Integrand size = 21, antiderivative size = 230 \[ \int \frac {\sin ^6(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) x}{16 a^7}-\frac {2 (a-b)^{5/2} b (a+b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^7 d}+\frac {\left (16 b \left (a^2-b^2\right )^2-a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right ) \sin (c+d x)}{16 a^6 d}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right ) \sin ^3(c+d x)}{24 a^4 d}+\frac {(6 b-5 a \cos (c+d x)) \sin ^5(c+d x)}{30 a^2 d} \]
1/16*(5*a^6-30*a^4*b^2+40*a^2*b^4-16*b^6)*x/a^7-2*(a-b)^(5/2)*b*(a+b)^(5/2 )*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^7/d+1/16*(16*b*(a^ 2-b^2)^2-a*(5*a^4-14*a^2*b^2+8*b^4)*cos(d*x+c))*sin(d*x+c)/a^6/d+1/24*(8*b *(a^2-b^2)-a*(5*a^2-6*b^2)*cos(d*x+c))*sin(d*x+c)^3/a^4/d+1/30*(6*b-5*a*co s(d*x+c))*sin(d*x+c)^5/a^2/d
Time = 2.32 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.17 \[ \int \frac {\sin ^6(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {300 a^6 c-1800 a^4 b^2 c+2400 a^2 b^4 c-960 b^6 c+300 a^6 d x-1800 a^4 b^2 d x+2400 a^2 b^4 d x-960 b^6 d x+1920 b \left (a^2-b^2\right )^{5/2} \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+120 a b \left (11 a^4-18 a^2 b^2+8 b^4\right ) \sin (c+d x)-15 \left (15 a^6-32 a^4 b^2+16 a^2 b^4\right ) \sin (2 (c+d x))-140 a^5 b \sin (3 (c+d x))+80 a^3 b^3 \sin (3 (c+d x))+45 a^6 \sin (4 (c+d x))-30 a^4 b^2 \sin (4 (c+d x))+12 a^5 b \sin (5 (c+d x))-5 a^6 \sin (6 (c+d x))}{960 a^7 d} \]
(300*a^6*c - 1800*a^4*b^2*c + 2400*a^2*b^4*c - 960*b^6*c + 300*a^6*d*x - 1 800*a^4*b^2*d*x + 2400*a^2*b^4*d*x - 960*b^6*d*x + 1920*b*(a^2 - b^2)^(5/2 )*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 120*a*b*(11*a^4 - 18*a^2*b^2 + 8*b^4)*Sin[c + d*x] - 15*(15*a^6 - 32*a^4*b^2 + 16*a^2*b^4)* Sin[2*(c + d*x)] - 140*a^5*b*Sin[3*(c + d*x)] + 80*a^3*b^3*Sin[3*(c + d*x) ] + 45*a^6*Sin[4*(c + d*x)] - 30*a^4*b^2*Sin[4*(c + d*x)] + 12*a^5*b*Sin[5 *(c + d*x)] - 5*a^6*Sin[6*(c + d*x)])/(960*a^7*d)
Time = 1.29 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.14, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {3042, 4360, 25, 25, 3042, 3344, 25, 3042, 3344, 27, 3042, 3344, 3042, 3214, 3042, 3138, 221}
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.
| \(\displaystyle \int \frac {\sin ^6(c+d x)}{a+b \sec (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^6}{a-b \csc \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\sin ^6(c+d x) \cos (c+d x)}{-a \cos (c+d x)-b}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cos (c+d x) \sin ^6(c+d x)}{b+a \cos (c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\sin ^6(c+d x) \cos (c+d x)}{a \cos (c+d x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \cos \left (c+d x+\frac {\pi }{2}\right )^6}{a \sin \left (c+d x+\frac {\pi }{2}\right )+b}dx\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {\int -\frac {\left (a b-\left (5 a^2-6 b^2\right ) \cos (c+d x)\right ) \sin ^4(c+d x)}{b+a \cos (c+d x)}dx}{6 a^2}+\frac {\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}-\frac {\int \frac {\left (a b-\left (5 a^2-6 b^2\right ) \cos (c+d x)\right ) \sin ^4(c+d x)}{b+a \cos (c+d x)}dx}{6 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}-\frac {\int \frac {\cos \left (c+d x+\frac {\pi }{2}\right )^4 \left (a b+\left (6 b^2-5 a^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{6 a^2}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}-\frac {\frac {\int \frac {3 \left (a b \left (3 a^2-2 b^2\right )-\left (5 a^4-14 b^2 a^2+8 b^4\right ) \cos (c+d x)\right ) \sin ^2(c+d x)}{b+a \cos (c+d x)}dx}{4 a^2}-\frac {\sin ^3(c+d x) \left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right )}{4 a^2 d}}{6 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}-\frac {\frac {3 \int \frac {\left (a b \left (3 a^2-2 b^2\right )-\left (5 a^4-14 b^2 a^2+8 b^4\right ) \cos (c+d x)\right ) \sin ^2(c+d x)}{b+a \cos (c+d x)}dx}{4 a^2}-\frac {\sin ^3(c+d x) \left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right )}{4 a^2 d}}{6 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}-\frac {\frac {3 \int \frac {\cos \left (c+d x+\frac {\pi }{2}\right )^2 \left (a b \left (3 a^2-2 b^2\right )+\left (-5 a^4+14 b^2 a^2-8 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{4 a^2}-\frac {\sin ^3(c+d x) \left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right )}{4 a^2 d}}{6 a^2}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}-\frac {\frac {3 \left (\frac {\int \frac {a b \left (11 a^4-18 b^2 a^2+8 b^4\right )-\left (5 a^6-30 b^2 a^4+40 b^4 a^2-16 b^6\right ) \cos (c+d x)}{b+a \cos (c+d x)}dx}{2 a^2}-\frac {\sin (c+d x) \left (16 b \left (a^2-b^2\right )^2-a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right )}{2 a^2 d}\right )}{4 a^2}-\frac {\sin ^3(c+d x) \left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right )}{4 a^2 d}}{6 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}-\frac {\frac {3 \left (\frac {\int \frac {a b \left (11 a^4-18 b^2 a^2+8 b^4\right )+\left (-5 a^6+30 b^2 a^4-40 b^4 a^2+16 b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a^2}-\frac {\sin (c+d x) \left (16 b \left (a^2-b^2\right )^2-a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right )}{2 a^2 d}\right )}{4 a^2}-\frac {\sin ^3(c+d x) \left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right )}{4 a^2 d}}{6 a^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}-\frac {\frac {3 \left (\frac {\frac {16 b \left (a^2-b^2\right )^3 \int \frac {1}{b+a \cos (c+d x)}dx}{a}-\frac {x \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right )}{a}}{2 a^2}-\frac {\sin (c+d x) \left (16 b \left (a^2-b^2\right )^2-a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right )}{2 a^2 d}\right )}{4 a^2}-\frac {\sin ^3(c+d x) \left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right )}{4 a^2 d}}{6 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}-\frac {\frac {3 \left (\frac {\frac {16 b \left (a^2-b^2\right )^3 \int \frac {1}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}-\frac {x \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right )}{a}}{2 a^2}-\frac {\sin (c+d x) \left (16 b \left (a^2-b^2\right )^2-a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right )}{2 a^2 d}\right )}{4 a^2}-\frac {\sin ^3(c+d x) \left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right )}{4 a^2 d}}{6 a^2}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}-\frac {\frac {3 \left (\frac {\frac {32 b \left (a^2-b^2\right )^3 \int \frac {1}{-\left ((a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}-\frac {x \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right )}{a}}{2 a^2}-\frac {\sin (c+d x) \left (16 b \left (a^2-b^2\right )^2-a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right )}{2 a^2 d}\right )}{4 a^2}-\frac {\sin ^3(c+d x) \left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right )}{4 a^2 d}}{6 a^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}-\frac {\frac {3 \left (\frac {\frac {32 b \left (a^2-b^2\right )^3 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}-\frac {x \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right )}{a}}{2 a^2}-\frac {\sin (c+d x) \left (16 b \left (a^2-b^2\right )^2-a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right )}{2 a^2 d}\right )}{4 a^2}-\frac {\sin ^3(c+d x) \left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right )}{4 a^2 d}}{6 a^2}\) |
((6*b - 5*a*Cos[c + d*x])*Sin[c + d*x]^5)/(30*a^2*d) - (-1/4*((8*b*(a^2 - b^2) - a*(5*a^2 - 6*b^2)*Cos[c + d*x])*Sin[c + d*x]^3)/(a^2*d) + (3*((-((( 5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*x)/a) + (32*b*(a^2 - b^2)^3*ArcT anh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b ]*d))/(2*a^2) - ((16*b*(a^2 - b^2)^2 - a*(5*a^4 - 14*a^2*b^2 + 8*b^4)*Cos[ c + d*x])*Sin[c + d*x])/(2*a^2*d)))/(4*a^2))/(6*a^2)
3.3.3.3.1 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/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*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]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(440\) vs. \(2(213)=426\).
Time = 1.56 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.92
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 b \left (a +b \right )^{3} \left (a -b \right )^{3} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{7} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\frac {2 \left (\left (\frac {5}{16} a^{6}+a^{5} b -\frac {7}{8} a^{4} b^{2}-2 a^{3} b^{3}+\frac {1}{2} a^{2} b^{4}+a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (\frac {19}{3} a^{5} b -\frac {29}{8} a^{4} b^{2}+\frac {3}{2} a^{2} b^{4}+5 a \,b^{5}+\frac {85}{48} a^{6}-\frac {34}{3} a^{3} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (\frac {86}{5} a^{5} b -\frac {11}{4} a^{4} b^{2}-24 a^{3} b^{3}+a^{2} b^{4}+10 a \,b^{5}+\frac {33}{8} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-\frac {33}{8} a^{6}+\frac {11}{4} a^{4} b^{2}-a^{2} b^{4}+\frac {86}{5} a^{5} b -24 a^{3} b^{3}+10 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (\frac {19}{3} a^{5} b +\frac {29}{8} a^{4} b^{2}-\frac {34}{3} a^{3} b^{3}-\frac {3}{2} a^{2} b^{4}+5 a \,b^{5}-\frac {85}{48} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{5} b -2 a^{3} b^{3}+a \,b^{5}-\frac {5}{16} a^{6}+\frac {7}{8} a^{4} b^{2}-\frac {1}{2} a^{2} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}+\frac {\left (5 a^{6}-30 a^{4} b^{2}+40 a^{2} b^{4}-16 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{7}}}{d}\) | \(441\) |
| default | \(\frac {-\frac {2 b \left (a +b \right )^{3} \left (a -b \right )^{3} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{7} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\frac {2 \left (\left (\frac {5}{16} a^{6}+a^{5} b -\frac {7}{8} a^{4} b^{2}-2 a^{3} b^{3}+\frac {1}{2} a^{2} b^{4}+a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (\frac {19}{3} a^{5} b -\frac {29}{8} a^{4} b^{2}+\frac {3}{2} a^{2} b^{4}+5 a \,b^{5}+\frac {85}{48} a^{6}-\frac {34}{3} a^{3} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (\frac {86}{5} a^{5} b -\frac {11}{4} a^{4} b^{2}-24 a^{3} b^{3}+a^{2} b^{4}+10 a \,b^{5}+\frac {33}{8} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-\frac {33}{8} a^{6}+\frac {11}{4} a^{4} b^{2}-a^{2} b^{4}+\frac {86}{5} a^{5} b -24 a^{3} b^{3}+10 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (\frac {19}{3} a^{5} b +\frac {29}{8} a^{4} b^{2}-\frac {34}{3} a^{3} b^{3}-\frac {3}{2} a^{2} b^{4}+5 a \,b^{5}-\frac {85}{48} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{5} b -2 a^{3} b^{3}+a \,b^{5}-\frac {5}{16} a^{6}+\frac {7}{8} a^{4} b^{2}-\frac {1}{2} a^{2} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}+\frac {\left (5 a^{6}-30 a^{4} b^{2}+40 a^{2} b^{4}-16 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{7}}}{d}\) | \(441\) |
| risch | \(\frac {3 \sin \left (4 d x +4 c \right )}{64 a d}-\frac {15 x \,b^{2}}{8 a^{3}}+\frac {5 x \,b^{4}}{2 a^{5}}-\frac {x \,b^{6}}{a^{7}}-\frac {\sin \left (6 d x +6 c \right )}{192 d a}-\frac {15 \sin \left (2 d x +2 c \right )}{64 d a}-\frac {\sin \left (4 d x +4 c \right ) b^{2}}{32 a^{3} d}-\frac {7 b \sin \left (3 d x +3 c \right )}{48 a^{2} d}-\frac {i b^{5} {\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{6}}-\frac {11 i b \,{\mathrm e}^{i \left (d x +c \right )}}{16 d \,a^{2}}+\frac {9 i b^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d \,a^{4}}+\frac {11 i b \,{\mathrm e}^{-i \left (d x +c \right )}}{16 d \,a^{2}}-\frac {9 i b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d \,a^{4}}+\frac {i b^{5} {\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{6}}-\frac {\sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{d \,a^{3}}+\frac {2 \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{d \,a^{5}}-\frac {\sqrt {a^{2}-b^{2}}\, b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{d \,a^{7}}+\frac {\sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{d \,a^{3}}-\frac {2 \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{d \,a^{5}}+\frac {\sqrt {a^{2}-b^{2}}\, b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{d \,a^{7}}+\frac {b \sin \left (5 d x +5 c \right )}{80 a^{2} d}+\frac {b^{3} \sin \left (3 d x +3 c \right )}{12 a^{4} d}+\frac {\sin \left (2 d x +2 c \right ) b^{2}}{2 d \,a^{3}}-\frac {\sin \left (2 d x +2 c \right ) b^{4}}{4 d \,a^{5}}+\frac {5 x}{16 a}\) | \(645\) |
1/d*(-2*b*(a+b)^3*(a-b)^3/a^7/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*d* x+1/2*c)/((a-b)*(a+b))^(1/2))+2/a^7*(((5/16*a^6+a^5*b-7/8*a^4*b^2-2*a^3*b^ 3+1/2*a^2*b^4+a*b^5)*tan(1/2*d*x+1/2*c)^11+(19/3*a^5*b-29/8*a^4*b^2+3/2*a^ 2*b^4+5*a*b^5+85/48*a^6-34/3*a^3*b^3)*tan(1/2*d*x+1/2*c)^9+(86/5*a^5*b-11/ 4*a^4*b^2-24*a^3*b^3+a^2*b^4+10*a*b^5+33/8*a^6)*tan(1/2*d*x+1/2*c)^7+(-33/ 8*a^6+11/4*a^4*b^2-a^2*b^4+86/5*a^5*b-24*a^3*b^3+10*a*b^5)*tan(1/2*d*x+1/2 *c)^5+(19/3*a^5*b+29/8*a^4*b^2-34/3*a^3*b^3-3/2*a^2*b^4+5*a*b^5-85/48*a^6) *tan(1/2*d*x+1/2*c)^3+(a^5*b-2*a^3*b^3+a*b^5-5/16*a^6+7/8*a^4*b^2-1/2*a^2* b^4)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^6+1/16*(5*a^6-30*a^4*b^2 +40*a^2*b^4-16*b^6)*arctan(tan(1/2*d*x+1/2*c))))
Time = 0.33 (sec) , antiderivative size = 553, normalized size of antiderivative = 2.40 \[ \int \frac {\sin ^6(c+d x)}{a+b \sec (c+d x)} \, dx=\left [\frac {15 \, {\left (5 \, a^{6} - 30 \, a^{4} b^{2} + 40 \, a^{2} b^{4} - 16 \, b^{6}\right )} d x + 120 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - {\left (40 \, a^{6} \cos \left (d x + c\right )^{5} - 48 \, a^{5} b \cos \left (d x + c\right )^{4} - 368 \, a^{5} b + 560 \, a^{3} b^{3} - 240 \, a b^{5} - 10 \, {\left (13 \, a^{6} - 6 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (11 \, a^{5} b - 5 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (11 \, a^{6} - 18 \, a^{4} b^{2} + 8 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a^{7} d}, \frac {15 \, {\left (5 \, a^{6} - 30 \, a^{4} b^{2} + 40 \, a^{2} b^{4} - 16 \, b^{6}\right )} d x - 240 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left (40 \, a^{6} \cos \left (d x + c\right )^{5} - 48 \, a^{5} b \cos \left (d x + c\right )^{4} - 368 \, a^{5} b + 560 \, a^{3} b^{3} - 240 \, a b^{5} - 10 \, {\left (13 \, a^{6} - 6 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (11 \, a^{5} b - 5 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (11 \, a^{6} - 18 \, a^{4} b^{2} + 8 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a^{7} d}\right ] \]
[1/240*(15*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*d*x + 120*(a^4*b - 2 *a^2*b^3 + b^5)*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*co s(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - (40*a^6*cos(d*x + c)^5 - 48*a^5*b*cos(d*x + c)^4 - 368*a^5*b + 560*a^3*b^3 - 240*a*b^5 - 10*(13*a^6 - 6*a^4*b^2)*cos(d*x + c)^3 + 16*(11*a^5*b - 5*a^3*b^3)*cos(d*x + c)^2 + 15*(11*a^6 - 18*a^4*b^2 + 8*a^2*b^4)*cos(d*x + c))*sin(d*x + c)) /(a^7*d), 1/240*(15*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*d*x - 240*( a^4*b - 2*a^2*b^3 + b^5)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos( d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - (40*a^6*cos(d*x + c)^5 - 48*a^ 5*b*cos(d*x + c)^4 - 368*a^5*b + 560*a^3*b^3 - 240*a*b^5 - 10*(13*a^6 - 6* a^4*b^2)*cos(d*x + c)^3 + 16*(11*a^5*b - 5*a^3*b^3)*cos(d*x + c)^2 + 15*(1 1*a^6 - 18*a^4*b^2 + 8*a^2*b^4)*cos(d*x + c))*sin(d*x + c))/(a^7*d)]
\[ \int \frac {\sin ^6(c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {\sin ^{6}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
Exception generated. \[ \int \frac {\sin ^6(c+d x)}{a+b \sec (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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*a^2-4*b^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 781 vs. \(2 (212) = 424\).
Time = 0.35 (sec) , antiderivative size = 781, normalized size of antiderivative = 3.40 \[ \int \frac {\sin ^6(c+d x)}{a+b \sec (c+d x)} \, dx=\text {Too large to display} \]
1/240*(15*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*(d*x + c)/a^7 - 480*( a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn (-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/s qrt(-a^2 + b^2)))/(sqrt(-a^2 + b^2)*a^7) + 2*(75*a^5*tan(1/2*d*x + 1/2*c)^ 11 + 240*a^4*b*tan(1/2*d*x + 1/2*c)^11 - 210*a^3*b^2*tan(1/2*d*x + 1/2*c)^ 11 - 480*a^2*b^3*tan(1/2*d*x + 1/2*c)^11 + 120*a*b^4*tan(1/2*d*x + 1/2*c)^ 11 + 240*b^5*tan(1/2*d*x + 1/2*c)^11 + 425*a^5*tan(1/2*d*x + 1/2*c)^9 + 15 20*a^4*b*tan(1/2*d*x + 1/2*c)^9 - 870*a^3*b^2*tan(1/2*d*x + 1/2*c)^9 - 272 0*a^2*b^3*tan(1/2*d*x + 1/2*c)^9 + 360*a*b^4*tan(1/2*d*x + 1/2*c)^9 + 1200 *b^5*tan(1/2*d*x + 1/2*c)^9 + 990*a^5*tan(1/2*d*x + 1/2*c)^7 + 4128*a^4*b* tan(1/2*d*x + 1/2*c)^7 - 660*a^3*b^2*tan(1/2*d*x + 1/2*c)^7 - 5760*a^2*b^3 *tan(1/2*d*x + 1/2*c)^7 + 240*a*b^4*tan(1/2*d*x + 1/2*c)^7 + 2400*b^5*tan( 1/2*d*x + 1/2*c)^7 - 990*a^5*tan(1/2*d*x + 1/2*c)^5 + 4128*a^4*b*tan(1/2*d *x + 1/2*c)^5 + 660*a^3*b^2*tan(1/2*d*x + 1/2*c)^5 - 5760*a^2*b^3*tan(1/2* d*x + 1/2*c)^5 - 240*a*b^4*tan(1/2*d*x + 1/2*c)^5 + 2400*b^5*tan(1/2*d*x + 1/2*c)^5 - 425*a^5*tan(1/2*d*x + 1/2*c)^3 + 1520*a^4*b*tan(1/2*d*x + 1/2* c)^3 + 870*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 2720*a^2*b^3*tan(1/2*d*x + 1/2 *c)^3 - 360*a*b^4*tan(1/2*d*x + 1/2*c)^3 + 1200*b^5*tan(1/2*d*x + 1/2*c)^3 - 75*a^5*tan(1/2*d*x + 1/2*c) + 240*a^4*b*tan(1/2*d*x + 1/2*c) + 210*a^3* b^2*tan(1/2*d*x + 1/2*c) - 480*a^2*b^3*tan(1/2*d*x + 1/2*c) - 120*a*b^4...
Time = 16.12 (sec) , antiderivative size = 3341, normalized size of antiderivative = 14.53 \[ \int \frac {\sin ^6(c+d x)}{a+b \sec (c+d x)} \, dx=\text {Too large to display} \]
(atan(((((((42*a^21*b - 10*a^22 + 32*a^14*b^8 - 48*a^15*b^7 - 80*a^16*b^6 + 140*a^17*b^5 + 52*a^18*b^4 - 134*a^19*b^3 + 6*a^20*b^2)/a^18 - (tan(c/2 + (d*x)/2)*(512*a^16*b + 512*a^14*b^3 - 1024*a^15*b^2)*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i))/(128*a^19))*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i))/(16*a^7) + (tan(c/2 + (d*x)/2)*(1024*a*b^14 - 75*a^14*b + 2 5*a^15 - 512*b^15 + 2048*a^2*b^13 - 5120*a^3*b^12 - 2560*a^4*b^11 + 10240* a^5*b^10 - 10240*a^7*b^8 + 2540*a^8*b^7 + 5180*a^9*b^6 - 2064*a^10*b^5 - 1 136*a^11*b^4 + 619*a^12*b^3 + 31*a^13*b^2))/(8*a^12))*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i)*1i)/(16*a^7) - (((((42*a^21*b - 10*a^22 + 32*a^ 14*b^8 - 48*a^15*b^7 - 80*a^16*b^6 + 140*a^17*b^5 + 52*a^18*b^4 - 134*a^19 *b^3 + 6*a^20*b^2)/a^18 + (tan(c/2 + (d*x)/2)*(512*a^16*b + 512*a^14*b^3 - 1024*a^15*b^2)*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i))/(128*a^19) )*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i))/(16*a^7) - (tan(c/2 + (d *x)/2)*(1024*a*b^14 - 75*a^14*b + 25*a^15 - 512*b^15 + 2048*a^2*b^13 - 512 0*a^3*b^12 - 2560*a^4*b^11 + 10240*a^5*b^10 - 10240*a^7*b^8 + 2540*a^8*b^7 + 5180*a^9*b^6 - 2064*a^10*b^5 - 1136*a^11*b^4 + 619*a^12*b^3 + 31*a^13*b ^2))/(8*a^12))*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i)*1i)/(16*a^7) )/(((25*a^19*b)/4 - 96*a*b^19 + 64*b^20 - 480*a^2*b^18 + 760*a^3*b^17 + 15 44*a^4*b^16 - 2628*a^5*b^15 - 2748*a^6*b^14 + 5179*a^7*b^13 + 2890*a^8*b^1 2 - 6359*a^9*b^11 - 1736*a^10*b^10 + (19951*a^11*b^9)/4 + (937*a^12*b^8...