Integrand size = 31, antiderivative size = 159 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {7}{16} a^3 (2 A+B) x-\frac {7 a^3 (2 A+B) \cos ^3(c+d x)}{24 d}+\frac {7 a^3 (2 A+B) \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a (2 A+B) \cos ^3(c+d x) (a+a \sin (c+d x))^2}{10 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {7 (2 A+B) \cos ^3(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{40 d} \] Output:
7/16*a^3*(2*A+B)*x-7/24*a^3*(2*A+B)*cos(d*x+c)^3/d+7/16*a^3*(2*A+B)*cos(d* x+c)*sin(d*x+c)/d-1/10*a*(2*A+B)*cos(d*x+c)^3*(a+a*sin(d*x+c))^2/d-1/6*B*c os(d*x+c)^3*(a+a*sin(d*x+c))^3/d-7/40*(2*A+B)*cos(d*x+c)^3*(a^3+a^3*sin(d* x+c))/d
Time = 1.61 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.92 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {a^3 \cos (c+d x) \left (284 A+212 B+\frac {420 (2 A+B) \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(c+d x)}}+16 (17 A+11 B) \cos (2 (c+d x))-12 (A+3 B) \cos (4 (c+d x))-330 A \sin (c+d x)-95 B \sin (c+d x)+90 A \sin (3 (c+d x))+110 B \sin (3 (c+d x))-5 B \sin (5 (c+d x))\right )}{480 d} \] Input:
Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]
Output:
-1/480*(a^3*Cos[c + d*x]*(284*A + 212*B + (420*(2*A + B)*ArcSin[Sqrt[1 - S in[c + d*x]]/Sqrt[2]])/Sqrt[Cos[c + d*x]^2] + 16*(17*A + 11*B)*Cos[2*(c + d*x)] - 12*(A + 3*B)*Cos[4*(c + d*x)] - 330*A*Sin[c + d*x] - 95*B*Sin[c + d*x] + 90*A*Sin[3*(c + d*x)] + 110*B*Sin[3*(c + d*x)] - 5*B*Sin[5*(c + d*x )]))/d
Time = 0.69 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {3042, 3339, 3042, 3157, 3042, 3157, 3042, 3148, 3042, 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.
| \(\displaystyle \int \cos ^2(c+d x) (a \sin (c+d x)+a)^3 (A+B \sin (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^2 (a \sin (c+d x)+a)^3 (A+B \sin (c+d x))dx\) |
| \(\Big \downarrow \) 3339 |
| \(\displaystyle \frac {1}{2} (2 A+B) \int \cos ^2(c+d x) (\sin (c+d x) a+a)^3dx-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} (2 A+B) \int \cos (c+d x)^2 (\sin (c+d x) a+a)^3dx-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {1}{2} (2 A+B) \left (\frac {7}{5} a \int \cos ^2(c+d x) (\sin (c+d x) a+a)^2dx-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} (2 A+B) \left (\frac {7}{5} a \int \cos (c+d x)^2 (\sin (c+d x) a+a)^2dx-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {1}{2} (2 A+B) \left (\frac {7}{5} a \left (\frac {5}{4} a \int \cos ^2(c+d x) (\sin (c+d x) a+a)dx-\frac {\cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{4 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} (2 A+B) \left (\frac {7}{5} a \left (\frac {5}{4} a \int \cos (c+d x)^2 (\sin (c+d x) a+a)dx-\frac {\cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{4 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{2} (2 A+B) \left (\frac {7}{5} a \left (\frac {5}{4} a \left (a \int \cos ^2(c+d x)dx-\frac {a \cos ^3(c+d x)}{3 d}\right )-\frac {\cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{4 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} (2 A+B) \left (\frac {7}{5} a \left (\frac {5}{4} a \left (a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {a \cos ^3(c+d x)}{3 d}\right )-\frac {\cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{4 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{2} (2 A+B) \left (\frac {7}{5} a \left (\frac {5}{4} a \left (a \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {a \cos ^3(c+d x)}{3 d}\right )-\frac {\cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{4 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{2} (2 A+B) \left (\frac {7}{5} a \left (\frac {5}{4} a \left (a \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {a \cos ^3(c+d x)}{3 d}\right )-\frac {\cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{4 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
Input:
Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]
Output:
-1/6*(B*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3)/d + ((2*A + B)*(-1/5*(a*Cos [c + d*x]^3*(a + a*Sin[c + d*x])^2)/d + (7*a*(-1/4*(Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x]))/d + (5*a*(-1/3*(a*Cos[c + d*x]^3)/d + a*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d))))/4))/5))/2
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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers Q[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[ a^2 - b^2, 0] && NeQ[m + p + 1, 0]
Time = 143.98 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.75
| method | result | size |
| parallelrisch | \(\frac {\left (\frac {\left (-13 A -7 B \right ) \cos \left (3 d x +3 c \right )}{12}+\frac {\left (A +3 B \right ) \cos \left (5 d x +5 c \right )}{20}+\left (A -\frac {B}{16}\right ) \sin \left (2 d x +2 c \right )+\frac {\left (-3 A -\frac {7 B}{2}\right ) \sin \left (4 d x +4 c \right )}{8}+\frac {B \sin \left (6 d x +6 c \right )}{48}+\frac {\left (-7 A -5 B \right ) \cos \left (d x +c \right )}{2}+\frac {7 d x A}{2}+\frac {7 d x B}{4}-\frac {68 A}{15}-\frac {44 B}{15}\right ) a^{3}}{4 d}\) | \(120\) |
| risch | \(\frac {7 a^{3} x A}{8}+\frac {7 a^{3} B x}{16}-\frac {7 a^{3} A \cos \left (d x +c \right )}{8 d}-\frac {5 a^{3} \cos \left (d x +c \right ) B}{8 d}+\frac {a^{3} B \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{3} \cos \left (5 d x +5 c \right ) A}{80 d}+\frac {3 a^{3} \cos \left (5 d x +5 c \right ) B}{80 d}-\frac {3 \sin \left (4 d x +4 c \right ) a^{3} A}{32 d}-\frac {7 \sin \left (4 d x +4 c \right ) a^{3} B}{64 d}-\frac {13 a^{3} \cos \left (3 d x +3 c \right ) A}{48 d}-\frac {7 a^{3} \cos \left (3 d x +3 c \right ) B}{48 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} A}{4 d}-\frac {\sin \left (2 d x +2 c \right ) a^{3} B}{64 d}\) | \(208\) |
| derivativedivides | \(\frac {a^{3} A \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+a^{3} B \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{3} A \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+3 a^{3} B \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )-a^{3} A \cos \left (d x +c \right )^{3}+3 a^{3} B \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+a^{3} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {a^{3} B \cos \left (d x +c \right )^{3}}{3}}{d}\) | \(279\) |
| default | \(\frac {a^{3} A \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+a^{3} B \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{3} A \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+3 a^{3} B \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )-a^{3} A \cos \left (d x +c \right )^{3}+3 a^{3} B \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+a^{3} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {a^{3} B \cos \left (d x +c \right )^{3}}{3}}{d}\) | \(279\) |
| norman | \(\frac {\left (\frac {7}{8} a^{3} A +\frac {7}{16} a^{3} B \right ) x +\left (\frac {7}{8} a^{3} A +\frac {7}{16} a^{3} B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {21}{4} a^{3} A +\frac {21}{8} a^{3} B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {21}{4} a^{3} A +\frac {21}{8} a^{3} B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {35}{2} a^{3} A +\frac {35}{4} a^{3} B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {105}{8} a^{3} A +\frac {105}{16} a^{3} B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {105}{8} a^{3} A +\frac {105}{16} a^{3} B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {34 a^{3} A +22 a^{3} B}{15 d}-\frac {2 \left (3 a^{3} A +a^{3} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {\left (12 a^{3} A +4 a^{3} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}-\frac {4 \left (17 a^{3} A +11 a^{3} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {2 \left (19 a^{3} A +17 a^{3} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5 d}-\frac {\left (22 a^{3} A +18 a^{3} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}+\frac {a^{3} \left (2 A -7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {a^{3} \left (2 A -7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {a^{3} \left (26 A +37 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {a^{3} \left (26 A +37 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {a^{3} \left (162 A +73 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}-\frac {a^{3} \left (162 A +73 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\) | \(505\) |
| orering | \(\text {Expression too large to display}\) | \(5958\) |
Input:
int(cos(d*x+c)^2*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x,method=_RETURNVERBO SE)
Output:
1/4*(1/12*(-13*A-7*B)*cos(3*d*x+3*c)+1/20*(A+3*B)*cos(5*d*x+5*c)+(A-1/16*B )*sin(2*d*x+2*c)+1/8*(-3*A-7/2*B)*sin(4*d*x+4*c)+1/48*B*sin(6*d*x+6*c)+1/2 *(-7*A-5*B)*cos(d*x+c)+7/2*d*x*A+7/4*d*x*B-68/15*A-44/15*B)*a^3/d
Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.70 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {48 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} - 320 \, {\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{3} + 105 \, {\left (2 \, A + B\right )} a^{3} d x + 5 \, {\left (8 \, B a^{3} \cos \left (d x + c\right )^{5} - 2 \, {\left (18 \, A + 25 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 21 \, {\left (2 \, A + B\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:
integrate(cos(d*x+c)^2*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="f ricas")
Output:
1/240*(48*(A + 3*B)*a^3*cos(d*x + c)^5 - 320*(A + B)*a^3*cos(d*x + c)^3 + 105*(2*A + B)*a^3*d*x + 5*(8*B*a^3*cos(d*x + c)^5 - 2*(18*A + 25*B)*a^3*co s(d*x + c)^3 + 21*(2*A + B)*a^3*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (151) = 302\).
Time = 0.41 (sec) , antiderivative size = 588, normalized size of antiderivative = 3.70 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)**2*(a+a*sin(d*x+c))**3*(A+B*sin(d*x+c)),x)
Output:
Piecewise((3*A*a**3*x*sin(c + d*x)**4/8 + 3*A*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + A*a**3*x*sin(c + d*x)**2/2 + 3*A*a**3*x*cos(c + d*x)**4/8 + A*a**3*x*cos(c + d*x)**2/2 + 3*A*a**3*sin(c + d*x)**3*cos(c + d*x)/(8*d) - A*a**3*sin(c + d*x)**2*cos(c + d*x)**3/(3*d) - 3*A*a**3*sin(c + d*x)*cos (c + d*x)**3/(8*d) + A*a**3*sin(c + d*x)*cos(c + d*x)/(2*d) - 2*A*a**3*cos (c + d*x)**5/(15*d) - A*a**3*cos(c + d*x)**3/d + B*a**3*x*sin(c + d*x)**6/ 16 + 3*B*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 3*B*a**3*x*sin(c + d* x)**4/8 + 3*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 3*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + B*a**3*x*cos(c + d*x)**6/16 + 3*B*a**3*x*co s(c + d*x)**4/8 + B*a**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) - B*a**3*sin( c + d*x)**3*cos(c + d*x)**3/(6*d) + 3*B*a**3*sin(c + d*x)**3*cos(c + d*x)/ (8*d) - B*a**3*sin(c + d*x)**2*cos(c + d*x)**3/d - B*a**3*sin(c + d*x)*cos (c + d*x)**5/(16*d) - 3*B*a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) - 2*B*a* *3*cos(c + d*x)**5/(5*d) - B*a**3*cos(c + d*x)**3/(3*d), Ne(d, 0)), (x*(A + B*sin(c))*(a*sin(c) + a)**3*cos(c)**2, True))
Time = 0.04 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.25 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {960 \, A a^{3} \cos \left (d x + c\right )^{3} + 320 \, B a^{3} \cos \left (d x + c\right )^{3} - 64 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} A a^{3} - 90 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} A a^{3} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 192 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} B a^{3} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{3} - 90 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{3}}{960 \, d} \] Input:
integrate(cos(d*x+c)^2*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="m axima")
Output:
-1/960*(960*A*a^3*cos(d*x + c)^3 + 320*B*a^3*cos(d*x + c)^3 - 64*(3*cos(d* x + c)^5 - 5*cos(d*x + c)^3)*A*a^3 - 90*(4*d*x + 4*c - sin(4*d*x + 4*c))*A *a^3 - 240*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^3 - 192*(3*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*B*a^3 + 5*(4*sin(2*d*x + 2*c)^3 - 12*d*x - 12*c + 3*si n(4*d*x + 4*c))*B*a^3 - 90*(4*d*x + 4*c - sin(4*d*x + 4*c))*B*a^3)/d
Time = 0.21 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.04 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {B a^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {7}{16} \, {\left (2 \, A a^{3} + B a^{3}\right )} x + \frac {{\left (A a^{3} + 3 \, B a^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {{\left (13 \, A a^{3} + 7 \, B a^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {{\left (7 \, A a^{3} + 5 \, B a^{3}\right )} \cos \left (d x + c\right )}{8 \, d} - \frac {{\left (6 \, A a^{3} + 7 \, B a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (16 \, A a^{3} - B a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \] Input:
integrate(cos(d*x+c)^2*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="g iac")
Output:
1/192*B*a^3*sin(6*d*x + 6*c)/d + 7/16*(2*A*a^3 + B*a^3)*x + 1/80*(A*a^3 + 3*B*a^3)*cos(5*d*x + 5*c)/d - 1/48*(13*A*a^3 + 7*B*a^3)*cos(3*d*x + 3*c)/d - 1/8*(7*A*a^3 + 5*B*a^3)*cos(d*x + c)/d - 1/64*(6*A*a^3 + 7*B*a^3)*sin(4 *d*x + 4*c)/d + 1/64*(16*A*a^3 - B*a^3)*sin(2*d*x + 2*c)/d
Time = 35.36 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.84 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {7\,a^3\,\mathrm {atan}\left (\frac {7\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,A+B\right )}{8\,\left (\frac {7\,A\,a^3}{4}+\frac {7\,B\,a^3}{8}\right )}\right )\,\left (2\,A+B\right )}{8\,d}-\frac {\frac {34\,A\,a^3}{15}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A\,a^3}{4}-\frac {7\,B\,a^3}{8}\right )+\frac {22\,B\,a^3}{15}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (6\,A\,a^3+2\,B\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (12\,A\,a^3+4\,B\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {A\,a^3}{4}-\frac {7\,B\,a^3}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (22\,A\,a^3+18\,B\,a^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {13\,A\,a^3}{2}+\frac {37\,B\,a^3}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {13\,A\,a^3}{2}+\frac {37\,B\,a^3}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {38\,A\,a^3}{5}+\frac {34\,B\,a^3}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {68\,A\,a^3}{3}+\frac {44\,B\,a^3}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {27\,A\,a^3}{4}+\frac {73\,B\,a^3}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {27\,A\,a^3}{4}+\frac {73\,B\,a^3}{24}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {7\,a^3\,\left (2\,A+B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d} \] Input:
int(cos(c + d*x)^2*(A + B*sin(c + d*x))*(a + a*sin(c + d*x))^3,x)
Output:
(7*a^3*atan((7*a^3*tan(c/2 + (d*x)/2)*(2*A + B))/(8*((7*A*a^3)/4 + (7*B*a^ 3)/8)))*(2*A + B))/(8*d) - ((34*A*a^3)/15 - tan(c/2 + (d*x)/2)*((A*a^3)/4 - (7*B*a^3)/8) + (22*B*a^3)/15 + tan(c/2 + (d*x)/2)^10*(6*A*a^3 + 2*B*a^3) + tan(c/2 + (d*x)/2)^4*(12*A*a^3 + 4*B*a^3) + tan(c/2 + (d*x)/2)^11*((A*a ^3)/4 - (7*B*a^3)/8) + tan(c/2 + (d*x)/2)^8*(22*A*a^3 + 18*B*a^3) - tan(c/ 2 + (d*x)/2)^5*((13*A*a^3)/2 + (37*B*a^3)/4) + tan(c/2 + (d*x)/2)^7*((13*A *a^3)/2 + (37*B*a^3)/4) + tan(c/2 + (d*x)/2)^2*((38*A*a^3)/5 + (34*B*a^3)/ 5) + tan(c/2 + (d*x)/2)^6*((68*A*a^3)/3 + (44*B*a^3)/3) - tan(c/2 + (d*x)/ 2)^3*((27*A*a^3)/4 + (73*B*a^3)/24) + tan(c/2 + (d*x)/2)^9*((27*A*a^3)/4 + (73*B*a^3)/24))/(d*(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c/2 + (d*x)/2)^4 + 20 *tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 + 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) - (7*a^3*(2*A + B)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(8*d)
Time = 0.17 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.21 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {a^{3} \left (40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b +48 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a +144 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b +180 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a +170 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b +224 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a +32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b +30 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a -105 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b -272 \cos \left (d x +c \right ) a -176 \cos \left (d x +c \right ) b +210 a d x +272 a +105 b d x +176 b \right )}{240 d} \] Input:
int(cos(d*x+c)^2*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x)
Output:
(a**3*(40*cos(c + d*x)*sin(c + d*x)**5*b + 48*cos(c + d*x)*sin(c + d*x)**4 *a + 144*cos(c + d*x)*sin(c + d*x)**4*b + 180*cos(c + d*x)*sin(c + d*x)**3 *a + 170*cos(c + d*x)*sin(c + d*x)**3*b + 224*cos(c + d*x)*sin(c + d*x)**2 *a + 32*cos(c + d*x)*sin(c + d*x)**2*b + 30*cos(c + d*x)*sin(c + d*x)*a - 105*cos(c + d*x)*sin(c + d*x)*b - 272*cos(c + d*x)*a - 176*cos(c + d*x)*b + 210*a*d*x + 272*a + 105*b*d*x + 176*b))/(240*d)