Integrand size = 23, antiderivative size = 148 \[ \int \frac {\sin ^6(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\frac {x}{b^3}-\frac {\sqrt {a} \left (8 a^2+20 a b+15 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{8 b^3 (a+b)^{5/2} d}+\frac {a \tan ^3(c+d x)}{4 b (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}+\frac {a (4 a+7 b) \tan (c+d x)}{8 b^2 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )} \]
x/b^3-1/8*(8*a^2+20*a*b+15*b^2)*arctan((a+b)^(1/2)*tan(d*x+c)/a^(1/2))*a^( 1/2)/b^3/(a+b)^(5/2)/d+1/4*a*tan(d*x+c)^3/b/(a+b)/d/(a+(a+b)*tan(d*x+c)^2) ^2+1/8*a*(4*a+7*b)*tan(d*x+c)/b^2/(a+b)^2/d/(a+(a+b)*tan(d*x+c)^2)
Time = 12.31 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.91 \[ \int \frac {\sin ^6(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\frac {8 (c+d x)-\frac {\sqrt {a} \left (8 a^2+20 a b+15 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{(a+b)^{5/2}}+\frac {a b \left (8 a^2+20 a b+9 b^2-3 b (2 a+3 b) \cos (2 (c+d x))\right ) \sin (2 (c+d x))}{(a+b)^2 (2 a+b-b \cos (2 (c+d x)))^2}}{8 b^3 d} \]
(8*(c + d*x) - (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcTan[(Sqrt[a + b]*Tan[ c + d*x])/Sqrt[a]])/(a + b)^(5/2) + (a*b*(8*a^2 + 20*a*b + 9*b^2 - 3*b*(2* a + 3*b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/((a + b)^2*(2*a + b - b*Cos[2 *(c + d*x)])^2))/(8*b^3*d)
Time = 0.41 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 3666, 372, 440, 397, 216, 218}
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.
| \(\displaystyle \int \frac {\sin ^6(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^6}{\left (a+b \sin (c+d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 3666 |
| \(\displaystyle \frac {\int \frac {\tan ^6(c+d x)}{\left (\tan ^2(c+d x)+1\right ) \left ((a+b) \tan ^2(c+d x)+a\right )^3}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\frac {a \tan ^3(c+d x)}{4 b (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )^2}-\frac {\int \frac {\tan ^2(c+d x) \left (3 a-(a+4 b) \tan ^2(c+d x)\right )}{\left (\tan ^2(c+d x)+1\right ) \left ((a+b) \tan ^2(c+d x)+a\right )^2}d\tan (c+d x)}{4 b (a+b)}}{d}\) |
| \(\Big \downarrow \) 440 |
| \(\displaystyle \frac {\frac {a \tan ^3(c+d x)}{4 b (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )^2}-\frac {\frac {\int \frac {a (4 a+7 b)-\left (4 a^2+9 b a+8 b^2\right ) \tan ^2(c+d x)}{\left (\tan ^2(c+d x)+1\right ) \left ((a+b) \tan ^2(c+d x)+a\right )}d\tan (c+d x)}{2 b (a+b)}-\frac {a (4 a+7 b) \tan (c+d x)}{2 b (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )}}{4 b (a+b)}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {a \tan ^3(c+d x)}{4 b (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )^2}-\frac {\frac {\frac {a \left (8 a^2+20 a b+15 b^2\right ) \int \frac {1}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{b}-\frac {8 (a+b)^2 \int \frac {1}{\tan ^2(c+d x)+1}d\tan (c+d x)}{b}}{2 b (a+b)}-\frac {a (4 a+7 b) \tan (c+d x)}{2 b (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )}}{4 b (a+b)}}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {a \tan ^3(c+d x)}{4 b (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )^2}-\frac {\frac {\frac {a \left (8 a^2+20 a b+15 b^2\right ) \int \frac {1}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{b}-\frac {8 (a+b)^2 \arctan (\tan (c+d x))}{b}}{2 b (a+b)}-\frac {a (4 a+7 b) \tan (c+d x)}{2 b (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )}}{4 b (a+b)}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {a \tan ^3(c+d x)}{4 b (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )^2}-\frac {\frac {\frac {\sqrt {a} \left (8 a^2+20 a b+15 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{b \sqrt {a+b}}-\frac {8 (a+b)^2 \arctan (\tan (c+d x))}{b}}{2 b (a+b)}-\frac {a (4 a+7 b) \tan (c+d x)}{2 b (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )}}{4 b (a+b)}}{d}\) |
((a*Tan[c + d*x]^3)/(4*b*(a + b)*(a + (a + b)*Tan[c + d*x]^2)^2) - (((-8*( a + b)^2*ArcTan[Tan[c + d*x]])/b + (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcT an[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(b*Sqrt[a + b]))/(2*b*(a + b)) - ( a*(4*a + 7*b)*Tan[c + d*x])/(2*b*(a + b)*(a + (a + b)*Tan[c + d*x]^2)))/(4 *b*(a + b)))/d
3.2.6.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ )*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[g*(b*e - a*f)*(g*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] - Simp[ g^2/(2*b*(b*c - a*d)*(p + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f)*(m - 1) + (d*(b*e - a*f)*(m + 2*q + 1) - b*2*(c *f - d*e)*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, q}, x] && LtQ[p, -1] && GtQ[m, 1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1)) , x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] & & IntegerQ[p]
Time = 1.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.07
| method | result | size |
| derivativedivides | \(\frac {\frac {\arctan \left (\tan \left (d x +c \right )\right )}{b^{3}}-\frac {a \left (\frac {-\frac {\left (4 a +9 b \right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{8 \left (a +b \right )}-\frac {a b \left (4 a +7 b \right ) \tan \left (d x +c \right )}{8 \left (a^{2}+2 a b +b^{2}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{2}}+\frac {\left (8 a^{2}+20 a b +15 b^{2}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a \left (a +b \right )}}\right )}{b^{3}}}{d}\) | \(158\) |
| default | \(\frac {\frac {\arctan \left (\tan \left (d x +c \right )\right )}{b^{3}}-\frac {a \left (\frac {-\frac {\left (4 a +9 b \right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{8 \left (a +b \right )}-\frac {a b \left (4 a +7 b \right ) \tan \left (d x +c \right )}{8 \left (a^{2}+2 a b +b^{2}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{2}}+\frac {\left (8 a^{2}+20 a b +15 b^{2}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a \left (a +b \right )}}\right )}{b^{3}}}{d}\) | \(158\) |
| risch | \(\frac {x}{b^{3}}-\frac {i a \left (-16 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-28 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-9 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+48 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+120 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+90 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+27 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-32 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-68 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-27 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+6 a \,b^{2}+9 b^{3}\right )}{4 b^{3} \left (a +b \right )^{2} d \left (-b \,{\mathrm e}^{4 i \left (d x +c \right )}+4 a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}-b \right )^{2}}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right ) a^{2}}{2 \left (a +b \right )^{3} d \,b^{3}}-\frac {5 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right ) a}{4 \left (a +b \right )^{3} d \,b^{2}}-\frac {15 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right )}{16 \left (a +b \right )^{3} d b}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right ) a^{2}}{2 \left (a +b \right )^{3} d \,b^{3}}+\frac {5 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right ) a}{4 \left (a +b \right )^{3} d \,b^{2}}+\frac {15 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right )}{16 \left (a +b \right )^{3} d b}\) | \(551\) |
1/d*(1/b^3*arctan(tan(d*x+c))-a/b^3*((-1/8*(4*a+9*b)*b/(a+b)*tan(d*x+c)^3- 1/8*a*b*(4*a+7*b)/(a^2+2*a*b+b^2)*tan(d*x+c))/(a*tan(d*x+c)^2+tan(d*x+c)^2 *b+a)^2+1/8*(8*a^2+20*a*b+15*b^2)/(a^2+2*a*b+b^2)/(a*(a+b))^(1/2)*arctan(( a+b)*tan(d*x+c)/(a*(a+b))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (134) = 268\).
Time = 0.31 (sec) , antiderivative size = 950, normalized size of antiderivative = 6.42 \[ \int \frac {\sin ^6(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
[1/32*(32*(a^2*b^2 + 2*a*b^3 + b^4)*d*x*cos(d*x + c)^4 - 64*(a^3*b + 3*a^2 *b^2 + 3*a*b^3 + b^4)*d*x*cos(d*x + c)^2 + 32*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x + ((8*a^2*b^2 + 20*a*b^3 + 15*b^4)*cos(d*x + c)^4 + 8* a^4 + 36*a^3*b + 63*a^2*b^2 + 50*a*b^3 + 15*b^4 - 2*(8*a^3*b + 28*a^2*b^2 + 35*a*b^3 + 15*b^4)*cos(d*x + c)^2)*sqrt(-a/(a + b))*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a^2 + 3*a*b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a /(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) - 4*(3*(2*a^2*b^2 + 3*a*b^3)*co s(d*x + c)^3 - (4*a^3*b + 13*a^2*b^2 + 9*a*b^3)*cos(d*x + c))*sin(d*x + c) )/((a^2*b^5 + 2*a*b^6 + b^7)*d*cos(d*x + c)^4 - 2*(a^3*b^4 + 3*a^2*b^5 + 3 *a*b^6 + b^7)*d*cos(d*x + c)^2 + (a^4*b^3 + 4*a^3*b^4 + 6*a^2*b^5 + 4*a*b^ 6 + b^7)*d), 1/16*(16*(a^2*b^2 + 2*a*b^3 + b^4)*d*x*cos(d*x + c)^4 - 32*(a ^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*x*cos(d*x + c)^2 + 16*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x + ((8*a^2*b^2 + 20*a*b^3 + 15*b^4)*cos(d*x + c)^4 + 8*a^4 + 36*a^3*b + 63*a^2*b^2 + 50*a*b^3 + 15*b^4 - 2*(8*a^3*b + 28*a^2*b^2 + 35*a*b^3 + 15*b^4)*cos(d*x + c)^2)*sqrt(a/(a + b))*arctan(1/ 2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*x + c)*sin(d *x + c))) - 2*(3*(2*a^2*b^2 + 3*a*b^3)*cos(d*x + c)^3 - (4*a^3*b + 13*a^2* b^2 + 9*a*b^3)*cos(d*x + c))*sin(d*x + c))/((a^2*b^5 + 2*a*b^6 + b^7)*d...
Timed out. \[ \int \frac {\sin ^6(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\text {Timed out} \]
Time = 0.37 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.58 \[ \int \frac {\sin ^6(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=-\frac {\frac {{\left (8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right )} \sqrt {{\left (a + b\right )} a}} - \frac {{\left (4 \, a^{3} + 13 \, a^{2} b + 9 \, a b^{2}\right )} \tan \left (d x + c\right )^{3} + {\left (4 \, a^{3} + 7 \, a^{2} b\right )} \tan \left (d x + c\right )}{a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} + {\left (a^{4} b^{2} + 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} + 4 \, a b^{5} + b^{6}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right )} \tan \left (d x + c\right )^{2}} - \frac {8 \, {\left (d x + c\right )}}{b^{3}}}{8 \, d} \]
-1/8*((8*a^3 + 20*a^2*b + 15*a*b^2)*arctan((a + b)*tan(d*x + c)/sqrt((a + b)*a))/((a^2*b^3 + 2*a*b^4 + b^5)*sqrt((a + b)*a)) - ((4*a^3 + 13*a^2*b + 9*a*b^2)*tan(d*x + c)^3 + (4*a^3 + 7*a^2*b)*tan(d*x + c))/(a^4*b^2 + 2*a^3 *b^3 + a^2*b^4 + (a^4*b^2 + 4*a^3*b^3 + 6*a^2*b^4 + 4*a*b^5 + b^6)*tan(d*x + c)^4 + 2*(a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*tan(d*x + c)^2) - 8* (d*x + c)/b^3)/d
Time = 0.40 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.51 \[ \int \frac {\sin ^6(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=-\frac {\frac {{\left (8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{{\left (a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right )} \sqrt {a^{2} + a b}} - \frac {4 \, a^{3} \tan \left (d x + c\right )^{3} + 13 \, a^{2} b \tan \left (d x + c\right )^{3} + 9 \, a b^{2} \tan \left (d x + c\right )^{3} + 4 \, a^{3} \tan \left (d x + c\right ) + 7 \, a^{2} b \tan \left (d x + c\right )}{{\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} {\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )}^{2}} - \frac {8 \, {\left (d x + c\right )}}{b^{3}}}{8 \, d} \]
-1/8*((8*a^3 + 20*a^2*b + 15*a*b^2)*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/((a^2* b^3 + 2*a*b^4 + b^5)*sqrt(a^2 + a*b)) - (4*a^3*tan(d*x + c)^3 + 13*a^2*b*t an(d*x + c)^3 + 9*a*b^2*tan(d*x + c)^3 + 4*a^3*tan(d*x + c) + 7*a^2*b*tan( d*x + c))/((a^2*b^2 + 2*a*b^3 + b^4)*(a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)^2) - 8*(d*x + c)/b^3)/d
Time = 17.72 (sec) , antiderivative size = 3189, normalized size of antiderivative = 21.55 \[ \int \frac {\sin ^6(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
((tan(c + d*x)^3*(9*a*b + 4*a^2))/(8*(a*b^2 + b^3)) + (a*tan(c + d*x)*(7*a *b + 4*a^2))/(8*(2*a*b^3 + b^4 + a^2*b^2)))/(d*(tan(c + d*x)^4*(2*a*b + a^ 2 + b^2) + a^2 + tan(c + d*x)^2*(2*a*b + 2*a^2))) - atan(((((((7*a*b^10)/2 + (25*a^2*b^9)/2 + (33*a^3*b^8)/2 + (19*a^4*b^7)/2 + 2*a^5*b^6)*1i)/(2*(3 *a*b^8 + b^9 + 3*a^2*b^7 + a^3*b^6)) - (tan(c + d*x)*(1792*a*b^11 + 256*b^ 12 + 5120*a^2*b^10 + 7680*a^3*b^9 + 6400*a^4*b^8 + 2816*a^5*b^7 + 512*a^6* b^6))/(128*b^3*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)))/(2*b^3) + (tan(c + d*x)*(384*a*b^5 + 704*a^5*b + 128*a^6 + 64*b^6 + 1185*a^2*b^4 + 1880*a^3*b ^3 + 1600*a^4*b^2))/(64*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)))/b^3 - (((( (7*a*b^10)/2 + (25*a^2*b^9)/2 + (33*a^3*b^8)/2 + (19*a^4*b^7)/2 + 2*a^5*b^ 6)*1i)/(2*(3*a*b^8 + b^9 + 3*a^2*b^7 + a^3*b^6)) + (tan(c + d*x)*(1792*a*b ^11 + 256*b^12 + 5120*a^2*b^10 + 7680*a^3*b^9 + 6400*a^4*b^8 + 2816*a^5*b^ 7 + 512*a^6*b^6))/(128*b^3*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)))/(2*b^3) - (tan(c + d*x)*(384*a*b^5 + 704*a^5*b + 128*a^6 + 64*b^6 + 1185*a^2*b^4 + 1880*a^3*b^3 + 1600*a^4*b^2))/(64*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)) )/b^3)/(((15*a*b^4)/4 + (19*a^4*b)/4 + a^5 + (295*a^2*b^3)/32 + (19*a^3*b^ 2)/2)/(3*a*b^8 + b^9 + 3*a^2*b^7 + a^3*b^6) + ((((((7*a*b^10)/2 + (25*a^2* b^9)/2 + (33*a^3*b^8)/2 + (19*a^4*b^7)/2 + 2*a^5*b^6)*1i)/(2*(3*a*b^8 + b^ 9 + 3*a^2*b^7 + a^3*b^6)) - (tan(c + d*x)*(1792*a*b^11 + 256*b^12 + 5120*a ^2*b^10 + 7680*a^3*b^9 + 6400*a^4*b^8 + 2816*a^5*b^7 + 512*a^6*b^6))/(1...