\(\int \cos ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx\) [996]

Optimal result

Integrand size = 31, antiderivative size = 200 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {9}{128} a^3 (8 A+3 B) x-\frac {3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}+\frac {9 a^3 (8 A+3 B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {3 a^3 (8 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {a (8 A+3 B) \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac {3 (8 A+3 B) \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d} \]

[Out]

9/128*a^3*(8*A+3*B)*x-3/80*a^3*(8*A+3*B)*cos(d*x+c)^5/d+9/128*a^3*(8*A+3*B)*cos(d*x+c)*sin(d*x+c)/d+3/64*a^3*(
8*A+3*B)*cos(d*x+c)^3*sin(d*x+c)/d-1/56*a*(8*A+3*B)*cos(d*x+c)^5*(a+a*sin(d*x+c))^2/d-1/8*B*cos(d*x+c)^5*(a+a*
sin(d*x+c))^3/d-3/112*(8*A+3*B)*cos(d*x+c)^5*(a^3+a^3*sin(d*x+c))/d

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2939, 2757, 2748, 2715, 8} \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}-\frac {3 (8 A+3 B) \cos ^5(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{112 d}+\frac {3 a^3 (8 A+3 B) \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {9 a^3 (8 A+3 B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {9}{128} a^3 x (8 A+3 B)-\frac {a (8 A+3 B) \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{56 d}-\frac {B \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d} \]

[In]

Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]

[Out]

(9*a^3*(8*A + 3*B)*x)/128 - (3*a^3*(8*A + 3*B)*Cos[c + d*x]^5)/(80*d) + (9*a^3*(8*A + 3*B)*Cos[c + d*x]*Sin[c
+ d*x])/(128*d) + (3*a^3*(8*A + 3*B)*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (a*(8*A + 3*B)*Cos[c + d*x]^5*(a +
a*Sin[c + d*x])^2)/(56*d) - (B*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3)/(8*d) - (3*(8*A + 3*B)*Cos[c + d*x]^5*(a
^3 + a^3*Sin[c + d*x]))/(112*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[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])

Rule 2757

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] + Dist[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] && IntegersQ[2*m, 2*p]

Rule 2939

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
] + Dist[(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] /; F
reeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}+\frac {1}{8} (8 A+3 B) \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx \\ & = -\frac {a (8 A+3 B) \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}+\frac {1}{56} (9 a (8 A+3 B)) \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = -\frac {a (8 A+3 B) \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac {3 (8 A+3 B) \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d}+\frac {1}{16} \left (3 a^2 (8 A+3 B)\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}-\frac {a (8 A+3 B) \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac {3 (8 A+3 B) \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d}+\frac {1}{16} \left (3 a^3 (8 A+3 B)\right ) \int \cos ^4(c+d x) \, dx \\ & = -\frac {3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}+\frac {3 a^3 (8 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {a (8 A+3 B) \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac {3 (8 A+3 B) \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d}+\frac {1}{64} \left (9 a^3 (8 A+3 B)\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}+\frac {9 a^3 (8 A+3 B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {3 a^3 (8 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {a (8 A+3 B) \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac {3 (8 A+3 B) \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d}+\frac {1}{128} \left (9 a^3 (8 A+3 B)\right ) \int 1 \, dx \\ & = \frac {9}{128} a^3 (8 A+3 B) x-\frac {3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}+\frac {9 a^3 (8 A+3 B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {3 a^3 (8 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {a (8 A+3 B) \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac {3 (8 A+3 B) \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.52 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.92 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {a^3 \cos (c+d x) \left (4576 A+2976 B+\frac {2520 (8 A+3 B) \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(c+d x)}}+16 (373 A+223 B) \cos (2 (c+d x))+32 (41 A+11 B) \cos (4 (c+d x))-80 A \cos (6 (c+d x))-240 B \cos (6 (c+d x))-10640 A \sin (c+d x)-3045 B \sin (c+d x)+1365 B \sin (3 (c+d x))+560 A \sin (5 (c+d x))+595 B \sin (5 (c+d x))-35 B \sin (7 (c+d x))\right )}{17920 d} \]

[In]

Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]

[Out]

-1/17920*(a^3*Cos[c + d*x]*(4576*A + 2976*B + (2520*(8*A + 3*B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]])/Sqrt[C
os[c + d*x]^2] + 16*(373*A + 223*B)*Cos[2*(c + d*x)] + 32*(41*A + 11*B)*Cos[4*(c + d*x)] - 80*A*Cos[6*(c + d*x
)] - 240*B*Cos[6*(c + d*x)] - 10640*A*Sin[c + d*x] - 3045*B*Sin[c + d*x] + 1365*B*Sin[3*(c + d*x)] + 560*A*Sin
[5*(c + d*x)] + 595*B*Sin[5*(c + d*x)] - 35*B*Sin[7*(c + d*x)]))/d

Maple [A] (verified)

Time = 1.20 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.76

method	result	size
parallelrisch	\(\frac {a^{3} \left (\left (-91 A -49 B \right ) \cos \left (3 d x +3 c \right )+\left (-\frac {77 A}{5}-\frac {7 B}{5}\right ) \cos \left (5 d x +5 c \right )+\left (A +3 B \right ) \cos \left (7 d x +7 c \right )+\left (133 A +21 B \right ) \sin \left (2 d x +2 c \right )+\left (-7 A -\frac {49 B}{2}\right ) \sin \left (4 d x +4 c \right )+\left (-7 A -7 B \right ) \sin \left (6 d x +6 c \right )+\frac {7 B \sin \left (8 d x +8 c \right )}{16}+\left (-189 A -119 B \right ) \cos \left (d x +c \right )+252 d x A +\frac {189 d x B}{2}-\frac {1472 A}{5}-\frac {832 B}{5}\right )}{448 d}\)	\(152\)
risch	\(\frac {9 a^{3} x A}{16}+\frac {27 a^{3} B x}{128}-\frac {27 A \,a^{3} \cos \left (d x +c \right )}{64 d}-\frac {17 a^{3} \cos \left (d x +c \right ) B}{64 d}+\frac {\sin \left (8 d x +8 c \right ) B \,a^{3}}{1024 d}+\frac {a^{3} \cos \left (7 d x +7 c \right ) A}{448 d}+\frac {3 a^{3} \cos \left (7 d x +7 c \right ) B}{448 d}-\frac {\sin \left (6 d x +6 c \right ) A \,a^{3}}{64 d}-\frac {\sin \left (6 d x +6 c \right ) B \,a^{3}}{64 d}-\frac {11 a^{3} \cos \left (5 d x +5 c \right ) A}{320 d}-\frac {a^{3} \cos \left (5 d x +5 c \right ) B}{320 d}-\frac {\sin \left (4 d x +4 c \right ) A \,a^{3}}{64 d}-\frac {7 \sin \left (4 d x +4 c \right ) B \,a^{3}}{128 d}-\frac {13 a^{3} \cos \left (3 d x +3 c \right ) A}{64 d}-\frac {7 a^{3} \cos \left (3 d x +3 c \right ) B}{64 d}+\frac {19 \sin \left (2 d x +2 c \right ) A \,a^{3}}{64 d}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{3}}{64 d}\)	\(280\)
derivativedivides	\(\frac {A \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+B \,a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+3 A \,a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+3 B \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )-\frac {3 A \,a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+3 B \,a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {B \,a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5}}{d}\)	\(323\)
default	\(\frac {A \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+B \,a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+3 A \,a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+3 B \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )-\frac {3 A \,a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+3 B \,a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {B \,a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5}}{d}\)	\(323\)
norman	\(\frac {\frac {a^{3} \left (56 A -27 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {5 a^{3} \left (120 A +61 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {a^{3} \left (440 A +437 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {46 A \,a^{3}+26 B \,a^{3}}{35 d}+\left (\frac {9}{16} A \,a^{3}+\frac {27}{128} B \,a^{3}\right ) x +\left (\frac {9}{16} A \,a^{3}+\frac {27}{128} B \,a^{3}\right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2 \left (109 A \,a^{3}+79 B \,a^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {2 \left (79 A \,a^{3}+69 B \,a^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}-\frac {2 \left (15 A \,a^{3}+5 B \,a^{3}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (3 A \,a^{3}+B \,a^{3}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (23 A \,a^{3}+13 B \,a^{3}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (37 A \,a^{3}+7 B \,a^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {2 \left (11 A \,a^{3}+9 B \,a^{3}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\left (\frac {63}{2} A \,a^{3}+\frac {189}{16} B \,a^{3}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {315}{8} A \,a^{3}+\frac {945}{64} B \,a^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {63}{2} A \,a^{3}+\frac {189}{16} B \,a^{3}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {63}{4} A \,a^{3}+\frac {189}{32} B \,a^{3}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {9}{2} A \,a^{3}+\frac {27}{16} B \,a^{3}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {9}{2} A \,a^{3}+\frac {27}{16} B \,a^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {63}{4} A \,a^{3}+\frac {189}{32} B \,a^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a^{3} \left (104 A +919 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a^{3} \left (440 A +437 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {5 a^{3} \left (120 A +61 B \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a^{3} \left (56 A -27 B \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a^{3} \left (104 A +919 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\)	\(667\)

[In]

int(cos(d*x+c)^4*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/448*a^3*((-91*A-49*B)*cos(3*d*x+3*c)+(-77/5*A-7/5*B)*cos(5*d*x+5*c)+(A+3*B)*cos(7*d*x+7*c)+(133*A+21*B)*sin(
2*d*x+2*c)+(-7*A-49/2*B)*sin(4*d*x+4*c)+(-7*A-7*B)*sin(6*d*x+6*c)+7/16*B*sin(8*d*x+8*c)+(-189*A-119*B)*cos(d*x
+c)+252*d*x*A+189/2*d*x*B-1472/5*A-832/5*B)/d

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.68 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {640 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{7} - 3584 \, {\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{5} + 315 \, {\left (8 \, A + 3 \, B\right )} a^{3} d x + 35 \, {\left (16 \, B a^{3} \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, A + 11 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} + 6 \, {\left (8 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 9 \, {\left (8 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4480 \, d} \]

[In]

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/4480*(640*(A + 3*B)*a^3*cos(d*x + c)^7 - 3584*(A + B)*a^3*cos(d*x + c)^5 + 315*(8*A + 3*B)*a^3*d*x + 35*(16*
B*a^3*cos(d*x + c)^7 - 8*(8*A + 11*B)*a^3*cos(d*x + c)^5 + 6*(8*A + 3*B)*a^3*cos(d*x + c)^3 + 9*(8*A + 3*B)*a^
3*cos(d*x + c))*sin(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (189) = 378\).

Time = 0.73 (sec) , antiderivative size = 823, normalized size of antiderivative = 4.12 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {3 A a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {9 A a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 A a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 A a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 A a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 A a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 A a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {A a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {3 A a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {A a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 A a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {5 A a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {2 A a^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {3 A a^{3} \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {3 B a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {3 B a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {3 B a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {9 B a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {9 B a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 B a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {9 B a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 B a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 B a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 B a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {11 B a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {3 B a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {11 B a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} + \frac {B a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {3 B a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 B a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {3 B a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {6 B a^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {B a^{3} \cos ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\left (c \right )}\right ) \left (a \sin {\left (c \right )} + a\right )^{3} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**4*(a+a*sin(d*x+c))**3*(A+B*sin(d*x+c)),x)

[Out]

Piecewise((3*A*a**3*x*sin(c + d*x)**6/16 + 9*A*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 3*A*a**3*x*sin(c +
d*x)**4/8 + 9*A*a**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 3*A*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*A
*a**3*x*cos(c + d*x)**6/16 + 3*A*a**3*x*cos(c + d*x)**4/8 + 3*A*a**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) + A*a
**3*sin(c + d*x)**3*cos(c + d*x)**3/(2*d) + 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)**5/(5*d) - 3*A*a**3*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 5*A*a**3*sin(c + d*x)*cos(c + d*x)**
3/(8*d) - 2*A*a**3*cos(c + d*x)**7/(35*d) - 3*A*a**3*cos(c + d*x)**5/(5*d) + 3*B*a**3*x*sin(c + d*x)**8/128 +
3*B*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 3*B*a**3*x*sin(c + d*x)**6/16 + 9*B*a**3*x*sin(c + d*x)**4*cos
(c + d*x)**4/64 + 9*B*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 3*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**6/3
2 + 9*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 3*B*a**3*x*cos(c + d*x)**8/128 + 3*B*a**3*x*cos(c + d*x)**
6/16 + 3*B*a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 11*B*a**3*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) + 3*B
*a**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) - 11*B*a**3*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) + B*a**3*sin(c +
d*x)**3*cos(c + d*x)**3/(2*d) - 3*B*a**3*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 3*B*a**3*sin(c + d*x)*cos(c
+ d*x)**7/(128*d) - 3*B*a**3*sin(c + d*x)*cos(c + d*x)**5/(16*d) - 6*B*a**3*cos(c + d*x)**7/(35*d) - B*a**3*co
s(c + d*x)**5/(5*d), Ne(d, 0)), (x*(A + B*sin(c))*(a*sin(c) + a)**3*cos(c)**4, True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.16 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {21504 \, A a^{3} \cos \left (d x + c\right )^{5} + 7168 \, B a^{3} \cos \left (d x + c\right )^{5} - 1024 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} A a^{3} - 560 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} A a^{3} - 1120 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 3072 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} B a^{3} - 560 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{3} - 35 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{3}}{35840 \, d} \]

[In]

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/35840*(21504*A*a^3*cos(d*x + c)^5 + 7168*B*a^3*cos(d*x + c)^5 - 1024*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*
A*a^3 - 560*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*A*a^3 - 1120*(12*d*x + 12*c + sin(4*d*
x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3 - 3072*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*B*a^3 - 560*(4*sin(2*d*x + 2
*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*B*a^3 - 35*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))
*B*a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.64 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.08 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {B a^{3} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {9}{128} \, {\left (8 \, A a^{3} + 3 \, B a^{3}\right )} x + \frac {{\left (A a^{3} + 3 \, B a^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {{\left (11 \, A a^{3} + B a^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (13 \, A a^{3} + 7 \, B a^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {{\left (27 \, A a^{3} + 17 \, B a^{3}\right )} \cos \left (d x + c\right )}{64 \, d} - \frac {{\left (A a^{3} + B a^{3}\right )} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} - \frac {{\left (2 \, A a^{3} + 7 \, B a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (19 \, A a^{3} + 3 \, B a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]

[In]

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

1/1024*B*a^3*sin(8*d*x + 8*c)/d + 9/128*(8*A*a^3 + 3*B*a^3)*x + 1/448*(A*a^3 + 3*B*a^3)*cos(7*d*x + 7*c)/d - 1
/320*(11*A*a^3 + B*a^3)*cos(5*d*x + 5*c)/d - 1/64*(13*A*a^3 + 7*B*a^3)*cos(3*d*x + 3*c)/d - 1/64*(27*A*a^3 + 1
7*B*a^3)*cos(d*x + c)/d - 1/64*(A*a^3 + B*a^3)*sin(6*d*x + 6*c)/d - 1/128*(2*A*a^3 + 7*B*a^3)*sin(4*d*x + 4*c)
/d + 1/64*(19*A*a^3 + 3*B*a^3)*sin(2*d*x + 2*c)/d

Mupad [B] (verification not implemented)

Time = 12.21 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.92 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {9\,a^3\,\mathrm {atan}\left (\frac {9\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,A+3\,B\right )}{64\,\left (\frac {9\,A\,a^3}{8}+\frac {27\,B\,a^3}{64}\right )}\right )\,\left (8\,A+3\,B\right )}{64\,d}-\frac {9\,a^3\,\left (8\,A+3\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{64\,d}-\frac {\frac {46\,A\,a^3}{35}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {7\,A\,a^3}{8}-\frac {27\,B\,a^3}{64}\right )+\frac {26\,B\,a^3}{35}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (6\,A\,a^3+2\,B\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (30\,A\,a^3+10\,B\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (22\,A\,a^3+18\,B\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (46\,A\,a^3+26\,B\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {74\,A\,a^3}{5}+\frac {14\,B\,a^3}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {7\,A\,a^3}{8}-\frac {27\,B\,a^3}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {158\,A\,a^3}{35}+\frac {138\,B\,a^3}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {218\,A\,a^3}{5}+\frac {158\,B\,a^3}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {75\,A\,a^3}{8}+\frac {305\,B\,a^3}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {75\,A\,a^3}{8}+\frac {305\,B\,a^3}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {55\,A\,a^3}{8}+\frac {437\,B\,a^3}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {55\,A\,a^3}{8}+\frac {437\,B\,a^3}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {13\,A\,a^3}{8}+\frac {919\,B\,a^3}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {13\,A\,a^3}{8}+\frac {919\,B\,a^3}{64}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

[In]

int(cos(c + d*x)^4*(A + B*sin(c + d*x))*(a + a*sin(c + d*x))^3,x)

[Out]

(9*a^3*atan((9*a^3*tan(c/2 + (d*x)/2)*(8*A + 3*B))/(64*((9*A*a^3)/8 + (27*B*a^3)/64)))*(8*A + 3*B))/(64*d) - (
9*a^3*(8*A + 3*B)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(64*d) - ((46*A*a^3)/35 - tan(c/2 + (d*x)/2)*((7*A*a^3
)/8 - (27*B*a^3)/64) + (26*B*a^3)/35 + tan(c/2 + (d*x)/2)^14*(6*A*a^3 + 2*B*a^3) + tan(c/2 + (d*x)/2)^10*(30*A
*a^3 + 10*B*a^3) + tan(c/2 + (d*x)/2)^12*(22*A*a^3 + 18*B*a^3) + tan(c/2 + (d*x)/2)^8*(46*A*a^3 + 26*B*a^3) +
tan(c/2 + (d*x)/2)^4*((74*A*a^3)/5 + (14*B*a^3)/5) + tan(c/2 + (d*x)/2)^15*((7*A*a^3)/8 - (27*B*a^3)/64) + tan
(c/2 + (d*x)/2)^2*((158*A*a^3)/35 + (138*B*a^3)/35) + tan(c/2 + (d*x)/2)^6*((218*A*a^3)/5 + (158*B*a^3)/5) - t
an(c/2 + (d*x)/2)^3*((75*A*a^3)/8 + (305*B*a^3)/64) + tan(c/2 + (d*x)/2)^13*((75*A*a^3)/8 + (305*B*a^3)/64) -
tan(c/2 + (d*x)/2)^5*((55*A*a^3)/8 + (437*B*a^3)/64) + tan(c/2 + (d*x)/2)^11*((55*A*a^3)/8 + (437*B*a^3)/64) +
tan(c/2 + (d*x)/2)^7*((13*A*a^3)/8 + (919*B*a^3)/64) - tan(c/2 + (d*x)/2)^9*((13*A*a^3)/8 + (919*B*a^3)/64))/
(d*(8*tan(c/2 + (d*x)/2)^2 + 28*tan(c/2 + (d*x)/2)^4 + 56*tan(c/2 + (d*x)/2)^6 + 70*tan(c/2 + (d*x)/2)^8 + 56*
tan(c/2 + (d*x)/2)^10 + 28*tan(c/2 + (d*x)/2)^12 + 8*tan(c/2 + (d*x)/2)^14 + tan(c/2 + (d*x)/2)^16 + 1))