3.1256 \(\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=525 \[ \frac{\left (-1435 a^4 b^2+588 a^2 b^4+840 a^6-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac{2 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^9 d}-\frac{\left (-30 a^2 b^2+20 a^4+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}+\frac{\left (-340 a^2 b^2+224 a^4+105 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{140 a^2 b^4 d}-\frac{\left (-37 a^2 b^2+24 a^4+12 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{12 a b^5 d}+\frac{\left (-441 a^2 b^2+280 a^4+150 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{105 b^6 d}-\frac{a \left (-52 a^2 b^2+32 a^4+19 b^4\right ) \sin (c+d x) \cos (c+d x)}{8 b^7 d}+\frac{a x \left (-120 a^4 b^2+60 a^2 b^4+64 a^6-5 b^6\right )}{8 b^9}-\frac{3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))} \]

[Out]

(a*(64*a^6 - 120*a^4*b^2 + 60*a^2*b^4 - 5*b^6)*x)/(8*b^9) - (2*a^2*(8*a^2 - 3*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b
+ a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^9*d) + ((840*a^6 - 1435*a^4*b^2 + 588*a^2*b^4 - 15*b^6)*Cos[c + d*
x])/(105*b^8*d) - (a*(32*a^4 - 52*a^2*b^2 + 19*b^4)*Cos[c + d*x]*Sin[c + d*x])/(8*b^7*d) + ((280*a^4 - 441*a^2
*b^2 + 150*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(105*b^6*d) - ((24*a^4 - 37*a^2*b^2 + 12*b^4)*Cos[c + d*x]*Sin[c
+ d*x]^3)/(12*a*b^5*d) + ((224*a^4 - 340*a^2*b^2 + 105*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(140*a^2*b^4*d) + (Co
s[c + d*x]*Sin[c + d*x]^4)/(4*a*d*(a + b*Sin[c + d*x])) - (3*b*Cos[c + d*x]*Sin[c + d*x]^5)/(20*a^2*d*(a + b*S
in[c + d*x])) - ((20*a^4 - 30*a^2*b^2 + 9*b^4)*Cos[c + d*x]*Sin[c + d*x]^5)/(15*a^2*b^3*d*(a + b*Sin[c + d*x])
) - (4*a*Cos[c + d*x]*Sin[c + d*x]^6)/(21*b^2*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*Sin[c + d*x]^7)/(7*b*d*(
a + b*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.9091, antiderivative size = 525, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2896, 3047, 3049, 3023, 2735, 2660, 618, 204} \[ \frac{\left (-1435 a^4 b^2+588 a^2 b^4+840 a^6-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac{2 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^9 d}-\frac{\left (-30 a^2 b^2+20 a^4+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}+\frac{\left (-340 a^2 b^2+224 a^4+105 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{140 a^2 b^4 d}-\frac{\left (-37 a^2 b^2+24 a^4+12 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{12 a b^5 d}+\frac{\left (-441 a^2 b^2+280 a^4+150 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{105 b^6 d}-\frac{a \left (-52 a^2 b^2+32 a^4+19 b^4\right ) \sin (c+d x) \cos (c+d x)}{8 b^7 d}+\frac{a x \left (-120 a^4 b^2+60 a^2 b^4+64 a^6-5 b^6\right )}{8 b^9}-\frac{3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]

[Out]

(a*(64*a^6 - 120*a^4*b^2 + 60*a^2*b^4 - 5*b^6)*x)/(8*b^9) - (2*a^2*(8*a^2 - 3*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b
+ a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^9*d) + ((840*a^6 - 1435*a^4*b^2 + 588*a^2*b^4 - 15*b^6)*Cos[c + d*
x])/(105*b^8*d) - (a*(32*a^4 - 52*a^2*b^2 + 19*b^4)*Cos[c + d*x]*Sin[c + d*x])/(8*b^7*d) + ((280*a^4 - 441*a^2
*b^2 + 150*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(105*b^6*d) - ((24*a^4 - 37*a^2*b^2 + 12*b^4)*Cos[c + d*x]*Sin[c
+ d*x]^3)/(12*a*b^5*d) + ((224*a^4 - 340*a^2*b^2 + 105*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(140*a^2*b^4*d) + (Co
s[c + d*x]*Sin[c + d*x]^4)/(4*a*d*(a + b*Sin[c + d*x])) - (3*b*Cos[c + d*x]*Sin[c + d*x]^5)/(20*a^2*d*(a + b*S
in[c + d*x])) - ((20*a^4 - 30*a^2*b^2 + 9*b^4)*Cos[c + d*x]*Sin[c + d*x]^5)/(15*a^2*b^3*d*(a + b*Sin[c + d*x])
) - (4*a*Cos[c + d*x]*Sin[c + d*x]^6)/(21*b^2*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*Sin[c + d*x]^7)/(7*b*d*(
a + b*Sin[c + d*x]))

Rule 2896

Int[cos[(e_.) + (f_.)*(x_)]^6*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1))/(a*d*f*(n + 1)), x] +
(Dist[1/(a^2*b^2*d^2*(n + 1)*(n + 2)*(m + n + 5)*(m + n + 6)), Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*
x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2*n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m +
n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e +
f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4)*(m + n + 5)*(m + n + 6) - a^2*b^2*(
n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))*Sin[e + f*x]^2, x], x], x] - Simp[(b*(m + n + 2)*Cos[e + f*x]*(d*
Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1))/(a^2*d^2*f*(n + 1)*(n + 2)), x] - Simp[(a*(n + 5)*Cos[e +
f*x]*(d*Sin[e + f*x])^(n + 3)*(a + b*Sin[e + f*x])^(m + 1))/(b^2*d^3*f*(m + n + 5)*(m + n + 6)), x] + Simp[(Co
s[e + f*x]*(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^(m + 1))/(b*d^4*f*(m + n + 6)), x]) /; FreeQ[{a, b, d
, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0]
&& NeQ[m + n + 6, 0] &&  !IGtQ[m, 0]

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(
c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e +
f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2735

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 2660

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 618

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac{3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}+\frac{\int \frac{\sin ^5(c+d x) \left (6 \left (160 a^4-245 a^2 b^2+84 b^4\right )-4 a b \left (10 a^2-21 b^2\right ) \sin (c+d x)-10 \left (112 a^4-180 a^2 b^2+63 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{840 a^2 b^2}\\ &=\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac{3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac{\int \frac{\sin ^4(c+d x) \left (-280 \left (20 a^6-50 a^4 b^2+39 a^2 b^4-9 b^6\right )+10 a b \left (16 a^4-37 a^2 b^2+21 b^4\right ) \sin (c+d x)+30 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{840 a^2 b^3 \left (a^2-b^2\right )}\\ &=\frac{\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac{3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac{\int \frac{\sin ^3(c+d x) \left (120 a \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right )-80 a^2 b \left (14 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x)-1400 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4200 a^2 b^4 \left (a^2-b^2\right )}\\ &=-\frac{\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac{\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac{3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac{\int \frac{\sin ^2(c+d x) \left (-4200 a^2 \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right )+120 a^3 b \left (56 a^4-121 a^2 b^2+65 b^4\right ) \sin (c+d x)+480 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{16800 a^2 b^5 \left (a^2-b^2\right )}\\ &=\frac{\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac{\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac{\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac{3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac{\int \frac{\sin (c+d x) \left (960 a^3 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right )-120 a^2 b \left (280 a^6-637 a^4 b^2+417 a^2 b^4-60 b^6\right ) \sin (c+d x)-12600 a^3 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{50400 a^2 b^6 \left (a^2-b^2\right )}\\ &=-\frac{a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac{\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac{\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac{\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac{3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac{\int \frac{-12600 a^4 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right )+120 a^3 b \left (1120 a^6-2716 a^4 b^2+2001 a^2 b^4-405 b^6\right ) \sin (c+d x)+960 a^2 \left (840 a^8-2275 a^6 b^2+2023 a^4 b^4-603 a^2 b^6+15 b^8\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{100800 a^2 b^7 \left (a^2-b^2\right )}\\ &=\frac{\left (840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac{a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac{\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac{\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac{\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac{3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac{\int \frac{-12600 a^4 b \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right )-12600 a^3 \left (64 a^8-184 a^6 b^2+180 a^4 b^4-65 a^2 b^6+5 b^8\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{100800 a^2 b^8 \left (a^2-b^2\right )}\\ &=\frac{a \left (64 a^6-120 a^4 b^2+60 a^2 b^4-5 b^6\right ) x}{8 b^9}+\frac{\left (840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac{a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac{\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac{\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac{\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac{3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac{\left (a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^2\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{b^9}\\ &=\frac{a \left (64 a^6-120 a^4 b^2+60 a^2 b^4-5 b^6\right ) x}{8 b^9}+\frac{\left (840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac{a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac{\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac{\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac{\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac{3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac{\left (2 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^2\right ) \operatorname{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^9 d}\\ &=\frac{a \left (64 a^6-120 a^4 b^2+60 a^2 b^4-5 b^6\right ) x}{8 b^9}+\frac{\left (840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac{a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac{\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac{\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac{\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac{3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}+\frac{\left (4 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^2\right ) \operatorname{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^9 d}\\ &=\frac{a \left (64 a^6-120 a^4 b^2+60 a^2 b^4-5 b^6\right ) x}{8 b^9}-\frac{2 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b^9 d}+\frac{\left (840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac{a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac{\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac{\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac{\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac{3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 8.86982, size = 531, normalized size = 1.01 \[ \frac{\frac{26880 a^6 b^2 \sin (2 (c+d x))-201600 a^5 b^3 c \sin (c+d x)-201600 a^5 b^3 d x \sin (c+d x)-45920 a^4 b^4 \sin (2 (c+d x))-1120 a^4 b^4 \sin (4 (c+d x))+100800 a^3 b^5 c \sin (c+d x)+100800 a^3 b^5 d x \sin (c+d x)+18480 a^2 b^6 \sin (2 (c+d x))+1428 a^2 b^6 \sin (4 (c+d x))+112 a^2 b^6 \sin (6 (c+d x))-336 a^3 b^5 \cos (5 (c+d x))+840 a b \left (-224 a^4 b^2+98 a^2 b^4+128 a^6-5 b^6\right ) \cos (c+d x)+70 \left (-96 a^3 b^5+64 a^5 b^3+27 a b^7\right ) \cos (3 (c+d x))-201600 a^6 b^2 c+100800 a^4 b^4 c-8400 a^2 b^6 c-201600 a^6 b^2 d x+100800 a^4 b^4 d x-8400 a^2 b^6 d x+107520 a^7 b c \sin (c+d x)+107520 a^7 b d x \sin (c+d x)+107520 a^8 c+107520 a^8 d x-8400 a b^7 c \sin (c+d x)-8400 a b^7 d x \sin (c+d x)+350 a b^7 \cos (5 (c+d x))+40 a b^7 \cos (7 (c+d x))-210 b^8 \sin (2 (c+d x))-210 b^8 \sin (4 (c+d x))-90 b^8 \sin (6 (c+d x))-15 b^8 \sin (8 (c+d x))}{a+b \sin (c+d x)}-26880 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{13440 b^9 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]

[Out]

(-26880*a^2*(8*a^2 - 3*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + (107520*a^8*c
- 201600*a^6*b^2*c + 100800*a^4*b^4*c - 8400*a^2*b^6*c + 107520*a^8*d*x - 201600*a^6*b^2*d*x + 100800*a^4*b^4
*d*x - 8400*a^2*b^6*d*x + 840*a*b*(128*a^6 - 224*a^4*b^2 + 98*a^2*b^4 - 5*b^6)*Cos[c + d*x] + 70*(64*a^5*b^3 -
96*a^3*b^5 + 27*a*b^7)*Cos[3*(c + d*x)] - 336*a^3*b^5*Cos[5*(c + d*x)] + 350*a*b^7*Cos[5*(c + d*x)] + 40*a*b^
7*Cos[7*(c + d*x)] + 107520*a^7*b*c*Sin[c + d*x] - 201600*a^5*b^3*c*Sin[c + d*x] + 100800*a^3*b^5*c*Sin[c + d*
x] - 8400*a*b^7*c*Sin[c + d*x] + 107520*a^7*b*d*x*Sin[c + d*x] - 201600*a^5*b^3*d*x*Sin[c + d*x] + 100800*a^3*
b^5*d*x*Sin[c + d*x] - 8400*a*b^7*d*x*Sin[c + d*x] + 26880*a^6*b^2*Sin[2*(c + d*x)] - 45920*a^4*b^4*Sin[2*(c +
d*x)] + 18480*a^2*b^6*Sin[2*(c + d*x)] - 210*b^8*Sin[2*(c + d*x)] - 1120*a^4*b^4*Sin[4*(c + d*x)] + 1428*a^2*
b^6*Sin[4*(c + d*x)] - 210*b^8*Sin[4*(c + d*x)] + 112*a^2*b^6*Sin[6*(c + d*x)] - 90*b^8*Sin[6*(c + d*x)] - 15*
b^8*Sin[8*(c + d*x)])/(a + b*Sin[c + d*x]))/(13440*b^9*d)

________________________________________________________________________________________

Maple [B]  time = 0.141, size = 2076, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x)

[Out]

14/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^12*a^6+18/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+
1/2*c)^12*a^2+28/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^3*a^3-28/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^7
*tan(1/2*d*x+1/2*c)^11*a^3-30/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^12*a^4-30/d/b^7/(1+tan(1/2*d
*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^5*a^5+280/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^6*a^6+29/d/b^5
/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^5*a^3-85/12/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)
^5*a+210/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^4*a^6-330/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/
2*d*x+1/2*c)^4*a^4+176/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^6*a^2-6/d/b^2/(1+tan(1/2*d*x+1/2*c)
^2)^7*tan(1/2*d*x+1/2*c)^4-70/3/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^7*a^4+46/5/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^7*a^2
+2/d*a^7/b^8/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)-4/d*a^5/b^6/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d
*x+1/2*c)*b+a)+2/d*a^3/b^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)+16/d/b^9*arctan(tan(1/2*d*x+1/2*c
))*a^7+15/d/b^5*arctan(tan(1/2*d*x+1/2*c))*a^3-5/4/d/b^3*arctan(tan(1/2*d*x+1/2*c))*a-30/d/b^7*arctan(tan(1/2*
d*x+1/2*c))*a^5+14/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^7*a^6-2/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^
12-10/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^8+6/d*a^2/b^3/(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))+7/3/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^11*a-24/d/b^7/(1+ta
n(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^3*a^5+146/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^8*a^2+3
0/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^9*a^5+210/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1
/2*c)^8*a^6+606/5/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^4*a^2-1360/3/d/b^6/(1+tan(1/2*d*x+1/2*c)
^2)^7*tan(1/2*d*x+1/2*c)^6*a^4+84/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^2*a^6-400/3/d/b^6/(1+tan
(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^2*a^4+232/5/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^2*a^2-
6/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)*a^5+9/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c
)*a^3-11/4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)*a+84/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d
*x+1/2*c)^10*a^6-160/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^10*a^4+72/d/b^4/(1+tan(1/2*d*x+1/2*c)
^2)^7*tan(1/2*d*x+1/2*c)^10*a^2+24/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^11*a^5+85/12/d/b^3/(1+t
an(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^9*a+6/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^13*a^5-9/d
/b^5/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^13*a^3-1090/3/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+
1/2*c)^8*a^4-7/3/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^3*a-29/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^7*t
an(1/2*d*x+1/2*c)^9*a^3+2/d*a^6/b^7/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)*tan(1/2*d*x+1/2*c)-4/d*a
^4/b^5/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)*tan(1/2*d*x+1/2*c)+2/d*a^2/b^3/(tan(1/2*d*x+1/2*c)^2*
a+2*tan(1/2*d*x+1/2*c)*b+a)*tan(1/2*d*x+1/2*c)-16/d*a^8/b^9/(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))+38/d*a^6/b^7/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-2
8/d*a^4/b^5/(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))-2/7/d/b^2/(1+tan(1/2*d*x+
1/2*c)^2)^7+11/4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^13*a

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.637, size = 2032, normalized size = 3.87 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

[1/840*(160*a*b^7*cos(d*x + c)^7 - 14*(24*a^3*b^5 - 5*a*b^7)*cos(d*x + c)^5 + 35*(32*a^5*b^3 - 36*a^3*b^5 + 5*
a*b^7)*cos(d*x + c)^3 + 105*(64*a^8 - 120*a^6*b^2 + 60*a^4*b^4 - 5*a^2*b^6)*d*x + 420*(8*a^7 - 11*a^5*b^2 + 3*
a^3*b^4 + (8*a^6*b - 11*a^4*b^3 + 3*a^2*b^5)*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)) + 105*(64*a^7*b - 120*a^5*b^3 + 60*a^3*b^5 - 5*a*b^7)*cos(d*x +
c) - (120*b^8*cos(d*x + c)^7 - 224*a^2*b^6*cos(d*x + c)^5 + 70*(8*a^4*b^4 - 7*a^2*b^6)*cos(d*x + c)^3 - 105*(
64*a^7*b - 120*a^5*b^3 + 60*a^3*b^5 - 5*a*b^7)*d*x - 105*(32*a^6*b^2 - 52*a^4*b^4 + 19*a^2*b^6)*cos(d*x + c))*
sin(d*x + c))/(b^10*d*sin(d*x + c) + a*b^9*d), 1/840*(160*a*b^7*cos(d*x + c)^7 - 14*(24*a^3*b^5 - 5*a*b^7)*cos
(d*x + c)^5 + 35*(32*a^5*b^3 - 36*a^3*b^5 + 5*a*b^7)*cos(d*x + c)^3 + 105*(64*a^8 - 120*a^6*b^2 + 60*a^4*b^4 -
5*a^2*b^6)*d*x + 840*(8*a^7 - 11*a^5*b^2 + 3*a^3*b^4 + (8*a^6*b - 11*a^4*b^3 + 3*a^2*b^5)*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))) + 105*(64*a^7*b - 120*a^5*b^3 + 60*a^3
*b^5 - 5*a*b^7)*cos(d*x + c) - (120*b^8*cos(d*x + c)^7 - 224*a^2*b^6*cos(d*x + c)^5 + 70*(8*a^4*b^4 - 7*a^2*b^
6)*cos(d*x + c)^3 - 105*(64*a^7*b - 120*a^5*b^3 + 60*a^3*b^5 - 5*a*b^7)*d*x - 105*(32*a^6*b^2 - 52*a^4*b^4 + 1
9*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/(b^10*d*sin(d*x + c) + a*b^9*d)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**3/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.27625, size = 1303, normalized size = 2.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/840*(105*(64*a^7 - 120*a^5*b^2 + 60*a^3*b^4 - 5*a*b^6)*(d*x + c)/b^9 - 1680*(8*a^8 - 19*a^6*b^2 + 14*a^4*b^4
- 3*a^2*b^6)*(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^9) + 1680*(a^6*b*tan(1/2*d*x + 1/2*c) - 2*a^4*b^3*tan(1/2*d*x + 1/2*c) + a^2*b^5*tan(1/2*d
*x + 1/2*c) + a^7 - 2*a^5*b^2 + a^3*b^4)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*b^8) + 2*(
2520*a^5*b*tan(1/2*d*x + 1/2*c)^13 - 3780*a^3*b^3*tan(1/2*d*x + 1/2*c)^13 + 1155*a*b^5*tan(1/2*d*x + 1/2*c)^13
+ 5880*a^6*tan(1/2*d*x + 1/2*c)^12 - 12600*a^4*b^2*tan(1/2*d*x + 1/2*c)^12 + 7560*a^2*b^4*tan(1/2*d*x + 1/2*c
)^12 - 840*b^6*tan(1/2*d*x + 1/2*c)^12 + 10080*a^5*b*tan(1/2*d*x + 1/2*c)^11 - 11760*a^3*b^3*tan(1/2*d*x + 1/2
*c)^11 + 980*a*b^5*tan(1/2*d*x + 1/2*c)^11 + 35280*a^6*tan(1/2*d*x + 1/2*c)^10 - 67200*a^4*b^2*tan(1/2*d*x + 1
/2*c)^10 + 30240*a^2*b^4*tan(1/2*d*x + 1/2*c)^10 + 12600*a^5*b*tan(1/2*d*x + 1/2*c)^9 - 12180*a^3*b^3*tan(1/2*
d*x + 1/2*c)^9 + 2975*a*b^5*tan(1/2*d*x + 1/2*c)^9 + 88200*a^6*tan(1/2*d*x + 1/2*c)^8 - 152600*a^4*b^2*tan(1/2
*d*x + 1/2*c)^8 + 61320*a^2*b^4*tan(1/2*d*x + 1/2*c)^8 - 4200*b^6*tan(1/2*d*x + 1/2*c)^8 + 117600*a^6*tan(1/2*
d*x + 1/2*c)^6 - 190400*a^4*b^2*tan(1/2*d*x + 1/2*c)^6 + 73920*a^2*b^4*tan(1/2*d*x + 1/2*c)^6 - 12600*a^5*b*ta
n(1/2*d*x + 1/2*c)^5 + 12180*a^3*b^3*tan(1/2*d*x + 1/2*c)^5 - 2975*a*b^5*tan(1/2*d*x + 1/2*c)^5 + 88200*a^6*ta
n(1/2*d*x + 1/2*c)^4 - 138600*a^4*b^2*tan(1/2*d*x + 1/2*c)^4 + 50904*a^2*b^4*tan(1/2*d*x + 1/2*c)^4 - 2520*b^6
*tan(1/2*d*x + 1/2*c)^4 - 10080*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 11760*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 - 980*a*b^
5*tan(1/2*d*x + 1/2*c)^3 + 35280*a^6*tan(1/2*d*x + 1/2*c)^2 - 56000*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 + 19488*a^2
*b^4*tan(1/2*d*x + 1/2*c)^2 - 2520*a^5*b*tan(1/2*d*x + 1/2*c) + 3780*a^3*b^3*tan(1/2*d*x + 1/2*c) - 1155*a*b^5
*tan(1/2*d*x + 1/2*c) + 5880*a^6 - 9800*a^4*b^2 + 3864*a^2*b^4 - 120*b^6)/((tan(1/2*d*x + 1/2*c)^2 + 1)^7*b^8)
)/d