Integrand size = 21, antiderivative size = 197 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {5 a \left (4 a^2-3 b^2\right ) x}{2 b^6}-\frac {5 \left (4 a^4-5 a^2 b^2+b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^6 \sqrt {a^2-b^2} d}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {5 \cos ^3(c+d x) (4 a+b \sin (c+d x))}{6 b^3 d (a+b \sin (c+d x))}+\frac {5 \cos (c+d x) \left (4 a^2-b^2-2 a b \sin (c+d x)\right )}{2 b^5 d} \]
5/2*a*(4*a^2-3*b^2)*x/b^6-1/2*cos(d*x+c)^5/b/d/(a+b*sin(d*x+c))^2-5/6*cos( d*x+c)^3*(4*a+b*sin(d*x+c))/b^3/d/(a+b*sin(d*x+c))+5/2*cos(d*x+c)*(4*a^2-b ^2-2*a*b*sin(d*x+c))/b^5/d-5*(4*a^4-5*a^2*b^2+b^4)*arctan((b+a*tan(1/2*d*x +1/2*c))/(a^2-b^2)^(1/2))/b^6/d/(a^2-b^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(3889\) vs. \(2(197)=394\).
Time = 7.57 (sec) , antiderivative size = 3889, normalized size of antiderivative = 19.74 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Result too large to show} \]
(Cos[c + d*x]^5*(-1/2*(b*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(7/2)*( b/(a + b) - (b*Sin[c + d*x])/(a + b))^(7/2))/(((a*b)/(a - b) - b^2/(a - b) )*((a*b)/(a + b) + b^2/(a + b))*(a + b*Sin[c + d*x])^2) - ((-3*a*b^3*(-(b/ (a - b)) - (b*Sin[c + d*x])/(a - b))^(7/2)*(b/(a + b) - (b*Sin[c + d*x])/( a + b))^(7/2))/((a^2 - b^2)*((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))*(a + b*Sin[c + d*x])) - ((144*Sqrt[2]*a*b^5*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(7/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)] *(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((7 *(3/(16*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3) + 1/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1)))/ 12 + (35*b^4*(((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/b - ((a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^2)/(3*b^2) + (2*(a - b)^3 *(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^3)/(15*b^3) - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(S qrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*S qrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))/(128 *(a - b)^4*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^4*(1 + ((a - b)*(-(b/ (a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3)))/(7*(a - b)*(a + b)^4*Sqr t[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) + (((18*a^2*b^5)...
Time = 0.96 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.10, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3172, 3042, 3342, 25, 3042, 3344, 27, 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.
| \(\displaystyle \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{(a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3172 |
| \(\displaystyle -\frac {5 \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 b}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 \int \frac {\cos (c+d x)^4 \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 b}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3342 |
| \(\displaystyle -\frac {5 \left (\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}-\frac {\int -\frac {\cos ^2(c+d x) (b+4 a \sin (c+d x))}{a+b \sin (c+d x)}dx}{b^2}\right )}{2 b}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5 \left (\frac {\int \frac {\cos ^2(c+d x) (b+4 a \sin (c+d x))}{a+b \sin (c+d x)}dx}{b^2}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}\right )}{2 b}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 \left (\frac {\int \frac {\cos (c+d x)^2 (b+4 a \sin (c+d x))}{a+b \sin (c+d x)}dx}{b^2}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}\right )}{2 b}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {\int -\frac {2 \left (b \left (2 a^2-b^2\right )+a \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{2 b^2}-\frac {\cos (c+d x) \left (4 a^2-2 a b \sin (c+d x)-b^2\right )}{b^2 d}}{b^2}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}\right )}{2 b}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 \left (\frac {-\frac {\int \frac {b \left (2 a^2-b^2\right )+a \left (4 a^2-3 b^2\right ) \sin (c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\cos (c+d x) \left (4 a^2-2 a b \sin (c+d x)-b^2\right )}{b^2 d}}{b^2}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}\right )}{2 b}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 \left (\frac {-\frac {\int \frac {b \left (2 a^2-b^2\right )+a \left (4 a^2-3 b^2\right ) \sin (c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\cos (c+d x) \left (4 a^2-2 a b \sin (c+d x)-b^2\right )}{b^2 d}}{b^2}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}\right )}{2 b}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {5 \left (\frac {-\frac {\frac {a x \left (4 a^2-3 b^2\right )}{b}-\frac {\left (4 a^4-5 a^2 b^2+b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}}{b^2}-\frac {\cos (c+d x) \left (4 a^2-2 a b \sin (c+d x)-b^2\right )}{b^2 d}}{b^2}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}\right )}{2 b}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 \left (\frac {-\frac {\frac {a x \left (4 a^2-3 b^2\right )}{b}-\frac {\left (4 a^4-5 a^2 b^2+b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}}{b^2}-\frac {\cos (c+d x) \left (4 a^2-2 a b \sin (c+d x)-b^2\right )}{b^2 d}}{b^2}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}\right )}{2 b}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {5 \left (\frac {-\frac {\frac {a x \left (4 a^2-3 b^2\right )}{b}-\frac {2 \left (4 a^4-5 a^2 b^2+b^4\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}}{b^2}-\frac {\cos (c+d x) \left (4 a^2-2 a b \sin (c+d x)-b^2\right )}{b^2 d}}{b^2}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}\right )}{2 b}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {5 \left (\frac {-\frac {\frac {4 \left (4 a^4-5 a^2 b^2+b^4\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {a x \left (4 a^2-3 b^2\right )}{b}}{b^2}-\frac {\cos (c+d x) \left (4 a^2-2 a b \sin (c+d x)-b^2\right )}{b^2 d}}{b^2}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}\right )}{2 b}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {5 \left (\frac {-\frac {\cos (c+d x) \left (4 a^2-2 a b \sin (c+d x)-b^2\right )}{b^2 d}-\frac {\frac {a x \left (4 a^2-3 b^2\right )}{b}-\frac {2 \left (4 a^4-5 a^2 b^2+b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d \sqrt {a^2-b^2}}}{b^2}}{b^2}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}\right )}{2 b}-\frac {\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}\) |
-1/2*Cos[c + d*x]^5/(b*d*(a + b*Sin[c + d*x])^2) - (5*((Cos[c + d*x]^3*(4* a + b*Sin[c + d*x]))/(3*b^2*d*(a + b*Sin[c + d*x])) + (-(((a*(4*a^2 - 3*b^ 2)*x)/b - (2*(4*a^4 - 5*a^2*b^2 + b^4)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2]) /(2*Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d))/b^2) - (Cos[c + d*x]*(4*a^2 - b^2 - 2*a*b*Sin[c + d*x]))/(b^2*d))/b^2))/(2*b)
3.5.56.3.1 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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + 1))), x] + Simp[g^2*((p - 1)/(b*(m + 1))) Int[(g*Cos [e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; Fre eQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && I ntegersQ[2*m, 2*p]
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*C os[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d*p + b*d*(m + 1)*Sin[e + f*x])/(b^2*f*(m + 1)*(m + p + 1))), x] + Simp[g^2*(( p - 1)/(b^2*(m + 1)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin [e + f*x])^(m + 1)*Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Sin[e + f*x ], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && Lt Q[m, -1] && GtQ[p, 1] && NeQ[m + p + 1, 0] && IntegerQ[2*m]
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. \(377\) vs. \(2(184)=368\).
Time = 3.41 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.92
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (\frac {-\frac {b^{2} \left (7 a^{4}-5 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {b \left (8 a^{6}+9 a^{4} b^{2}-15 a^{2} b^{4}-2 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}}-\frac {b^{2} \left (25 a^{4}-23 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-4 b \,a^{4}+\frac {7 b^{3} a^{2}}{2}+\frac {b^{5}}{2}}{{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {5 \left (4 a^{4}-5 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{6}}+\frac {\frac {2 \left (\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{2}+\left (6 a^{2} b -3 b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (12 a^{2} b -4 b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}}{2}+6 a^{2} b -\frac {7 b^{3}}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+5 a \left (4 a^{2}-3 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{6}}}{d}\) | \(378\) |
| default | \(\frac {-\frac {2 \left (\frac {-\frac {b^{2} \left (7 a^{4}-5 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {b \left (8 a^{6}+9 a^{4} b^{2}-15 a^{2} b^{4}-2 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}}-\frac {b^{2} \left (25 a^{4}-23 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-4 b \,a^{4}+\frac {7 b^{3} a^{2}}{2}+\frac {b^{5}}{2}}{{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {5 \left (4 a^{4}-5 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{6}}+\frac {\frac {2 \left (\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{2}+\left (6 a^{2} b -3 b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (12 a^{2} b -4 b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}}{2}+6 a^{2} b -\frac {7 b^{3}}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+5 a \left (4 a^{2}-3 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{6}}}{d}\) | \(378\) |
| risch | \(\frac {10 a^{3} x}{b^{6}}-\frac {15 a x}{2 b^{4}}-\frac {{\mathrm e}^{3 i \left (d x +c \right )}}{24 b^{3} d}+\frac {3 i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{4} d}+\frac {3 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{b^{5} d}-\frac {9 \,{\mathrm e}^{i \left (d x +c \right )}}{8 b^{3} d}+\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{b^{5} d}-\frac {9 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 b^{3} d}-\frac {3 i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{4} d}-\frac {{\mathrm e}^{-3 i \left (d x +c \right )}}{24 b^{3} d}-\frac {i \left (-10 i a^{4} b \,{\mathrm e}^{3 i \left (d x +c \right )}+11 i b^{3} a^{2} {\mathrm e}^{3 i \left (d x +c \right )}-i b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+26 i a^{4} b \,{\mathrm e}^{i \left (d x +c \right )}-25 i a^{2} b^{3} {\mathrm e}^{i \left (d x +c \right )}-i b^{5} {\mathrm e}^{i \left (d x +c \right )}+18 a^{5} {\mathrm e}^{2 i \left (d x +c \right )}-9 a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-9 a^{3} b^{2}+9 a \,b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+i b \right )^{2} d \,b^{6}}+\frac {10 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{2}}{d \,b^{6}}-\frac {5 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{2 d \,b^{4}}-\frac {10 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{2}}{d \,b^{6}}+\frac {5 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{2 d \,b^{4}}\) | \(574\) |
1/d*(-2/b^6*((-1/2*b^2*(7*a^4-5*a^2*b^2-2*b^4)/a*tan(1/2*d*x+1/2*c)^3-1/2* b*(8*a^6+9*a^4*b^2-15*a^2*b^4-2*b^6)/a^2*tan(1/2*d*x+1/2*c)^2-1/2*b^2*(25* a^4-23*a^2*b^2-2*b^4)/a*tan(1/2*d*x+1/2*c)-4*b*a^4+7/2*b^3*a^2+1/2*b^5)/(a *tan(1/2*d*x+1/2*c)^2+2*b*tan(1/2*d*x+1/2*c)+a)^2+5/2*(4*a^4-5*a^2*b^2+b^4 )/(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^6*((3/2*tan(1/2*d*x+1/2*c)^5*a*b^2+(6*a^2*b-3*b^3)*tan(1/2*d*x+1/2*c )^4+(12*a^2*b-4*b^3)*tan(1/2*d*x+1/2*c)^2-3/2*tan(1/2*d*x+1/2*c)*a*b^2+6*a ^2*b-7/3*b^3)/(1+tan(1/2*d*x+1/2*c)^2)^3+5/2*a*(4*a^2-3*b^2)*arctan(tan(1/ 2*d*x+1/2*c))))
Time = 0.34 (sec) , antiderivative size = 752, normalized size of antiderivative = 3.82 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\left [-\frac {4 \, b^{5} \cos \left (d x + c\right )^{5} - 30 \, {\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} d x \cos \left (d x + c\right )^{2} - 20 \, {\left (2 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} + 30 \, {\left (4 \, a^{5} + a^{3} b^{2} - 3 \, a b^{4}\right )} d x - 15 \, {\left (4 \, a^{4} + 3 \, a^{2} b^{2} - b^{4} - {\left (4 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (4 \, a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\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 ) + 30 \, {\left (4 \, a^{4} b - a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right ) + 10 \, {\left (a b^{4} \cos \left (d x + c\right )^{3} + 6 \, {\left (4 \, a^{4} b - 3 \, a^{2} b^{3}\right )} d x + 6 \, {\left (3 \, a^{3} b^{2} - 2 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (b^{8} d \cos \left (d x + c\right )^{2} - 2 \, a b^{7} d \sin \left (d x + c\right ) - {\left (a^{2} b^{6} + b^{8}\right )} d\right )}}, -\frac {2 \, b^{5} \cos \left (d x + c\right )^{5} - 15 \, {\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} d x \cos \left (d x + c\right )^{2} - 10 \, {\left (2 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (4 \, a^{5} + a^{3} b^{2} - 3 \, a b^{4}\right )} d x + 15 \, {\left (4 \, a^{4} + 3 \, a^{2} b^{2} - b^{4} - {\left (4 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (4 \, a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\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 ) + 15 \, {\left (4 \, a^{4} b - a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right ) + 5 \, {\left (a b^{4} \cos \left (d x + c\right )^{3} + 6 \, {\left (4 \, a^{4} b - 3 \, a^{2} b^{3}\right )} d x + 6 \, {\left (3 \, a^{3} b^{2} - 2 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (b^{8} d \cos \left (d x + c\right )^{2} - 2 \, a b^{7} d \sin \left (d x + c\right ) - {\left (a^{2} b^{6} + b^{8}\right )} d\right )}}\right ] \]
[-1/12*(4*b^5*cos(d*x + c)^5 - 30*(4*a^3*b^2 - 3*a*b^4)*d*x*cos(d*x + c)^2 - 20*(2*a^2*b^3 - b^5)*cos(d*x + c)^3 + 30*(4*a^5 + a^3*b^2 - 3*a*b^4)*d* x - 15*(4*a^4 + 3*a^2*b^2 - b^4 - (4*a^2*b^2 - b^4)*cos(d*x + c)^2 + 2*(4* a^3*b - a*b^3)*sin(d*x + c))*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)) + 30*(4*a^4*b - a^2*b^3 - b^5)*cos(d*x + c) + 10*(a*b^4*co s(d*x + c)^3 + 6*(4*a^4*b - 3*a^2*b^3)*d*x + 6*(3*a^3*b^2 - 2*a*b^4)*cos(d *x + c))*sin(d*x + c))/(b^8*d*cos(d*x + c)^2 - 2*a*b^7*d*sin(d*x + c) - (a ^2*b^6 + b^8)*d), -1/6*(2*b^5*cos(d*x + c)^5 - 15*(4*a^3*b^2 - 3*a*b^4)*d* x*cos(d*x + c)^2 - 10*(2*a^2*b^3 - b^5)*cos(d*x + c)^3 + 15*(4*a^5 + a^3*b ^2 - 3*a*b^4)*d*x + 15*(4*a^4 + 3*a^2*b^2 - b^4 - (4*a^2*b^2 - b^4)*cos(d* x + c)^2 + 2*(4*a^3*b - a*b^3)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*si n(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) + 15*(4*a^4*b - a^2*b^3 - b^5)*cos(d*x + c) + 5*(a*b^4*cos(d*x + c)^3 + 6*(4*a^4*b - 3*a^2*b^3)*d*x + 6*(3*a^3*b^2 - 2*a*b^4)*cos(d*x + c))*sin(d*x + c))/(b^8*d*cos(d*x + c)^ 2 - 2*a*b^7*d*sin(d*x + c) - (a^2*b^6 + b^8)*d)]
Timed out. \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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. 457 vs. \(2 (182) = 364\).
Time = 0.38 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.32 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {15 \, {\left (4 \, a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{b^{6}} - \frac {30 \, {\left (4 \, a^{4} - 5 \, a^{2} b^{2} + 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^{6}} + \frac {2 \, {\left (9 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 18 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 72 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a^{2} - 14 \, b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} b^{5}} + \frac {6 \, {\left (7 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 25 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 23 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, a^{6} - 7 \, a^{4} b^{2} - a^{2} b^{4}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2} a^{2} b^{5}}}{6 \, d} \]
1/6*(15*(4*a^3 - 3*a*b^2)*(d*x + c)/b^6 - 30*(4*a^4 - 5*a^2*b^2 + 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^6) + 2*(9*a*b*tan(1/2*d*x + 1/2*c)^ 5 + 36*a^2*tan(1/2*d*x + 1/2*c)^4 - 18*b^2*tan(1/2*d*x + 1/2*c)^4 + 72*a^2 *tan(1/2*d*x + 1/2*c)^2 - 24*b^2*tan(1/2*d*x + 1/2*c)^2 - 9*a*b*tan(1/2*d* x + 1/2*c) + 36*a^2 - 14*b^2)/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*b^5) + 6*(7* a^5*b*tan(1/2*d*x + 1/2*c)^3 - 5*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 - 2*a*b^5* tan(1/2*d*x + 1/2*c)^3 + 8*a^6*tan(1/2*d*x + 1/2*c)^2 + 9*a^4*b^2*tan(1/2* d*x + 1/2*c)^2 - 15*a^2*b^4*tan(1/2*d*x + 1/2*c)^2 - 2*b^6*tan(1/2*d*x + 1 /2*c)^2 + 25*a^5*b*tan(1/2*d*x + 1/2*c) - 23*a^3*b^3*tan(1/2*d*x + 1/2*c) - 2*a*b^5*tan(1/2*d*x + 1/2*c) + 8*a^6 - 7*a^4*b^2 - a^2*b^4)/((a*tan(1/2* d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2*a^2*b^5))/d
Time = 8.02 (sec) , antiderivative size = 1226, normalized size of antiderivative = 6.22 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
(atanh((1000*a^2*(b^2 - a^2)^(1/2))/(1000*a^2*b - (5000*a^4)/b + (4000*a^6 )/b^3 - 10000*a^3*tan(c/2 + (d*x)/2) + 2000*a*b^2*tan(c/2 + (d*x)/2) + (80 00*a^5*tan(c/2 + (d*x)/2))/b^2) - (4000*a^4*(b^2 - a^2)^(1/2))/(1000*a^2*b ^3 - 5000*a^4*b + (4000*a^6)/b + 8000*a^5*tan(c/2 + (d*x)/2) + 2000*a*b^4* tan(c/2 + (d*x)/2) - 10000*a^3*b^2*tan(c/2 + (d*x)/2)) + (2000*a*tan(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2))/(1000*a^2 - (5000*a^4)/b^2 + (4000*a^6)/b^4 - (10000*a^3*tan(c/2 + (d*x)/2))/b + (8000*a^5*tan(c/2 + (d*x)/2))/b^3 + 20 00*a*b*tan(c/2 + (d*x)/2)) - (9000*a^3*tan(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2 ))/(1000*a^2*b^2 - 5000*a^4 + (4000*a^6)/b^2 + 2000*a*b^3*tan(c/2 + (d*x)/ 2) - 10000*a^3*b*tan(c/2 + (d*x)/2) + (8000*a^5*tan(c/2 + (d*x)/2))/b) + ( 4000*a^5*tan(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2))/(4000*a^6 + 1000*a^2*b^4 - 5000*a^4*b^2 + 2000*a*b^5*tan(c/2 + (d*x)/2) + 8000*a^5*b*tan(c/2 + (d*x)/ 2) - 10000*a^3*b^3*tan(c/2 + (d*x)/2)))*(20*a^2*(b^2 - a^2)^(1/2) - 5*b^2* (b^2 - a^2)^(1/2)))/(b^6*d) - ((3*b^4 - 60*a^4 + 35*a^2*b^2)/(3*b^5) + (ta n(c/2 + (d*x)/2)*(6*b^4 - 210*a^4 + 125*a^2*b^2))/(3*a*b^4) - (tan(c/2 + ( d*x)/2)^8*(20*a^6 - 2*b^6 - 15*a^2*b^4 + 15*a^4*b^2))/(a^2*b^5) - (2*tan(c /2 + (d*x)/2)^6*(40*a^6 - 3*b^6 - 35*a^2*b^4 + 30*a^4*b^2))/(a^2*b^5) - (2 *tan(c/2 + (d*x)/2)^2*(120*a^6 - 3*b^6 - 55*a^2*b^4 + 10*a^4*b^2))/(3*a^2* b^5) - (2*tan(c/2 + (d*x)/2)^4*(180*a^6 - 9*b^6 - 120*a^2*b^4 + 95*a^4*b^2 ))/(3*a^2*b^5) + (tan(c/2 + (d*x)/2)^9*(2*b^4 - 10*a^4 + 5*a^2*b^2))/(a...