Integrand size = 27, antiderivative size = 159 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\left (8 a^4-12 a^2 b^2+3 b^4\right ) x}{8 b^5}-\frac {2 a \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 d}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}+\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^4 d} \]
1/8*(8*a^4-12*a^2*b^2+3*b^4)*x/b^5-2*a*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2 *d*x+1/2*c))/(a^2-b^2)^(1/2))/b^5/d-1/12*cos(d*x+c)^3*(4*a-3*b*sin(d*x+c)) /b^2/d+1/8*cos(d*x+c)*(8*a*(a^2-b^2)-b*(4*a^2-3*b^2)*sin(d*x+c))/b^4/d
Time = 0.76 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-192 a \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+24 a b \left (4 a^2-5 b^2\right ) \cos (c+d x)-8 a b^3 \cos (3 (c+d x))+3 \left (4 \left (8 a^4-12 a^2 b^2+3 b^4\right ) (c+d x)+\left (-8 a^2 b^2+8 b^4\right ) \sin (2 (c+d x))+b^4 \sin (4 (c+d x))\right )}{96 b^5 d} \]
(-192*a*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 24*a*b*(4*a^2 - 5*b^2)*Cos[c + d*x] - 8*a*b^3*Cos[3*(c + d*x)] + 3*(4*( 8*a^4 - 12*a^2*b^2 + 3*b^4)*(c + d*x) + (-8*a^2*b^2 + 8*b^4)*Sin[2*(c + d* x)] + b^4*Sin[4*(c + d*x)]))/(96*b^5*d)
Time = 0.76 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3344, 25, 3042, 3344, 25, 3042, 3214, 3042, 3139, 1083, 217}
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.
| \(\displaystyle \int \frac {\sin (c+d x) \cos ^4(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x) \cos (c+d x)^4}{a+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {\int -\frac {\cos ^2(c+d x) \left (a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{4 b^2}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\cos ^2(c+d x) \left (a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{4 b^2}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\cos (c+d x)^2 \left (a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{4 b^2}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle -\frac {\frac {\int -\frac {a b \left (4 a^2-5 b^2\right )+\left (8 a^4-12 b^2 a^2+3 b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)}dx}{2 b^2}-\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{2 b^2 d}}{4 b^2}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int \frac {a b \left (4 a^2-5 b^2\right )+\left (8 a^4-12 b^2 a^2+3 b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)}dx}{2 b^2}-\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{2 b^2 d}}{4 b^2}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {a b \left (4 a^2-5 b^2\right )+\left (8 a^4-12 b^2 a^2+3 b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)}dx}{2 b^2}-\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{2 b^2 d}}{4 b^2}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {-\frac {\frac {x \left (8 a^4-12 a^2 b^2+3 b^4\right )}{b}-\frac {8 a \left (a^2-b^2\right )^2 \int \frac {1}{a+b \sin (c+d x)}dx}{b}}{2 b^2}-\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{2 b^2 d}}{4 b^2}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {x \left (8 a^4-12 a^2 b^2+3 b^4\right )}{b}-\frac {8 a \left (a^2-b^2\right )^2 \int \frac {1}{a+b \sin (c+d x)}dx}{b}}{2 b^2}-\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{2 b^2 d}}{4 b^2}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {-\frac {\frac {x \left (8 a^4-12 a^2 b^2+3 b^4\right )}{b}-\frac {16 a \left (a^2-b^2\right )^2 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}}{2 b^2}-\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{2 b^2 d}}{4 b^2}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {-\frac {\frac {32 a \left (a^2-b^2\right )^2 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {x \left (8 a^4-12 a^2 b^2+3 b^4\right )}{b}}{2 b^2}-\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{2 b^2 d}}{4 b^2}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {-\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{2 b^2 d}-\frac {\frac {x \left (8 a^4-12 a^2 b^2+3 b^4\right )}{b}-\frac {16 a \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d}}{2 b^2}}{4 b^2}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}\) |
-1/12*(Cos[c + d*x]^3*(4*a - 3*b*Sin[c + d*x]))/(b^2*d) - (-1/2*(((8*a^4 - 12*a^2*b^2 + 3*b^4)*x)/b - (16*a*(a^2 - b^2)^(3/2)*ArcTan[(2*b + 2*a*Tan[ (c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(b*d))/b^2 - (Cos[c + d*x]*(8*a*(a^2 - b^2) - b*(4*a^2 - 3*b^2)*Sin[c + d*x]))/(2*b^2*d))/(4*b^2)
3.14.3.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*(g* Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d* p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Simp[g^2*( (p - 1)/(b^2*(m + p)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si n[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^ 2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1 , 0] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(308\) vs. \(2(148)=296\).
Time = 0.65 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.94
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{5} \sqrt {a^{2}-b^{2}}}+\frac {\frac {2 \left (\left (\frac {1}{2} a^{2} b^{2}-\frac {5}{8} b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{3} b -2 a \,b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a^{2} b^{2}+\frac {3}{8} b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 a^{3} b -4 a \,b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{2} b^{2}-\frac {3}{8} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 a^{3} b -\frac {10}{3} a \,b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{2} b^{2}+\frac {5}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{3} b -\frac {4 a \,b^{3}}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\left (8 a^{4}-12 a^{2} b^{2}+3 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{5}}}{d}\) | \(309\) |
| default | \(\frac {-\frac {2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{5} \sqrt {a^{2}-b^{2}}}+\frac {\frac {2 \left (\left (\frac {1}{2} a^{2} b^{2}-\frac {5}{8} b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{3} b -2 a \,b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a^{2} b^{2}+\frac {3}{8} b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 a^{3} b -4 a \,b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{2} b^{2}-\frac {3}{8} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 a^{3} b -\frac {10}{3} a \,b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{2} b^{2}+\frac {5}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{3} b -\frac {4 a \,b^{3}}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\left (8 a^{4}-12 a^{2} b^{2}+3 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{5}}}{d}\) | \(309\) |
| risch | \(\frac {x \,a^{4}}{b^{5}}-\frac {3 x \,a^{2}}{2 b^{3}}+\frac {3 x}{8 b}+\frac {a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 b^{4} d}-\frac {5 a \,{\mathrm e}^{i \left (d x +c \right )}}{8 d \,b^{2}}+\frac {a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{4} d}-\frac {5 a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 b^{2} d}-\frac {i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{d \,b^{5}}+\frac {i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{d \,b^{3}}+\frac {i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{d \,b^{5}}-\frac {i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{d \,b^{3}}+\frac {\sin \left (4 d x +4 c \right )}{32 b d}-\frac {a \cos \left (3 d x +3 c \right )}{12 b^{2} d}-\frac {\sin \left (2 d x +2 c \right ) a^{2}}{4 b^{3} d}+\frac {\sin \left (2 d x +2 c \right )}{4 b d}\) | \(461\) |
1/d*(-2*a*(a^4-2*a^2*b^2+b^4)/b^5/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2* d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))+2/b^5*(((1/2*a^2*b^2-5/8*b^4)*tan(1/2*d*x +1/2*c)^7+(a^3*b-2*a*b^3)*tan(1/2*d*x+1/2*c)^6+(1/2*a^2*b^2+3/8*b^4)*tan(1 /2*d*x+1/2*c)^5+(3*a^3*b-4*a*b^3)*tan(1/2*d*x+1/2*c)^4+(-1/2*a^2*b^2-3/8*b ^4)*tan(1/2*d*x+1/2*c)^3+(3*a^3*b-10/3*a*b^3)*tan(1/2*d*x+1/2*c)^2+(-1/2*a ^2*b^2+5/8*b^4)*tan(1/2*d*x+1/2*c)+a^3*b-4/3*a*b^3)/(1+tan(1/2*d*x+1/2*c)^ 2)^4+1/8*(8*a^4-12*a^2*b^2+3*b^4)*arctan(tan(1/2*d*x+1/2*c))))
Time = 0.43 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.60 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\left [-\frac {8 \, a b^{3} \cos \left (d x + c\right )^{3} - 3 \, {\left (8 \, a^{4} - 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} d x + 12 \, {\left (a^{3} - a b^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 24 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right ) - 3 \, {\left (2 \, b^{4} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, b^{5} d}, -\frac {8 \, a b^{3} \cos \left (d x + c\right )^{3} - 3 \, {\left (8 \, a^{4} - 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} d x - 24 \, {\left (a^{3} - a b^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 24 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right ) - 3 \, {\left (2 \, b^{4} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, b^{5} d}\right ] \]
[-1/24*(8*a*b^3*cos(d*x + c)^3 - 3*(8*a^4 - 12*a^2*b^2 + 3*b^4)*d*x + 12*( a^3 - a*b^2)*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*s in(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c)) *sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 24*(a^3*b - a*b^3)*cos(d*x + c) - 3*(2*b^4*cos(d*x + c)^3 - (4*a^2*b^2 - 3*b^4)*cos(d*x + c))*sin(d*x + c))/(b^5*d), -1/24*(8*a*b^3*cos(d*x + c)^3 - 3*(8*a^4 - 12*a^2*b^2 + 3*b^4)*d*x - 24*(a^3 - a*b^2)*sqrt(a^2 - b^2)*a rctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 24*(a^3*b - a*b^3)*cos(d*x + c) - 3*(2*b^4*cos(d*x + c)^3 - (4*a^2*b^2 - 3*b^4)*cos(d* x + c))*sin(d*x + c))/(b^5*d)]
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (147) = 294\).
Time = 0.33 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.33 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {3 \, {\left (8 \, a^{4} - 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} {\left (d x + c\right )}}{b^{5}} - \frac {48 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{5}} + \frac {2 \, {\left (12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 96 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 80 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, a^{3} - 32 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} b^{4}}}{24 \, d} \]
1/24*(3*(8*a^4 - 12*a^2*b^2 + 3*b^4)*(d*x + c)/b^5 - 48*(a^5 - 2*a^3*b^2 + a*b^4)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*b^5) + 2*(12*a^2*b*tan(1/2 *d*x + 1/2*c)^7 - 15*b^3*tan(1/2*d*x + 1/2*c)^7 + 24*a^3*tan(1/2*d*x + 1/2 *c)^6 - 48*a*b^2*tan(1/2*d*x + 1/2*c)^6 + 12*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 9*b^3*tan(1/2*d*x + 1/2*c)^5 + 72*a^3*tan(1/2*d*x + 1/2*c)^4 - 96*a*b^2* tan(1/2*d*x + 1/2*c)^4 - 12*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 9*b^3*tan(1/2*d *x + 1/2*c)^3 + 72*a^3*tan(1/2*d*x + 1/2*c)^2 - 80*a*b^2*tan(1/2*d*x + 1/2 *c)^2 - 12*a^2*b*tan(1/2*d*x + 1/2*c) + 15*b^3*tan(1/2*d*x + 1/2*c) + 24*a ^3 - 32*a*b^2)/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*b^4))/d
Time = 13.15 (sec) , antiderivative size = 453, normalized size of antiderivative = 2.85 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,b\,d}+\frac {\sin \left (2\,c+2\,d\,x\right )}{4\,b\,d}+\frac {\sin \left (4\,c+4\,d\,x\right )}{32\,b\,d}-\frac {a\,\cos \left (3\,c+3\,d\,x\right )}{12\,b^2\,d}+\frac {a^3\,\cos \left (c+d\,x\right )}{b^4\,d}-\frac {3\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{b^3\,d}+\frac {2\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{b^5\,d}-\frac {a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,b^3\,d}-\frac {5\,a\,\cos \left (c+d\,x\right )}{4\,b^2\,d}-\frac {2\,a\,\mathrm {atanh}\left (\frac {2\,b^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}-a^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}+a\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^5+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b-2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b^2-4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^3+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^4+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^5}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}{b^5\,d} \]
(3*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(4*b*d) + sin(2*c + 2*d*x) /(4*b*d) + sin(4*c + 4*d*x)/(32*b*d) - (a*cos(3*c + 3*d*x))/(12*b^2*d) + ( a^3*cos(c + d*x))/(b^4*d) - (3*a^2*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x) /2)))/(b^3*d) + (2*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(b^5*d ) - (a^2*sin(2*c + 2*d*x))/(4*b^3*d) - (5*a*cos(c + d*x))/(4*b^2*d) - (2*a *atanh((2*b^2*sin(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2) - a^2*sin(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2) + a*b* cos(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(a^5*cos(c/2 + (d*x)/2) + 2*b^5*sin(c/2 + (d*x)/2) + a*b^4*cos(c/2 + (d*x)/2) + 2*a^4* b*sin(c/2 + (d*x)/2) - 2*a^3*b^2*cos(c/2 + (d*x)/2) - 4*a^2*b^3*sin(c/2 + (d*x)/2)))*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(b^5*d)