Integrand size = 27, antiderivative size = 228 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^7}+\frac {2 a \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^7 d}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d} \]
-1/16*(16*a^6-40*a^4*b^2+30*a^2*b^4-5*b^6)*x/b^7+2*a*(a^2-b^2)^(5/2)*arcta n((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^7/d-1/30*cos(d*x+c)^5*(6*a-5 *b*sin(d*x+c))/b^2/d+1/24*cos(d*x+c)^3*(8*a*(a^2-b^2)-b*(6*a^2-5*b^2)*sin( d*x+c))/b^4/d-1/16*cos(d*x+c)*(16*a*(a^2-b^2)^2-b*(8*a^4-14*a^2*b^2+5*b^4) *sin(d*x+c))/b^6/d
Time = 1.51 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.21 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-960 a^6 c+2400 a^4 b^2 c-1800 a^2 b^4 c+300 b^6 c-960 a^6 d x+2400 a^4 b^2 d x-1800 a^2 b^4 d x+300 b^6 d x+1920 a \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )-120 a b \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x)+20 \left (4 a^3 b^3-7 a b^5\right ) \cos (3 (c+d x))-12 a b^5 \cos (5 (c+d x))+240 a^4 b^2 \sin (2 (c+d x))-480 a^2 b^4 \sin (2 (c+d x))+225 b^6 \sin (2 (c+d x))-30 a^2 b^4 \sin (4 (c+d x))+45 b^6 \sin (4 (c+d x))+5 b^6 \sin (6 (c+d x))}{960 b^7 d} \]
(-960*a^6*c + 2400*a^4*b^2*c - 1800*a^2*b^4*c + 300*b^6*c - 960*a^6*d*x + 2400*a^4*b^2*d*x - 1800*a^2*b^4*d*x + 300*b^6*d*x + 1920*a*(a^2 - b^2)^(5/ 2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 120*a*b*(8*a^4 - 18* a^2*b^2 + 11*b^4)*Cos[c + d*x] + 20*(4*a^3*b^3 - 7*a*b^5)*Cos[3*(c + d*x)] - 12*a*b^5*Cos[5*(c + d*x)] + 240*a^4*b^2*Sin[2*(c + d*x)] - 480*a^2*b^4* Sin[2*(c + d*x)] + 225*b^6*Sin[2*(c + d*x)] - 30*a^2*b^4*Sin[4*(c + d*x)] + 45*b^6*Sin[4*(c + d*x)] + 5*b^6*Sin[6*(c + d*x)])/(960*b^7*d)
Time = 1.16 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.12, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3344, 25, 3042, 3344, 27, 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.
| \(\displaystyle \int \frac {\sin (c+d x) \cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x) \cos (c+d x)^6}{a+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {\int -\frac {\cos ^4(c+d x) \left (a b+\left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\cos ^4(c+d x) \left (a b+\left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\cos (c+d x)^4 \left (a b+\left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle -\frac {\frac {\int -\frac {3 \cos ^2(c+d x) \left (a b \left (2 a^2-3 b^2\right )+\left (8 a^4-14 b^2 a^2+5 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{4 b^2}-\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{4 b^2 d}}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {3 \int \frac {\cos ^2(c+d x) \left (a b \left (2 a^2-3 b^2\right )+\left (8 a^4-14 b^2 a^2+5 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{4 b^2}-\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{4 b^2 d}}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {3 \int \frac {\cos (c+d x)^2 \left (a b \left (2 a^2-3 b^2\right )+\left (8 a^4-14 b^2 a^2+5 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{4 b^2}-\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{4 b^2 d}}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle -\frac {-\frac {3 \left (\frac {\int -\frac {a b \left (8 a^4-18 b^2 a^2+11 b^4\right )+\left (16 a^6-40 b^2 a^4+30 b^4 a^2-5 b^6\right ) \sin (c+d x)}{a+b \sin (c+d x)}dx}{2 b^2}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{2 b^2 d}\right )}{4 b^2}-\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{4 b^2 d}}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {3 \left (-\frac {\int \frac {a b \left (8 a^4-18 b^2 a^2+11 b^4\right )+\left (16 a^6-40 b^2 a^4+30 b^4 a^2-5 b^6\right ) \sin (c+d x)}{a+b \sin (c+d x)}dx}{2 b^2}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{2 b^2 d}\right )}{4 b^2}-\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{4 b^2 d}}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {3 \left (-\frac {\int \frac {a b \left (8 a^4-18 b^2 a^2+11 b^4\right )+\left (16 a^6-40 b^2 a^4+30 b^4 a^2-5 b^6\right ) \sin (c+d x)}{a+b \sin (c+d x)}dx}{2 b^2}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{2 b^2 d}\right )}{4 b^2}-\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{4 b^2 d}}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {-\frac {3 \left (-\frac {\frac {x \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right )}{b}-\frac {16 a \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{b}}{2 b^2}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{2 b^2 d}\right )}{4 b^2}-\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{4 b^2 d}}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {3 \left (-\frac {\frac {x \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right )}{b}-\frac {16 a \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{b}}{2 b^2}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{2 b^2 d}\right )}{4 b^2}-\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{4 b^2 d}}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {-\frac {3 \left (-\frac {\frac {x \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right )}{b}-\frac {32 a \left (a^2-b^2\right )^3 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}}{2 b^2}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{2 b^2 d}\right )}{4 b^2}-\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{4 b^2 d}}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {-\frac {3 \left (-\frac {\frac {64 a \left (a^2-b^2\right )^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {x \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right )}{b}}{2 b^2}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{2 b^2 d}\right )}{4 b^2}-\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{4 b^2 d}}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {-\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{4 b^2 d}-\frac {3 \left (-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{2 b^2 d}-\frac {\frac {x \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right )}{b}-\frac {32 a \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d}}{2 b^2}\right )}{4 b^2}}{6 b^2}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}\) |
-1/30*(Cos[c + d*x]^5*(6*a - 5*b*Sin[c + d*x]))/(b^2*d) - (-1/4*(Cos[c + d *x]^3*(8*a*(a^2 - b^2) - b*(6*a^2 - 5*b^2)*Sin[c + d*x]))/(b^2*d) - (3*(-1 /2*(((16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*x)/b - (32*a*(a^2 - b^2)^( 5/2)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(b*d))/b^2 - (Cos[c + d*x]*(16*a*(a^2 - b^2)^2 - b*(8*a^4 - 14*a^2*b^2 + 5*b^4)*Sin[c + d*x]))/(2*b^2*d)))/(4*b^2))/(6*b^2)
3.14.22.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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. \(523\) vs. \(2(215)=430\).
Time = 1.02 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.30
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (\frac {\left (\frac {1}{2} a^{4} b^{2}-\frac {9}{8} a^{2} b^{4}+\frac {11}{16} b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{5} b -3 a^{3} b^{3}+3 a \,b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{2} a^{4} b^{2}-\frac {19}{8} a^{2} b^{4}-\frac {5}{48} b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5 a^{5} b -13 a^{3} b^{3}+9 a \,b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{4} b^{2}-\frac {5}{4} a^{2} b^{4}+\frac {15}{8} b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{5} b -\frac {70}{3} a^{3} b^{3}+\frac {46}{3} a \,b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{4} b^{2}+\frac {5}{4} a^{2} b^{4}-\frac {15}{8} b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{5} b -22 a^{3} b^{3}+14 a \,b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{2} a^{4} b^{2}+\frac {19}{8} a^{2} b^{4}+\frac {5}{48} b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5 a^{5} b -11 a^{3} b^{3}+\frac {31}{5} a \,b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{4} b^{2}+\frac {9}{8} a^{2} b^{4}-\frac {11}{16} b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{5} b -\frac {7 a^{3} b^{3}}{3}+\frac {23 a \,b^{5}}{15}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {\left (16 a^{6}-40 a^{4} b^{2}+30 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}\right )}{b^{7}}+\frac {2 a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\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^{7} \sqrt {a^{2}-b^{2}}}}{d}\) | \(524\) |
| default | \(\frac {-\frac {2 \left (\frac {\left (\frac {1}{2} a^{4} b^{2}-\frac {9}{8} a^{2} b^{4}+\frac {11}{16} b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{5} b -3 a^{3} b^{3}+3 a \,b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{2} a^{4} b^{2}-\frac {19}{8} a^{2} b^{4}-\frac {5}{48} b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5 a^{5} b -13 a^{3} b^{3}+9 a \,b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{4} b^{2}-\frac {5}{4} a^{2} b^{4}+\frac {15}{8} b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{5} b -\frac {70}{3} a^{3} b^{3}+\frac {46}{3} a \,b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{4} b^{2}+\frac {5}{4} a^{2} b^{4}-\frac {15}{8} b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{5} b -22 a^{3} b^{3}+14 a \,b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{2} a^{4} b^{2}+\frac {19}{8} a^{2} b^{4}+\frac {5}{48} b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5 a^{5} b -11 a^{3} b^{3}+\frac {31}{5} a \,b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{4} b^{2}+\frac {9}{8} a^{2} b^{4}-\frac {11}{16} b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{5} b -\frac {7 a^{3} b^{3}}{3}+\frac {23 a \,b^{5}}{15}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {\left (16 a^{6}-40 a^{4} b^{2}+30 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}\right )}{b^{7}}+\frac {2 a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\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^{7} \sqrt {a^{2}-b^{2}}}}{d}\) | \(524\) |
| risch | \(-\frac {2 i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{5}}+\frac {i \sqrt {a^{2}-b^{2}}\, a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{7}}+\frac {2 i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{5}}+\frac {5 x}{16 b}-\frac {i \sqrt {a^{2}-b^{2}}\, a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{7}}-\frac {11 a \,{\mathrm e}^{-i \left (d x +c \right )}}{16 b^{2} d}-\frac {\sin \left (2 d x +2 c \right ) a^{2}}{2 b^{3} d}-\frac {a \cos \left (5 d x +5 c \right )}{80 d \,b^{2}}-\frac {\sin \left (4 d x +4 c \right ) a^{2}}{32 b^{3} d}+\frac {a^{3} \cos \left (3 d x +3 c \right )}{12 d \,b^{4}}-\frac {7 a \cos \left (3 d x +3 c \right )}{48 d \,b^{2}}+\frac {\sin \left (2 d x +2 c \right ) a^{4}}{4 b^{5} d}+\frac {\sin \left (6 d x +6 c \right )}{192 b d}+\frac {3 \sin \left (4 d x +4 c \right )}{64 b d}-\frac {i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{3}}+\frac {i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{3}}-\frac {11 a \,{\mathrm e}^{i \left (d x +c \right )}}{16 d \,b^{2}}-\frac {x \,a^{6}}{b^{7}}+\frac {5 x \,a^{4}}{2 b^{5}}-\frac {15 x \,a^{2}}{8 b^{3}}+\frac {15 \sin \left (2 d x +2 c \right )}{64 b d}-\frac {a^{5} {\mathrm e}^{i \left (d x +c \right )}}{2 b^{6} d}+\frac {9 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 b^{4} d}+\frac {9 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 b^{4} d}-\frac {a^{5} {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{6} d}\) | \(638\) |
1/d*(-2/b^7*(((1/2*a^4*b^2-9/8*a^2*b^4+11/16*b^6)*tan(1/2*d*x+1/2*c)^11+(a ^5*b-3*a^3*b^3+3*a*b^5)*tan(1/2*d*x+1/2*c)^10+(3/2*a^4*b^2-19/8*a^2*b^4-5/ 48*b^6)*tan(1/2*d*x+1/2*c)^9+(5*a^5*b-13*a^3*b^3+9*a*b^5)*tan(1/2*d*x+1/2* c)^8+(a^4*b^2-5/4*a^2*b^4+15/8*b^6)*tan(1/2*d*x+1/2*c)^7+(10*a^5*b-70/3*a^ 3*b^3+46/3*a*b^5)*tan(1/2*d*x+1/2*c)^6+(-a^4*b^2+5/4*a^2*b^4-15/8*b^6)*tan (1/2*d*x+1/2*c)^5+(10*a^5*b-22*a^3*b^3+14*a*b^5)*tan(1/2*d*x+1/2*c)^4+(-3/ 2*a^4*b^2+19/8*a^2*b^4+5/48*b^6)*tan(1/2*d*x+1/2*c)^3+(5*a^5*b-11*a^3*b^3+ 31/5*a*b^5)*tan(1/2*d*x+1/2*c)^2+(-1/2*a^4*b^2+9/8*a^2*b^4-11/16*b^6)*tan( 1/2*d*x+1/2*c)+a^5*b-7/3*a^3*b^3+23/15*a*b^5)/(1+tan(1/2*d*x+1/2*c)^2)^6+1 /16*(16*a^6-40*a^4*b^2+30*a^2*b^4-5*b^6)*arctan(tan(1/2*d*x+1/2*c)))+2*a*( a^6-3*a^4*b^2+3*a^2*b^4-b^6)/b^7/(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)))
Time = 0.49 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.50 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\left [-\frac {48 \, a b^{5} \cos \left (d x + c\right )^{5} - 80 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (16 \, a^{6} - 40 \, a^{4} b^{2} + 30 \, a^{2} b^{4} - 5 \, b^{6}\right )} d x - 120 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 240 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - 5 \, {\left (8 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{2} b^{4} - 5 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{4} b^{2} - 14 \, a^{2} b^{4} + 5 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, b^{7} d}, -\frac {48 \, a b^{5} \cos \left (d x + c\right )^{5} - 80 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (16 \, a^{6} - 40 \, a^{4} b^{2} + 30 \, a^{2} b^{4} - 5 \, b^{6}\right )} d x + 240 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + 240 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - 5 \, {\left (8 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{2} b^{4} - 5 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{4} b^{2} - 14 \, a^{2} b^{4} + 5 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, b^{7} d}\right ] \]
[-1/240*(48*a*b^5*cos(d*x + c)^5 - 80*(a^3*b^3 - a*b^5)*cos(d*x + c)^3 + 1 5*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*d*x - 120*(a^5 - 2*a^3*b^2 + a*b^4)*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(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)) + 240* (a^5*b - 2*a^3*b^3 + a*b^5)*cos(d*x + c) - 5*(8*b^6*cos(d*x + c)^5 - 2*(6* a^2*b^4 - 5*b^6)*cos(d*x + c)^3 + 3*(8*a^4*b^2 - 14*a^2*b^4 + 5*b^6)*cos(d *x + c))*sin(d*x + c))/(b^7*d), -1/240*(48*a*b^5*cos(d*x + c)^5 - 80*(a^3* b^3 - a*b^5)*cos(d*x + c)^3 + 15*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6 )*d*x + 240*(a^5 - 2*a^3*b^2 + a*b^4)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) + 240*(a^5*b - 2*a^3*b^3 + a*b^5) *cos(d*x + c) - 5*(8*b^6*cos(d*x + c)^5 - 2*(6*a^2*b^4 - 5*b^6)*cos(d*x + c)^3 + 3*(8*a^4*b^2 - 14*a^2*b^4 + 5*b^6)*cos(d*x + c))*sin(d*x + c))/(b^7 *d)]
Timed out. \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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. 735 vs. \(2 (214) = 428\).
Time = 0.34 (sec) , antiderivative size = 735, normalized size of antiderivative = 3.22 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
-1/240*(15*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*(d*x + c)/b^7 - 480* (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sg n(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b ^2)*b^7) + 2*(120*a^4*b*tan(1/2*d*x + 1/2*c)^11 - 270*a^2*b^3*tan(1/2*d*x + 1/2*c)^11 + 165*b^5*tan(1/2*d*x + 1/2*c)^11 + 240*a^5*tan(1/2*d*x + 1/2* c)^10 - 720*a^3*b^2*tan(1/2*d*x + 1/2*c)^10 + 720*a*b^4*tan(1/2*d*x + 1/2* c)^10 + 360*a^4*b*tan(1/2*d*x + 1/2*c)^9 - 570*a^2*b^3*tan(1/2*d*x + 1/2*c )^9 - 25*b^5*tan(1/2*d*x + 1/2*c)^9 + 1200*a^5*tan(1/2*d*x + 1/2*c)^8 - 31 20*a^3*b^2*tan(1/2*d*x + 1/2*c)^8 + 2160*a*b^4*tan(1/2*d*x + 1/2*c)^8 + 24 0*a^4*b*tan(1/2*d*x + 1/2*c)^7 - 300*a^2*b^3*tan(1/2*d*x + 1/2*c)^7 + 450* b^5*tan(1/2*d*x + 1/2*c)^7 + 2400*a^5*tan(1/2*d*x + 1/2*c)^6 - 5600*a^3*b^ 2*tan(1/2*d*x + 1/2*c)^6 + 3680*a*b^4*tan(1/2*d*x + 1/2*c)^6 - 240*a^4*b*t an(1/2*d*x + 1/2*c)^5 + 300*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 - 450*b^5*tan(1 /2*d*x + 1/2*c)^5 + 2400*a^5*tan(1/2*d*x + 1/2*c)^4 - 5280*a^3*b^2*tan(1/2 *d*x + 1/2*c)^4 + 3360*a*b^4*tan(1/2*d*x + 1/2*c)^4 - 360*a^4*b*tan(1/2*d* x + 1/2*c)^3 + 570*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 25*b^5*tan(1/2*d*x + 1 /2*c)^3 + 1200*a^5*tan(1/2*d*x + 1/2*c)^2 - 2640*a^3*b^2*tan(1/2*d*x + 1/2 *c)^2 + 1488*a*b^4*tan(1/2*d*x + 1/2*c)^2 - 120*a^4*b*tan(1/2*d*x + 1/2*c) + 270*a^2*b^3*tan(1/2*d*x + 1/2*c) - 165*b^5*tan(1/2*d*x + 1/2*c) + 240*a ^5 - 560*a^3*b^2 + 368*a*b^4)/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*b^6))/d
Time = 14.54 (sec) , antiderivative size = 3683, normalized size of antiderivative = 16.15 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
- ((2*(23*a*b^4 + 15*a^5 - 35*a^3*b^2))/(15*b^6) + (4*tan(c/2 + (d*x)/2)^4 *(7*a*b^4 + 5*a^5 - 11*a^3*b^2))/b^6 + (2*tan(c/2 + (d*x)/2)^8*(9*a*b^4 + 5*a^5 - 13*a^3*b^2))/b^6 + (4*tan(c/2 + (d*x)/2)^6*(23*a*b^4 + 15*a^5 - 35 *a^3*b^2))/(3*b^6) + (2*tan(c/2 + (d*x)/2)^2*(31*a*b^4 + 25*a^5 - 55*a^3*b ^2))/(5*b^6) - (tan(c/2 + (d*x)/2)*(8*a^4 + 11*b^4 - 18*a^2*b^2))/(8*b^5) - (tan(c/2 + (d*x)/2)^5*(8*a^4 + 15*b^4 - 10*a^2*b^2))/(4*b^5) + (tan(c/2 + (d*x)/2)^7*(8*a^4 + 15*b^4 - 10*a^2*b^2))/(4*b^5) + (tan(c/2 + (d*x)/2)^ 11*(8*a^4 + 11*b^4 - 18*a^2*b^2))/(8*b^5) + (tan(c/2 + (d*x)/2)^3*(5*b^4 - 72*a^4 + 114*a^2*b^2))/(24*b^5) - (tan(c/2 + (d*x)/2)^9*(5*b^4 - 72*a^4 + 114*a^2*b^2))/(24*b^5) + (2*tan(c/2 + (d*x)/2)^10*(3*a*b^4 + a^5 - 3*a^3* b^2))/b^6)/(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)) - (atan((((((25*a^2*b^18)/8 - (75*a^4*b^16)/2 + (3 25*a^6*b^14)/2 - 320*a^8*b^12 + 320*a^10*b^10 - 160*a^12*b^8 + 32*a^14*b^6 )/b^17 - (((10*a*b^20 - 38*a^3*b^18 + 44*a^5*b^16 - 16*a^7*b^14)/b^17 - (( 32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^22 - 512*a^3*b^20))/(8*b^18))*(a ^6*16i - b^6*5i + a^2*b^4*30i - a^4*b^2*40i))/(16*b^7) + (tan(c/2 + (d*x)/ 2)*(512*a^2*b^20 - 1536*a^4*b^18 + 1536*a^6*b^16 - 512*a^8*b^14))/(8*b^18) )*(a^6*16i - b^6*5i + a^2*b^4*30i - a^4*b^2*40i))/(16*b^7) + (tan(c/2 + (d *x)/2)*(50*a*b^20 - 881*a^3*b^18 + 4436*a^5*b^16 - 10260*a^7*b^14 + 128...