\(\int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1268]

Optimal result

Integrand size = 27, antiderivative size = 237 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {15 \left (8 a^4-8 a^2 b^2+b^4\right ) x}{8 b^7}+\frac {15 a \left (2 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^7 \sqrt {a^2-b^2} d}+\frac {\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}+\frac {5 \cos ^3(c+d x) \left (4 a^2-b^2+a b \sin (c+d x)\right )}{4 b^4 d (a+b \sin (c+d x))}-\frac {15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d} \]

[Out]

-15/8*(8*a^4-8*a^2*b^2+b^4)*x/b^7+1/4*cos(d*x+c)^5*(3*a+b*sin(d*x+c))/b^2/d/(a+b*sin(d*x+c))^2+5/4*cos(d*x+c)^
3*(4*a^2-b^2+a*b*sin(d*x+c))/b^4/d/(a+b*sin(d*x+c))-15/8*cos(d*x+c)*(4*a*(2*a^2-b^2)-b*(4*a^2-b^2)*sin(d*x+c))
/b^6/d+15*a*(2*a^4-3*a^2*b^2+b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^7/d/(a^2-b^2)^(1/2)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2942, 2944, 2814, 2739, 632, 210} \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d}+\frac {5 \cos ^3(c+d x) \left (4 a^2+a b \sin (c+d x)-b^2\right )}{4 b^4 d (a+b \sin (c+d x))}+\frac {15 a \left (2 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^7 d \sqrt {a^2-b^2}}-\frac {15 x \left (8 a^4-8 a^2 b^2+b^4\right )}{8 b^7}+\frac {\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2} \]

[In]

Int[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

(-15*(8*a^4 - 8*a^2*b^2 + b^4)*x)/(8*b^7) + (15*a*(2*a^4 - 3*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sq
rt[a^2 - b^2]])/(b^7*Sqrt[a^2 - b^2]*d) + (Cos[c + d*x]^5*(3*a + b*Sin[c + d*x]))/(4*b^2*d*(a + b*Sin[c + d*x]
)^2) + (5*Cos[c + d*x]^3*(4*a^2 - b^2 + a*b*Sin[c + d*x]))/(4*b^4*d*(a + b*Sin[c + d*x])) - (15*Cos[c + d*x]*(
4*a*(2*a^2 - b^2) - b*(4*a^2 - b^2)*Sin[c + d*x]))/(8*b^6*d)

Rule 210

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])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[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]

Rule 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(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]

Rule 2942

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 + 1)*Sin[e + f*x])/(b^2*f*(m + 1)*(m + p + 1))), x] + Dist[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)*Si
n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && N
eQ[m + p + 1, 0] && IntegerQ[2*m]

Rule 2944

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] + Dist[g^2*((p - 1)/(b^2*(m + p)*(m + p +
1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}-\frac {5 \int \frac {\cos ^4(c+d x) (-2 b-6 a \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx}{8 b^2} \\ & = \frac {\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}+\frac {5 \cos ^3(c+d x) \left (4 a^2-b^2+a b \sin (c+d x)\right )}{4 b^4 d (a+b \sin (c+d x))}+\frac {5 \int \frac {\cos ^2(c+d x) \left (6 a b+6 \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{8 b^4} \\ & = \frac {\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}+\frac {5 \cos ^3(c+d x) \left (4 a^2-b^2+a b \sin (c+d x)\right )}{4 b^4 d (a+b \sin (c+d x))}-\frac {15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d}+\frac {5 \int \frac {-6 a b \left (4 a^2-3 b^2\right )-6 \left (8 a^4-8 a^2 b^2+b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{16 b^6} \\ & = -\frac {15 \left (8 a^4-8 a^2 b^2+b^4\right ) x}{8 b^7}+\frac {\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}+\frac {5 \cos ^3(c+d x) \left (4 a^2-b^2+a b \sin (c+d x)\right )}{4 b^4 d (a+b \sin (c+d x))}-\frac {15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d}+\frac {\left (15 a \left (2 a^4-3 a^2 b^2+b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 b^7} \\ & = -\frac {15 \left (8 a^4-8 a^2 b^2+b^4\right ) x}{8 b^7}+\frac {\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}+\frac {5 \cos ^3(c+d x) \left (4 a^2-b^2+a b \sin (c+d x)\right )}{4 b^4 d (a+b \sin (c+d x))}-\frac {15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d}+\frac {\left (15 a \left (2 a^4-3 a^2 b^2+b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^7 d} \\ & = -\frac {15 \left (8 a^4-8 a^2 b^2+b^4\right ) x}{8 b^7}+\frac {\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}+\frac {5 \cos ^3(c+d x) \left (4 a^2-b^2+a b \sin (c+d x)\right )}{4 b^4 d (a+b \sin (c+d x))}-\frac {15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d}-\frac {\left (30 a \left (2 a^4-3 a^2 b^2+b^4\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^7 d} \\ & = -\frac {15 \left (8 a^4-8 a^2 b^2+b^4\right ) x}{8 b^7}+\frac {15 a \left (2 a^4-3 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^7 \sqrt {a^2-b^2} d}+\frac {\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}+\frac {5 \cos ^3(c+d x) \left (4 a^2-b^2+a b \sin (c+d x)\right )}{4 b^4 d (a+b \sin (c+d x))}-\frac {15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1250\) vs. \(2(237)=474\).

Time = 6.65 (sec) , antiderivative size = 1250, normalized size of antiderivative = 5.27 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {18 \left (-8 (c+d x)+\frac {2 a \left (8 a^4-20 a^2 b^2+15 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {a b \left (4 a^2-3 b^2\right ) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))^2}-\frac {3 b \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}\right )}{b^3}-\frac {10 \left (\frac {6 a b \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {\cos (c+d x) \left (a \left (2 a^2+b^2\right )+b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^2}\right )}{(a-b)^2 (a+b)^2}+\frac {10 \left (-24 \left (-8 a^2+b^2\right ) (c+d x)-\frac {6 a \left (64 a^6-168 a^4 b^2+140 a^2 b^4-35 b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+96 a b \cos (c+d x)+\frac {a b \left (-16 a^4+20 a^2 b^2-5 b^4\right ) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))^2}+\frac {b \left (112 a^6-220 a^4 b^2+115 a^2 b^4-10 b^6\right ) \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}-8 b^2 \sin (2 (c+d x))\right )}{b^5}+\frac {\frac {12 a \left (640 a^8-1920 a^6 b^2+2016 a^4 b^4-840 a^2 b^6+105 b^8\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {-3840 a^{10} (c+d x)+7680 a^8 b^2 (c+d x)-2976 a^6 b^4 (c+d x)-1776 a^4 b^6 (c+d x)+960 a^2 b^8 (c+d x)-48 b^{10} (c+d x)-3840 a^9 b \cos (c+d x)+8640 a^7 b^3 \cos (c+d x)-5696 a^5 b^5 \cos (c+d x)+788 a^3 b^7 \cos (c+d x)+114 a b^9 \cos (c+d x)+1920 a^8 b^2 (c+d x) \cos (2 (c+d x))-4800 a^6 b^4 (c+d x) \cos (2 (c+d x))+3888 a^4 b^6 (c+d x) \cos (2 (c+d x))-1056 a^2 b^8 (c+d x) \cos (2 (c+d x))+48 b^{10} (c+d x) \cos (2 (c+d x))+320 a^7 b^3 \cos (3 (c+d x))-760 a^5 b^5 \cos (3 (c+d x))+560 a^3 b^7 \cos (3 (c+d x))-120 a b^9 \cos (3 (c+d x))-8 a^5 b^5 \cos (5 (c+d x))+16 a^3 b^7 \cos (5 (c+d x))-8 a b^9 \cos (5 (c+d x))-7680 a^9 b (c+d x) \sin (c+d x)+19200 a^7 b^3 (c+d x) \sin (c+d x)-15552 a^5 b^5 (c+d x) \sin (c+d x)+4224 a^3 b^7 (c+d x) \sin (c+d x)-192 a b^9 (c+d x) \sin (c+d x)-2880 a^8 b^2 \sin (2 (c+d x))+6880 a^6 b^4 \sin (2 (c+d x))-5182 a^4 b^6 \sin (2 (c+d x))+1221 a^2 b^8 \sin (2 (c+d x))-36 b^{10} \sin (2 (c+d x))-40 a^6 b^4 \sin (4 (c+d x))+88 a^4 b^6 \sin (4 (c+d x))-56 a^2 b^8 \sin (4 (c+d x))+8 b^{10} \sin (4 (c+d x))+2 a^4 b^6 \sin (6 (c+d x))-4 a^2 b^8 \sin (6 (c+d x))+2 b^{10} \sin (6 (c+d x))}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}}{b^7}}{256 d} \]

[In]

Integrate[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

((18*(-8*(c + d*x) + (2*a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2
- b^2)^(5/2) + (a*b*(4*a^2 - 3*b^2)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) - (3*b*(4*a^4 - 7*
a^2*b^2 + 2*b^4)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x]))))/b^3 - (10*((6*a*b*ArcTan[(b + a*Ta
n[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Cos[c + d*x]*(a*(2*a^2 + b^2) + b*(a^2 + 2*b^2)*Sin[c + d
*x]))/(a + b*Sin[c + d*x])^2))/((a - b)^2*(a + b)^2) + (10*(-24*(-8*a^2 + b^2)*(c + d*x) - (6*a*(64*a^6 - 168*
a^4*b^2 + 140*a^2*b^4 - 35*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 96*a*b*C
os[c + d*x] + (a*b*(-16*a^4 + 20*a^2*b^2 - 5*b^4)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) + (b*
(112*a^6 - 220*a^4*b^2 + 115*a^2*b^4 - 10*b^6)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])) - 8*b^
2*Sin[2*(c + d*x)]))/b^5 + ((12*a*(640*a^8 - 1920*a^6*b^2 + 2016*a^4*b^4 - 840*a^2*b^6 + 105*b^8)*ArcTan[(b +
a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (-3840*a^10*(c + d*x) + 7680*a^8*b^2*(c + d*x) - 297
6*a^6*b^4*(c + d*x) - 1776*a^4*b^6*(c + d*x) + 960*a^2*b^8*(c + d*x) - 48*b^10*(c + d*x) - 3840*a^9*b*Cos[c +
d*x] + 8640*a^7*b^3*Cos[c + d*x] - 5696*a^5*b^5*Cos[c + d*x] + 788*a^3*b^7*Cos[c + d*x] + 114*a*b^9*Cos[c + d*
x] + 1920*a^8*b^2*(c + d*x)*Cos[2*(c + d*x)] - 4800*a^6*b^4*(c + d*x)*Cos[2*(c + d*x)] + 3888*a^4*b^6*(c + d*x
)*Cos[2*(c + d*x)] - 1056*a^2*b^8*(c + d*x)*Cos[2*(c + d*x)] + 48*b^10*(c + d*x)*Cos[2*(c + d*x)] + 320*a^7*b^
3*Cos[3*(c + d*x)] - 760*a^5*b^5*Cos[3*(c + d*x)] + 560*a^3*b^7*Cos[3*(c + d*x)] - 120*a*b^9*Cos[3*(c + d*x)]
- 8*a^5*b^5*Cos[5*(c + d*x)] + 16*a^3*b^7*Cos[5*(c + d*x)] - 8*a*b^9*Cos[5*(c + d*x)] - 7680*a^9*b*(c + d*x)*S
in[c + d*x] + 19200*a^7*b^3*(c + d*x)*Sin[c + d*x] - 15552*a^5*b^5*(c + d*x)*Sin[c + d*x] + 4224*a^3*b^7*(c +
d*x)*Sin[c + d*x] - 192*a*b^9*(c + d*x)*Sin[c + d*x] - 2880*a^8*b^2*Sin[2*(c + d*x)] + 6880*a^6*b^4*Sin[2*(c +
d*x)] - 5182*a^4*b^6*Sin[2*(c + d*x)] + 1221*a^2*b^8*Sin[2*(c + d*x)] - 36*b^10*Sin[2*(c + d*x)] - 40*a^6*b^4
*Sin[4*(c + d*x)] + 88*a^4*b^6*Sin[4*(c + d*x)] - 56*a^2*b^8*Sin[4*(c + d*x)] + 8*b^10*Sin[4*(c + d*x)] + 2*a^
4*b^6*Sin[6*(c + d*x)] - 4*a^2*b^8*Sin[6*(c + d*x)] + 2*b^10*Sin[6*(c + d*x)])/((a^2 - b^2)^2*(a + b*Sin[c + d
*x])^2))/b^7)/(256*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(471\) vs. \(2(224)=448\).

Time = 2.79 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.99

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {\left (3 a^{2} b^{2}-\frac {9}{8} b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{3} b -9 a \,b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 a^{2} b^{2}-\frac {1}{8} b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{3} b -21 a \,b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a^{2} b^{2}+\frac {1}{8} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{3} b -19 a \,b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a^{2} b^{2}+\frac {9}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+10 a^{3} b -7 a \,b^{3}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {15 \left (8 a^{4}-8 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}\right )}{b^{7}}+\frac {\frac {2 \left (\left (-\frac {9}{2} a^{4} b^{2}+\frac {9}{2} a^{2} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {b \left (10 a^{6}+9 a^{4} b^{2}-21 a^{2} b^{4}+2 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {b^{2} \left (31 a^{4}-35 a^{2} b^{2}+4 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a b \left (10 a^{4}-11 a^{2} b^{2}+b^{4}\right )}{2}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {15 a \left (2 a^{4}-3 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{b^{7}}}{d}\)	\(472\)
default	\(\frac {-\frac {2 \left (\frac {\left (3 a^{2} b^{2}-\frac {9}{8} b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{3} b -9 a \,b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 a^{2} b^{2}-\frac {1}{8} b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{3} b -21 a \,b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a^{2} b^{2}+\frac {1}{8} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{3} b -19 a \,b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a^{2} b^{2}+\frac {9}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+10 a^{3} b -7 a \,b^{3}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {15 \left (8 a^{4}-8 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}\right )}{b^{7}}+\frac {\frac {2 \left (\left (-\frac {9}{2} a^{4} b^{2}+\frac {9}{2} a^{2} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {b \left (10 a^{6}+9 a^{4} b^{2}-21 a^{2} b^{4}+2 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {b^{2} \left (31 a^{4}-35 a^{2} b^{2}+4 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a b \left (10 a^{4}-11 a^{2} b^{2}+b^{4}\right )}{2}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {15 a \left (2 a^{4}-3 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{b^{7}}}{d}\)	\(472\)
risch	\(-\frac {15 x \,a^{4}}{b^{7}}+\frac {15 x \,a^{2}}{b^{5}}-\frac {15 x}{8 b^{3}}+\frac {a \,{\mathrm e}^{3 i \left (d x +c \right )}}{8 b^{4} d}-\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} a^{2}}{4 b^{5} d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{4 b^{3} d}-\frac {5 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{b^{6} d}+\frac {27 a \,{\mathrm e}^{i \left (d x +c \right )}}{8 b^{4} d}-\frac {5 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{b^{6} d}+\frac {27 a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 b^{4} d}+\frac {i \left (-12 i a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}+15 i a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-3 i a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+32 i a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-37 i a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}+5 i a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}+22 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-15 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+2 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-11 a^{4} b^{2}+13 a^{2} b^{4}-2 b^{6}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} d \,b^{7}}+\frac {15 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 )}{2 d \,b^{5}}+\frac {a \,{\mathrm e}^{-3 i \left (d x +c \right )}}{8 b^{4} d}+\frac {15 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^{7}}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{4 b^{3} d}-\frac {15 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^{7}}+\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{2}}{4 b^{5} d}-\frac {15 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 )}{2 d \,b^{5}}-\frac {\sin \left (4 d x +4 c \right )}{32 b^{3} d}\)	\(670\)

[In]

int(cos(d*x+c)^6*sin(d*x+c)/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-2/b^7*(((3*a^2*b^2-9/8*b^4)*tan(1/2*d*x+1/2*c)^7+(10*a^3*b-9*a*b^3)*tan(1/2*d*x+1/2*c)^6+(3*a^2*b^2-1/8*
b^4)*tan(1/2*d*x+1/2*c)^5+(30*a^3*b-21*a*b^3)*tan(1/2*d*x+1/2*c)^4+(-3*a^2*b^2+1/8*b^4)*tan(1/2*d*x+1/2*c)^3+(
30*a^3*b-19*a*b^3)*tan(1/2*d*x+1/2*c)^2+(-3*a^2*b^2+9/8*b^4)*tan(1/2*d*x+1/2*c)+10*a^3*b-7*a*b^3)/(1+tan(1/2*d
*x+1/2*c)^2)^4+15/8*(8*a^4-8*a^2*b^2+b^4)*arctan(tan(1/2*d*x+1/2*c)))+2/b^7*(((-9/2*a^4*b^2+9/2*a^2*b^4)*tan(1
/2*d*x+1/2*c)^3-1/2*b*(10*a^6+9*a^4*b^2-21*a^2*b^4+2*b^6)/a*tan(1/2*d*x+1/2*c)^2-1/2*b^2*(31*a^4-35*a^2*b^2+4*
b^4)*tan(1/2*d*x+1/2*c)-1/2*a*b*(10*a^4-11*a^2*b^2+b^4))/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+1
5/2*a*(2*a^4-3*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))))

Fricas [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 837, normalized size of antiderivative = 3.53 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\left [\frac {4 \, a b^{5} \cos \left (d x + c\right )^{5} - 15 \, {\left (8 \, a^{4} b^{2} - 8 \, a^{2} b^{4} + b^{6}\right )} d x \cos \left (d x + c\right )^{2} - 10 \, {\left (4 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, a^{6} - 7 \, a^{2} b^{4} + b^{6}\right )} d x + 30 \, {\left (2 \, a^{5} + a^{3} b^{2} - a b^{4} - {\left (2 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{4} b - a^{2} 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^{5} b - 2 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right ) - {\left (2 \, b^{6} \cos \left (d x + c\right )^{5} - 5 \, {\left (2 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} - 30 \, {\left (8 \, a^{5} b - 8 \, a^{3} b^{3} + a b^{5}\right )} d x - 15 \, {\left (12 \, a^{4} b^{2} - 11 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, {\left (b^{9} d \cos \left (d x + c\right )^{2} - 2 \, a b^{8} d \sin \left (d x + c\right ) - {\left (a^{2} b^{7} + b^{9}\right )} d\right )}}, \frac {4 \, a b^{5} \cos \left (d x + c\right )^{5} - 15 \, {\left (8 \, a^{4} b^{2} - 8 \, a^{2} b^{4} + b^{6}\right )} d x \cos \left (d x + c\right )^{2} - 10 \, {\left (4 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, a^{6} - 7 \, a^{2} b^{4} + b^{6}\right )} d x + 60 \, {\left (2 \, a^{5} + a^{3} b^{2} - a b^{4} - {\left (2 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{4} b - a^{2} 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 ) + 30 \, {\left (4 \, a^{5} b - 2 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right ) - {\left (2 \, b^{6} \cos \left (d x + c\right )^{5} - 5 \, {\left (2 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} - 30 \, {\left (8 \, a^{5} b - 8 \, a^{3} b^{3} + a b^{5}\right )} d x - 15 \, {\left (12 \, a^{4} b^{2} - 11 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, {\left (b^{9} d \cos \left (d x + c\right )^{2} - 2 \, a b^{8} d \sin \left (d x + c\right ) - {\left (a^{2} b^{7} + b^{9}\right )} d\right )}}\right ] \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/8*(4*a*b^5*cos(d*x + c)^5 - 15*(8*a^4*b^2 - 8*a^2*b^4 + b^6)*d*x*cos(d*x + c)^2 - 10*(4*a^3*b^3 - 3*a*b^5)*
cos(d*x + c)^3 + 15*(8*a^6 - 7*a^2*b^4 + b^6)*d*x + 30*(2*a^5 + a^3*b^2 - a*b^4 - (2*a^3*b^2 - a*b^4)*cos(d*x
+ c)^2 + 2*(2*a^4*b - a^2*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^5*b - 2*a^3*b^3 - a*b^5)*cos(d*x + c) - (2*b^6*cos(d*x + c)^5 - 5*(
2*a^2*b^4 - b^6)*cos(d*x + c)^3 - 30*(8*a^5*b - 8*a^3*b^3 + a*b^5)*d*x - 15*(12*a^4*b^2 - 11*a^2*b^4 + b^6)*co
s(d*x + c))*sin(d*x + c))/(b^9*d*cos(d*x + c)^2 - 2*a*b^8*d*sin(d*x + c) - (a^2*b^7 + b^9)*d), 1/8*(4*a*b^5*co
s(d*x + c)^5 - 15*(8*a^4*b^2 - 8*a^2*b^4 + b^6)*d*x*cos(d*x + c)^2 - 10*(4*a^3*b^3 - 3*a*b^5)*cos(d*x + c)^3 +
15*(8*a^6 - 7*a^2*b^4 + b^6)*d*x + 60*(2*a^5 + a^3*b^2 - a*b^4 - (2*a^3*b^2 - a*b^4)*cos(d*x + c)^2 + 2*(2*a^
4*b - a^2*b^3)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) + 30
*(4*a^5*b - 2*a^3*b^3 - a*b^5)*cos(d*x + c) - (2*b^6*cos(d*x + c)^5 - 5*(2*a^2*b^4 - b^6)*cos(d*x + c)^3 - 30*
(8*a^5*b - 8*a^3*b^3 + a*b^5)*d*x - 15*(12*a^4*b^2 - 11*a^2*b^4 + b^6)*cos(d*x + c))*sin(d*x + c))/(b^9*d*cos(
d*x + c)^2 - 2*a*b^8*d*sin(d*x + c) - (a^2*b^7 + b^9)*d)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (223) = 446\).

Time = 0.35 (sec) , antiderivative size = 581, normalized size of antiderivative = 2.45 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\frac {15 \, {\left (8 \, a^{4} - 8 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{b^{7}} - \frac {120 \, {\left (2 \, a^{5} - 3 \, a^{3} b^{2} + a b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{7}} + \frac {8 \, {\left (9 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, 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} - 21 \, 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} + 31 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 35 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 10 \, a^{6} - 11 \, 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 b^{6}} + \frac {2 \, {\left (24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 80 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 72 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 240 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 168 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 152 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 80 \, a^{3} - 56 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} b^{6}}}{8 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/8*(15*(8*a^4 - 8*a^2*b^2 + b^4)*(d*x + c)/b^7 - 120*(2*a^5 - 3*a^3*b^2 + a*b^4)*(pi*floor(1/2*(d*x + c)/pi
+ 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*b^7) + 8*(9*a^5*b*tan(1
/2*d*x + 1/2*c)^3 - 9*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 10*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 - 21*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 + 31*a^5*b*tan(1/2*d*x + 1/2*c) -
35*a^3*b^3*tan(1/2*d*x + 1/2*c) + 4*a*b^5*tan(1/2*d*x + 1/2*c) + 10*a^6 - 11*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*b^6) + 2*(24*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 9*b^3*tan(1/2*d*
x + 1/2*c)^7 + 80*a^3*tan(1/2*d*x + 1/2*c)^6 - 72*a*b^2*tan(1/2*d*x + 1/2*c)^6 + 24*a^2*b*tan(1/2*d*x + 1/2*c)
^5 - b^3*tan(1/2*d*x + 1/2*c)^5 + 240*a^3*tan(1/2*d*x + 1/2*c)^4 - 168*a*b^2*tan(1/2*d*x + 1/2*c)^4 - 24*a^2*b
*tan(1/2*d*x + 1/2*c)^3 + b^3*tan(1/2*d*x + 1/2*c)^3 + 240*a^3*tan(1/2*d*x + 1/2*c)^2 - 152*a*b^2*tan(1/2*d*x
+ 1/2*c)^2 - 24*a^2*b*tan(1/2*d*x + 1/2*c) + 9*b^3*tan(1/2*d*x + 1/2*c) + 80*a^3 - 56*a*b^2)/((tan(1/2*d*x + 1
/2*c)^2 + 1)^4*b^6))/d

Mupad [B] (verification not implemented)

Time = 16.76 (sec) , antiderivative size = 2529, normalized size of antiderivative = 10.67 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

int((cos(c + d*x)^6*sin(c + d*x))/(a + b*sin(c + d*x))^3,x)

[Out]

- ((a*b^4 + 30*a^5 - 25*a^3*b^2)/b^6 + (tan(c/2 + (d*x)/2)*(420*a^4 + 16*b^4 - 355*a^2*b^2))/(4*b^5) + (15*tan
(c/2 + (d*x)/2)^11*(4*a^4 - 3*a^2*b^2))/(4*b^5) + (15*tan(c/2 + (d*x)/2)^7*(68*a^4 + 2*b^4 - 55*a^2*b^2))/(2*b
^5) - (5*tan(c/2 + (d*x)/2)^9*(4*b^4 - 132*a^4 + 99*a^2*b^2))/(4*b^5) + (5*tan(c/2 + (d*x)/2)^5*(276*a^4 + 10*
b^4 - 235*a^2*b^2))/(2*b^5) + (5*tan(c/2 + (d*x)/2)^3*(348*a^4 + 20*b^4 - 301*a^2*b^2))/(4*b^5) + (tan(c/2 + (
d*x)/2)^10*(30*a^6 + 2*b^6 - 30*a^2*b^4 + 15*a^4*b^2))/(a*b^6) + (tan(c/2 + (d*x)/2)^2*(150*a^6 + 2*b^6 - 64*a
^2*b^4 - 45*a^4*b^2))/(a*b^6) + (2*tan(c/2 + (d*x)/2)^4*(150*a^6 + 4*b^6 - 110*a^2*b^4 + 15*a^4*b^2))/(a*b^6)
+ (tan(c/2 + (d*x)/2)^8*(150*a^6 + 8*b^6 - 165*a^2*b^4 + 75*a^4*b^2))/(a*b^6) + (2*tan(c/2 + (d*x)/2)^6*(5*a^2
+ 6*b^2)*(30*a^4 + b^4 - 25*a^2*b^2))/(a*b^6))/(d*(tan(c/2 + (d*x)/2)^2*(6*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^
10*(6*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^4*(15*a^2 + 16*b^2) + tan(c/2 + (d*x)/2)^8*(15*a^2 + 16*b^2) + tan(c/2
+ (d*x)/2)^6*(20*a^2 + 24*b^2) + a^2*tan(c/2 + (d*x)/2)^12 + a^2 + 20*a*b*tan(c/2 + (d*x)/2)^3 + 40*a*b*tan(c
/2 + (d*x)/2)^5 + 40*a*b*tan(c/2 + (d*x)/2)^7 + 20*a*b*tan(c/2 + (d*x)/2)^9 + 4*a*b*tan(c/2 + (d*x)/2)^11 + 4*
a*b*tan(c/2 + (d*x)/2))) - (atanh((3375*a^3*(b^2 - a^2)^(1/2))/(2*((3375*a^3*b)/2 - (10125*a^5)/(2*b) + (3375*
a^7)/b^3 - 10125*a^4*tan(c/2 + (d*x)/2) + 3375*a^2*b^2*tan(c/2 + (d*x)/2) + (6750*a^6*tan(c/2 + (d*x)/2))/b^2)
) - (3375*a^5*(b^2 - a^2)^(1/2))/((3375*a^3*b^3)/2 - (10125*a^5*b)/2 + (3375*a^7)/b + 6750*a^6*tan(c/2 + (d*x)
/2) + 3375*a^2*b^4*tan(c/2 + (d*x)/2) - 10125*a^4*b^2*tan(c/2 + (d*x)/2)) + (3375*a^2*tan(c/2 + (d*x)/2)*(b^2
- a^2)^(1/2))/((3375*a^3)/2 - (10125*a^5)/(2*b^2) + (3375*a^7)/b^4 + 3375*a^2*b*tan(c/2 + (d*x)/2) - (10125*a^
4*tan(c/2 + (d*x)/2))/b + (6750*a^6*tan(c/2 + (d*x)/2))/b^3) - (16875*a^4*tan(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)
)/(2*((3375*a^3*b^2)/2 - (10125*a^5)/2 + (3375*a^7)/b^2 - 10125*a^4*b*tan(c/2 + (d*x)/2) + 3375*a^2*b^3*tan(c/
2 + (d*x)/2) + (6750*a^6*tan(c/2 + (d*x)/2))/b)) + (3375*a^6*tan(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2))/(3375*a^7 +
(3375*a^3*b^4)/2 - (10125*a^5*b^2)/2 + 6750*a^6*b*tan(c/2 + (d*x)/2) + 3375*a^2*b^5*tan(c/2 + (d*x)/2) - 1012
5*a^4*b^3*tan(c/2 + (d*x)/2)))*(30*a^3*(b^2 - a^2)^(1/2) - 15*a*b^2*(b^2 - a^2)^(1/2)))/(b^7*d) - (atan((((a^4
*8i + b^4*1i - a^2*b^2*8i)*(((225*a^2*b^14)/2 - 1800*a^4*b^12 + 9000*a^6*b^10 - 14400*a^8*b^8 + 7200*a^10*b^6)
/b^17 + (tan(c/2 + (d*x)/2)*(450*a*b^16 - 11025*a^3*b^14 + 61200*a^5*b^12 - 122400*a^7*b^10 + 100800*a^9*b^8 -
28800*a^11*b^6))/(2*b^18) - (15*(a^4*8i + b^4*1i - a^2*b^2*8i)*((60*a*b^18 - 300*a^3*b^16 + 240*a^5*b^14)/b^1
7 - (15*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^22 - 128*a^3*b^20))/(2*b^18))*(a^4*8i + b^4*1i - a^2*b^2*8i
))/(8*b^7) + (tan(c/2 + (d*x)/2)*(960*a^2*b^18 - 2880*a^4*b^16 + 1920*a^6*b^14))/(2*b^18)))/(8*b^7))*15i)/(8*b
^7) + ((a^4*8i + b^4*1i - a^2*b^2*8i)*(((225*a^2*b^14)/2 - 1800*a^4*b^12 + 9000*a^6*b^10 - 14400*a^8*b^8 + 720
0*a^10*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(450*a*b^16 - 11025*a^3*b^14 + 61200*a^5*b^12 - 122400*a^7*b^10 + 10080
0*a^9*b^8 - 28800*a^11*b^6))/(2*b^18) + (15*(a^4*8i + b^4*1i - a^2*b^2*8i)*((60*a*b^18 - 300*a^3*b^16 + 240*a^
5*b^14)/b^17 + (15*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^22 - 128*a^3*b^20))/(2*b^18))*(a^4*8i + b^4*1i -
a^2*b^2*8i))/(8*b^7) + (tan(c/2 + (d*x)/2)*(960*a^2*b^18 - 2880*a^4*b^16 + 1920*a^6*b^14))/(2*b^18)))/(8*b^7)
)*15i)/(8*b^7))/((108000*a^13 - (10125*a^3*b^10)/2 + (124875*a^5*b^8)/2 - 246375*a^7*b^6 + 432000*a^9*b^4 - 35
1000*a^11*b^2)/b^17 - (15*(a^4*8i + b^4*1i - a^2*b^2*8i)*(((225*a^2*b^14)/2 - 1800*a^4*b^12 + 9000*a^6*b^10 -
14400*a^8*b^8 + 7200*a^10*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(450*a*b^16 - 11025*a^3*b^14 + 61200*a^5*b^12 - 1224
00*a^7*b^10 + 100800*a^9*b^8 - 28800*a^11*b^6))/(2*b^18) - (15*(a^4*8i + b^4*1i - a^2*b^2*8i)*((60*a*b^18 - 30
0*a^3*b^16 + 240*a^5*b^14)/b^17 - (15*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^22 - 128*a^3*b^20))/(2*b^18))
*(a^4*8i + b^4*1i - a^2*b^2*8i))/(8*b^7) + (tan(c/2 + (d*x)/2)*(960*a^2*b^18 - 2880*a^4*b^16 + 1920*a^6*b^14))
/(2*b^18)))/(8*b^7)))/(8*b^7) + (15*(a^4*8i + b^4*1i - a^2*b^2*8i)*(((225*a^2*b^14)/2 - 1800*a^4*b^12 + 9000*a
^6*b^10 - 14400*a^8*b^8 + 7200*a^10*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(450*a*b^16 - 11025*a^3*b^14 + 61200*a^5*b
^12 - 122400*a^7*b^10 + 100800*a^9*b^8 - 28800*a^11*b^6))/(2*b^18) + (15*(a^4*8i + b^4*1i - a^2*b^2*8i)*((60*a
*b^18 - 300*a^3*b^16 + 240*a^5*b^14)/b^17 + (15*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^22 - 128*a^3*b^20))
/(2*b^18))*(a^4*8i + b^4*1i - a^2*b^2*8i))/(8*b^7) + (tan(c/2 + (d*x)/2)*(960*a^2*b^18 - 2880*a^4*b^16 + 1920*
a^6*b^14))/(2*b^18)))/(8*b^7)))/(8*b^7) + (tan(c/2 + (d*x)/2)*(432000*a^14 + 3375*a^2*b^12 - 64125*a^4*b^10 +
438750*a^6*b^8 - 1350000*a^8*b^6 + 2052000*a^10*b^4 - 1512000*a^12*b^2))/b^18))*(a^4*8i + b^4*1i - a^2*b^2*8i)
*15i)/(4*b^7*d)