)
Time = 1.02 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.34 \[ \int \left (a+b \cos ^2(x)\right )^3 \, dx=a^3\,x+\frac {5\,b^3\,x}{16}+\frac {\left (72\,a^2\,b+54\,a\,b^2+15\,b^3\right )\,{\mathrm {tan}\left (x\right )}^5+\left (144\,a^2\,b+144\,a\,b^2+40\,b^3\right )\,{\mathrm {tan}\left (x\right )}^3+\left (72\,a^2\,b+90\,a\,b^2+33\,b^3\right )\,\mathrm {tan}\left (x\right )}{48\,{\mathrm {tan}\left (x\right )}^6+144\,{\mathrm {tan}\left (x\right )}^4+144\,{\mathrm {tan}\left (x\right )}^2+48}+\frac {9\,a\,b^2\,x}{8}+\frac {3\,a^2\,b\,x}{2} \] Input:
int((a + b*cos(x)^2)^3,x)
Output:
a^3*x + (5*b^3*x)/16 + (tan(x)^5*(54*a*b^2 + 72*a^2*b + 15*b^3) + tan(x)^3 *(144*a*b^2 + 144*a^2*b + 40*b^3) + tan(x)*(90*a*b^2 + 72*a^2*b + 33*b^3)) /(144*tan(x)^2 + 144*tan(x)^4 + 48*tan(x)^6 + 48) + (9*a*b^2*x)/8 + (3*a^2 *b*x)/2
Time = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02 \[ \int \left (a+b \cos ^2(x)\right )^3 \, dx=\frac {\cos \left (x \right ) \sin \left (x \right )^{5} b^{3}}{6}-\frac {3 \cos \left (x \right ) \sin \left (x \right )^{3} a \,b^{2}}{4}-\frac {13 \cos \left (x \right ) \sin \left (x \right )^{3} b^{3}}{24}+\frac {3 \cos \left (x \right ) \sin \left (x \right ) a^{2} b}{2}+\frac {15 \cos \left (x \right ) \sin \left (x \right ) a \,b^{2}}{8}+\frac {11 \cos \left (x \right ) \sin \left (x \right ) b^{3}}{16}+a^{3} x +\frac {3 a^{2} b x}{2}+\frac {9 a \,b^{2} x}{8}+\frac {5 b^{3} x}{16} \] Input:
int((a+b*cos(x)^2)^3,x)
Output:
(8*cos(x)*sin(x)**5*b**3 - 36*cos(x)*sin(x)**3*a*b**2 - 26*cos(x)*sin(x)** 3*b**3 + 72*cos(x)*sin(x)*a**2*b + 90*cos(x)*sin(x)*a*b**2 + 33*cos(x)*sin (x)*b**3 + 48*a**3*x + 72*a**2*b*x + 54*a*b**2*x + 15*b**3*x)/48