3.1266 \(\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=536 \[ -\frac{a \left (-985 a^2 b^2+840 a^4+213 b^4\right ) \cos (c+d x)}{30 b^8 d}+\frac{a \sqrt{a^2-b^2} \left (-47 a^2 b^2+56 a^4+6 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^9 d}-\frac{\left (-60 a^2 b^2+56 a^4+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{\left (-110 a^2 b^2+112 a^4+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\left (-169 a^2 b^2+168 a^4+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^5 d}-\frac{\left (-291 a^2 b^2+280 a^4+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{30 a b^6 d}+\frac{\left (-244 a^2 b^2+224 a^4+43 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^7 d}-\frac{x \left (-600 a^4 b^2+180 a^2 b^4+448 a^6-5 b^6\right )}{16 b^9}-\frac{b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2} \]

[Out]

-((448*a^6 - 600*a^4*b^2 + 180*a^2*b^4 - 5*b^6)*x)/(16*b^9) + (a*Sqrt[a^2 - b^2]*(56*a^4 - 47*a^2*b^2 + 6*b^4)
*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^9*d) - (a*(840*a^4 - 985*a^2*b^2 + 213*b^4)*Cos[c + d*x]
)/(30*b^8*d) + ((224*a^4 - 244*a^2*b^2 + 43*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^7*d) - ((280*a^4 - 291*a^2*b
^2 + 45*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(30*a*b^6*d) + ((168*a^4 - 169*a^2*b^2 + 24*b^4)*Cos[c + d*x]*Sin[c
+ d*x]^3)/(24*a^2*b^5*d) + (Cos[c + d*x]*Sin[c + d*x]^4)/(4*a*d*(a + b*Sin[c + d*x])^2) - (b*Cos[c + d*x]*Sin[
c + d*x]^5)/(10*a^2*d*(a + b*Sin[c + d*x])^2) - ((56*a^4 - 60*a^2*b^2 + 9*b^4)*Cos[c + d*x]*Sin[c + d*x]^5)/(6
0*a^2*b^3*d*(a + b*Sin[c + d*x])^2) - (4*a*Cos[c + d*x]*Sin[c + d*x]^6)/(15*b^2*d*(a + b*Sin[c + d*x])^2) + (C
os[c + d*x]*Sin[c + d*x]^7)/(6*b*d*(a + b*Sin[c + d*x])^2) - ((112*a^4 - 110*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Si
n[c + d*x]^4)/(20*a^2*b^4*d*(a + b*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 2.19938, antiderivative size = 536, 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} \[ -\frac{a \left (-985 a^2 b^2+840 a^4+213 b^4\right ) \cos (c+d x)}{30 b^8 d}+\frac{a \sqrt{a^2-b^2} \left (-47 a^2 b^2+56 a^4+6 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^9 d}-\frac{\left (-60 a^2 b^2+56 a^4+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{\left (-110 a^2 b^2+112 a^4+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\left (-169 a^2 b^2+168 a^4+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^5 d}-\frac{\left (-291 a^2 b^2+280 a^4+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{30 a b^6 d}+\frac{\left (-244 a^2 b^2+224 a^4+43 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^7 d}-\frac{x \left (-600 a^4 b^2+180 a^2 b^4+448 a^6-5 b^6\right )}{16 b^9}-\frac{b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((448*a^6 - 600*a^4*b^2 + 180*a^2*b^4 - 5*b^6)*x)/(16*b^9) + (a*Sqrt[a^2 - b^2]*(56*a^4 - 47*a^2*b^2 + 6*b^4)
*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^9*d) - (a*(840*a^4 - 985*a^2*b^2 + 213*b^4)*Cos[c + d*x]
)/(30*b^8*d) + ((224*a^4 - 244*a^2*b^2 + 43*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^7*d) - ((280*a^4 - 291*a^2*b
^2 + 45*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(30*a*b^6*d) + ((168*a^4 - 169*a^2*b^2 + 24*b^4)*Cos[c + d*x]*Sin[c
+ d*x]^3)/(24*a^2*b^5*d) + (Cos[c + d*x]*Sin[c + d*x]^4)/(4*a*d*(a + b*Sin[c + d*x])^2) - (b*Cos[c + d*x]*Sin[
c + d*x]^5)/(10*a^2*d*(a + b*Sin[c + d*x])^2) - ((56*a^4 - 60*a^2*b^2 + 9*b^4)*Cos[c + d*x]*Sin[c + d*x]^5)/(6
0*a^2*b^3*d*(a + b*Sin[c + d*x])^2) - (4*a*Cos[c + d*x]*Sin[c + d*x]^6)/(15*b^2*d*(a + b*Sin[c + d*x])^2) + (C
os[c + d*x]*Sin[c + d*x]^7)/(6*b*d*(a + b*Sin[c + d*x])^2) - ((112*a^4 - 110*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Si
n[c + d*x]^4)/(20*a^2*b^4*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))^3} \, dx &=\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^5(c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^7(c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac{\int \frac{\sin ^5(c+d x) \left (30 \left (32 a^4-35 a^2 b^2+6 b^4\right )-30 a b \left (2 a^2-3 b^2\right ) \sin (c+d x)-20 \left (56 a^4-65 a^2 b^2+12 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{600 a^2 b^2}\\ &=\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^5(c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (56 a^4-60 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^7(c+d x)}{6 b d (a+b \sin (c+d x))^2}-\frac{\int \frac{\sin ^4(c+d x) \left (-100 \left (56 a^6-116 a^4 b^2+69 a^2 b^4-9 b^6\right )+20 a b \left (16 a^4-31 a^2 b^2+15 b^4\right ) \sin (c+d x)+40 \left (168 a^6-353 a^4 b^2+215 a^2 b^4-30 b^6\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{1200 a^2 b^3 \left (a^2-b^2\right )}\\ &=\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^5(c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (56 a^4-60 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^7(c+d x)}{6 b d (a+b \sin (c+d x))^2}-\frac{\left (112 a^4-110 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\int \frac{\sin ^3(c+d x) \left (240 \left (a^2-b^2\right )^2 \left (112 a^4-110 a^2 b^2+15 b^4\right )-40 a b \left (28 a^2-15 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-200 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1200 a^2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (168 a^4-169 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^5(c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (56 a^4-60 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^7(c+d x)}{6 b d (a+b \sin (c+d x))^2}-\frac{\left (112 a^4-110 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\int \frac{\sin ^2(c+d x) \left (-600 a \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right )+840 a^2 b \left (8 a^2-5 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)+480 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4800 a^2 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (280 a^4-291 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a b^6 d}+\frac{\left (168 a^4-169 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^5(c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (56 a^4-60 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^7(c+d x)}{6 b d (a+b \sin (c+d x))^2}-\frac{\left (112 a^4-110 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\int \frac{\sin (c+d x) \left (960 a^2 \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right )-120 a^3 b \left (280 a^2-207 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-1800 a^2 \left (a^2-b^2\right )^2 \left (224 a^4-244 a^2 b^2+43 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{14400 a^2 b^6 \left (a^2-b^2\right )^2}\\ &=\frac{\left (224 a^4-244 a^2 b^2+43 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^7 d}-\frac{\left (280 a^4-291 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a b^6 d}+\frac{\left (168 a^4-169 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^5(c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (56 a^4-60 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^7(c+d x)}{6 b d (a+b \sin (c+d x))^2}-\frac{\left (112 a^4-110 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\int \frac{-1800 a^3 \left (a^2-b^2\right )^2 \left (224 a^4-244 a^2 b^2+43 b^4\right )+120 a^2 b \left (a^2-b^2\right )^2 \left (1120 a^4-996 a^2 b^2+75 b^4\right ) \sin (c+d x)+960 a^3 \left (a^2-b^2\right )^2 \left (840 a^4-985 a^2 b^2+213 b^4\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{28800 a^2 b^7 \left (a^2-b^2\right )^2}\\ &=-\frac{a \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{30 b^8 d}+\frac{\left (224 a^4-244 a^2 b^2+43 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^7 d}-\frac{\left (280 a^4-291 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a b^6 d}+\frac{\left (168 a^4-169 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^5(c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (56 a^4-60 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^7(c+d x)}{6 b d (a+b \sin (c+d x))^2}-\frac{\left (112 a^4-110 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\int \frac{-1800 a^3 b \left (a^2-b^2\right )^2 \left (224 a^4-244 a^2 b^2+43 b^4\right )-1800 a^2 \left (a^2-b^2\right )^2 \left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{28800 a^2 b^8 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right ) x}{16 b^9}-\frac{a \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{30 b^8 d}+\frac{\left (224 a^4-244 a^2 b^2+43 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^7 d}-\frac{\left (280 a^4-291 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a b^6 d}+\frac{\left (168 a^4-169 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^5(c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (56 a^4-60 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^7(c+d x)}{6 b d (a+b \sin (c+d x))^2}-\frac{\left (112 a^4-110 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\left (a \left (a^2-b^2\right ) \left (56 a^4-47 a^2 b^2+6 b^4\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{2 b^9}\\ &=-\frac{\left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right ) x}{16 b^9}-\frac{a \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{30 b^8 d}+\frac{\left (224 a^4-244 a^2 b^2+43 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^7 d}-\frac{\left (280 a^4-291 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a b^6 d}+\frac{\left (168 a^4-169 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^5(c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (56 a^4-60 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^7(c+d x)}{6 b d (a+b \sin (c+d x))^2}-\frac{\left (112 a^4-110 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\left (a \left (a^2-b^2\right ) \left (56 a^4-47 a^2 b^2+6 b^4\right )\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{\left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right ) x}{16 b^9}-\frac{a \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{30 b^8 d}+\frac{\left (224 a^4-244 a^2 b^2+43 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^7 d}-\frac{\left (280 a^4-291 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a b^6 d}+\frac{\left (168 a^4-169 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^5(c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (56 a^4-60 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^7(c+d x)}{6 b d (a+b \sin (c+d x))^2}-\frac{\left (112 a^4-110 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}-\frac{\left (2 a \left (a^2-b^2\right ) \left (56 a^4-47 a^2 b^2+6 b^4\right )\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{\left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right ) x}{16 b^9}+\frac{a \sqrt{a^2-b^2} \left (56 a^4-47 a^2 b^2+6 b^4\right ) \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{a \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{30 b^8 d}+\frac{\left (224 a^4-244 a^2 b^2+43 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^7 d}-\frac{\left (280 a^4-291 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a b^6 d}+\frac{\left (168 a^4-169 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^5(c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (56 a^4-60 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{4 a \cos (c+d x) \sin ^6(c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^7(c+d x)}{6 b d (a+b \sin (c+d x))^2}-\frac{\left (112 a^4-110 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [B]  time = 14.4196, size = 2015, normalized size = 3.76 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-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])))/(64*b^3*d) - (3*((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))/(256*(a - b)^2*(a + b)^2*d) - (3*((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) - 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)])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^2)))/(1024*b^7*d) - ((-60*a*(14336*a^10 - 49280*a^8*b^2 + 63360*a^6*
b^4 - 36960*a^4*b^6 + 9240*a^2*b^8 - 693*b^10)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(
5/2) + (430080*a^12*(c + d*x) - 1048320*a^10*b^2*(c + d*x) + 691200*a^8*b^4*(c + d*x) + 83040*a^6*b^6*(c + d*x
) - 198000*a^4*b^8*(c + d*x) + 43200*a^2*b^10*(c + d*x) - 1200*b^12*(c + d*x) + 430080*a^11*b*Cos[c + d*x] - 1
155840*a^9*b^3*Cos[c + d*x] + 1042880*a^7*b^5*Cos[c + d*x] - 332800*a^5*b^7*Cos[c + d*x] + 11060*a^3*b^9*Cos[c
+ d*x] + 4530*a*b^11*Cos[c + d*x] - 215040*a^10*b^2*(c + d*x)*Cos[2*(c + d*x)] + 631680*a^8*b^4*(c + d*x)*Cos
[2*(c + d*x)] - 661440*a^6*b^6*(c + d*x)*Cos[2*(c + d*x)] + 289200*a^4*b^8*(c + d*x)*Cos[2*(c + d*x)] - 45600*
a^2*b^10*(c + d*x)*Cos[2*(c + d*x)] + 1200*b^12*(c + d*x)*Cos[2*(c + d*x)] - 35840*a^9*b^3*Cos[3*(c + d*x)] +
100800*a^7*b^5*Cos[3*(c + d*x)] - 98424*a^5*b^7*Cos[3*(c + d*x)] + 37808*a^3*b^9*Cos[3*(c + d*x)] - 4344*a*b^1
1*Cos[3*(c + d*x)] + 896*a^7*b^5*Cos[5*(c + d*x)] - 2184*a^5*b^7*Cos[5*(c + d*x)] + 1680*a^3*b^9*Cos[5*(c + d*
x)] - 392*a*b^11*Cos[5*(c + d*x)] - 64*a^5*b^7*Cos[7*(c + d*x)] + 128*a^3*b^9*Cos[7*(c + d*x)] - 64*a*b^11*Cos
[7*(c + d*x)] + 860160*a^11*b*(c + d*x)*Sin[c + d*x] - 2526720*a^9*b^3*(c + d*x)*Sin[c + d*x] + 2645760*a^7*b^
5*(c + d*x)*Sin[c + d*x] - 1156800*a^5*b^7*(c + d*x)*Sin[c + d*x] + 182400*a^3*b^9*(c + d*x)*Sin[c + d*x] - 48
00*a*b^11*(c + d*x)*Sin[c + d*x] + 322560*a^10*b^2*Sin[2*(c + d*x)] - 911680*a^8*b^4*Sin[2*(c + d*x)] + 903680
*a^6*b^6*Sin[2*(c + d*x)] - 362830*a^4*b^8*Sin[2*(c + d*x)] + 49125*a^2*b^10*Sin[2*(c + d*x)] - 900*b^12*Sin[2
*(c + d*x)] + 4480*a^8*b^4*Sin[4*(c + d*x)] - 11816*a^6*b^6*Sin[4*(c + d*x)] + 10392*a^4*b^8*Sin[4*(c + d*x)]
- 3256*a^2*b^10*Sin[4*(c + d*x)] + 200*b^12*Sin[4*(c + d*x)] - 224*a^6*b^6*Sin[6*(c + d*x)] + 498*a^4*b^8*Sin[
6*(c + d*x)] - 324*a^2*b^10*Sin[6*(c + d*x)] + 50*b^12*Sin[6*(c + d*x)] + 20*a^4*b^8*Sin[8*(c + d*x)] - 40*a^2
*b^10*Sin[8*(c + d*x)] + 20*b^12*Sin[8*(c + d*x)])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^2))/(15360*b^9*d)

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Maple [B]  time = 0.161, size = 2174, normalized size = 4.1 \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))^3,x)

[Out]

5/8/d/b^3*arctan(tan(1/2*d*x+1/2*c))-15/4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7+15/4/d/b^3/(1+
tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5-5/24/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3+11/8/d
/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)+75/d/b^7*arctan(tan(1/2*d*x+1/2*c))*a^4-45/2/d/b^5*arctan(t
an(1/2*d*x+1/2*c))*a^2-46/5/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*a+140/3/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*a^3-42/d
/b^8/(1+tan(1/2*d*x+1/2*c)^2)^6*a^5-56/d/b^9*arctan(tan(1/2*d*x+1/2*c))*a^6-14/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)^2+19/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)^2-5/d*a^3/b^4/(t
an(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2-11/8/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11+
5/24/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9+1400/3/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x
+1/2*c)^6*a^3-10/d*a/b^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2-43/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)^2*tan(1/2*d*x+1/2*c)+59/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)^2*tan(1/2*d*x+1/2*c)-16/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)^2
*tan(1/2*d*x+1/2*c)+56/d*a^7/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))-103/
d*a^5/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))+53/d*a^3/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))-6/d*a/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))-420/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^4*a^5-45/d/b^7/(1+tan(
1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9*a^4+57/2/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9*a^2+44
0/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^4*a^3-15/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/
2*c)^11*a^4+27/2/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11*a^2-30/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^
6*tan(1/2*d*x+1/2*c)^7*a^4-420/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^6*a^5-210/d/b^8/(1+tan(1/2*
d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8*a^5+260/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8*a^3-13/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)^2*tan(1/2*d*x+1/2*c)^3+17/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)^2*tan(1/2*d*x+1/2*c)^3-4/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)^2*tan(1/2*d*x+1/2*c)^3-14/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)^2*tan(1/2*d*x+1/2
*c)^2-9/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*tan(1/2*d*x+1/2*c)^2+33/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)^2*tan(1/2*d*x+1/2*c)^2-92/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2
*d*x+1/2*c)^6*a+30/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5*a^4-15/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)
^6*tan(1/2*d*x+1/2*c)^5*a^2+45/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3*a^4-57/2/d/b^5/(1+tan(1/2
*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3*a^2-210/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^2*a^5+220/d/
b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^2*a^3-186/5/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2
*c)^2*a+15/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)*a^4-27/2/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1
/2*d*x+1/2*c)*a^2-42/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^10*a^5+60/d/b^6/(1+tan(1/2*d*x+1/2*c)
^2)^6*tan(1/2*d*x+1/2*c)^10*a^3-18/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^10*a+15/d/b^5/(1+tan(1/
2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7*a^2-84/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^4*a-54/d/b^4
/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8*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))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.9361, size = 2642, normalized size = 4.93 \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))^3,x, algorithm="fricas")

[Out]

[-1/240*(64*a*b^7*cos(d*x + c)^7 - 4*(56*a^3*b^5 - 19*a*b^7)*cos(d*x + c)^5 + 15*(448*a^6*b^2 - 600*a^4*b^4 +
180*a^2*b^6 - 5*b^8)*d*x*cos(d*x + c)^2 + 10*(224*a^5*b^3 - 244*a^3*b^5 + 43*a*b^7)*cos(d*x + c)^3 - 15*(448*a
^8 - 152*a^6*b^2 - 420*a^4*b^4 + 175*a^2*b^6 - 5*b^8)*d*x + 60*(56*a^7 + 9*a^5*b^2 - 41*a^3*b^4 + 6*a*b^6 - (5
6*a^5*b^2 - 47*a^3*b^4 + 6*a*b^6)*cos(d*x + c)^2 + 2*(56*a^6*b - 47*a^4*b^3 + 6*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)) - 30*(224*a^7*b -
188*a^5*b^3 - 32*a^3*b^5 + 19*a*b^7)*cos(d*x + c) - (40*b^8*cos(d*x + c)^7 - 2*(56*a^2*b^6 - 5*b^8)*cos(d*x +
c)^5 + 5*(112*a^4*b^4 - 94*a^2*b^6 + 5*b^8)*cos(d*x + c)^3 + 30*(448*a^7*b - 600*a^5*b^3 + 180*a^3*b^5 - 5*a*b
^7)*d*x + 15*(672*a^6*b^2 - 844*a^4*b^4 + 223*a^2*b^6 - 5*b^8)*cos(d*x + c))*sin(d*x + c))/(b^11*d*cos(d*x + c
)^2 - 2*a*b^10*d*sin(d*x + c) - (a^2*b^9 + b^11)*d), -1/240*(64*a*b^7*cos(d*x + c)^7 - 4*(56*a^3*b^5 - 19*a*b^
7)*cos(d*x + c)^5 + 15*(448*a^6*b^2 - 600*a^4*b^4 + 180*a^2*b^6 - 5*b^8)*d*x*cos(d*x + c)^2 + 10*(224*a^5*b^3
- 244*a^3*b^5 + 43*a*b^7)*cos(d*x + c)^3 - 15*(448*a^8 - 152*a^6*b^2 - 420*a^4*b^4 + 175*a^2*b^6 - 5*b^8)*d*x
- 120*(56*a^7 + 9*a^5*b^2 - 41*a^3*b^4 + 6*a*b^6 - (56*a^5*b^2 - 47*a^3*b^4 + 6*a*b^6)*cos(d*x + c)^2 + 2*(56*
a^6*b - 47*a^4*b^3 + 6*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)*co
s(d*x + c))) - 30*(224*a^7*b - 188*a^5*b^3 - 32*a^3*b^5 + 19*a*b^7)*cos(d*x + c) - (40*b^8*cos(d*x + c)^7 - 2*
(56*a^2*b^6 - 5*b^8)*cos(d*x + c)^5 + 5*(112*a^4*b^4 - 94*a^2*b^6 + 5*b^8)*cos(d*x + c)^3 + 30*(448*a^7*b - 60
0*a^5*b^3 + 180*a^3*b^5 - 5*a*b^7)*d*x + 15*(672*a^6*b^2 - 844*a^4*b^4 + 223*a^2*b^6 - 5*b^8)*cos(d*x + c))*si
n(d*x + c))/(b^11*d*cos(d*x + c)^2 - 2*a*b^10*d*sin(d*x + c) - (a^2*b^9 + b^11)*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))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.34767, size = 1307, normalized size = 2.44 \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))^3,x, algorithm="giac")

[Out]

-1/240*(15*(448*a^6 - 600*a^4*b^2 + 180*a^2*b^4 - 5*b^6)*(d*x + c)/b^9 - 240*(56*a^7 - 103*a^5*b^2 + 53*a^3*b^
4 - 6*a*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) + 240*(13*a^6*b*tan(1/2*d*x + 1/2*c)^3 - 17*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 + 4*a^2*b^5*t
an(1/2*d*x + 1/2*c)^3 + 14*a^7*tan(1/2*d*x + 1/2*c)^2 + 9*a^5*b^2*tan(1/2*d*x + 1/2*c)^2 - 33*a^3*b^4*tan(1/2*
d*x + 1/2*c)^2 + 10*a*b^6*tan(1/2*d*x + 1/2*c)^2 + 43*a^6*b*tan(1/2*d*x + 1/2*c) - 59*a^4*b^3*tan(1/2*d*x + 1/
2*c) + 16*a^2*b^5*tan(1/2*d*x + 1/2*c) + 14*a^7 - 19*a^5*b^2 + 5*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)^2*b^8) + 2*(1800*a^4*b*tan(1/2*d*x + 1/2*c)^11 - 1620*a^2*b^3*tan(1/2*d*x + 1/2*c)^11 +
165*b^5*tan(1/2*d*x + 1/2*c)^11 + 5040*a^5*tan(1/2*d*x + 1/2*c)^10 - 7200*a^3*b^2*tan(1/2*d*x + 1/2*c)^10 + 2
160*a*b^4*tan(1/2*d*x + 1/2*c)^10 + 5400*a^4*b*tan(1/2*d*x + 1/2*c)^9 - 3420*a^2*b^3*tan(1/2*d*x + 1/2*c)^9 -
25*b^5*tan(1/2*d*x + 1/2*c)^9 + 25200*a^5*tan(1/2*d*x + 1/2*c)^8 - 31200*a^3*b^2*tan(1/2*d*x + 1/2*c)^8 + 6480
*a*b^4*tan(1/2*d*x + 1/2*c)^8 + 3600*a^4*b*tan(1/2*d*x + 1/2*c)^7 - 1800*a^2*b^3*tan(1/2*d*x + 1/2*c)^7 + 450*
b^5*tan(1/2*d*x + 1/2*c)^7 + 50400*a^5*tan(1/2*d*x + 1/2*c)^6 - 56000*a^3*b^2*tan(1/2*d*x + 1/2*c)^6 + 11040*a
*b^4*tan(1/2*d*x + 1/2*c)^6 - 3600*a^4*b*tan(1/2*d*x + 1/2*c)^5 + 1800*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 - 450*b^
5*tan(1/2*d*x + 1/2*c)^5 + 50400*a^5*tan(1/2*d*x + 1/2*c)^4 - 52800*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 + 10080*a*b
^4*tan(1/2*d*x + 1/2*c)^4 - 5400*a^4*b*tan(1/2*d*x + 1/2*c)^3 + 3420*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 25*b^5*t
an(1/2*d*x + 1/2*c)^3 + 25200*a^5*tan(1/2*d*x + 1/2*c)^2 - 26400*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 + 4464*a*b^4*t
an(1/2*d*x + 1/2*c)^2 - 1800*a^4*b*tan(1/2*d*x + 1/2*c) + 1620*a^2*b^3*tan(1/2*d*x + 1/2*c) - 165*b^5*tan(1/2*
d*x + 1/2*c) + 5040*a^5 - 5600*a^3*b^2 + 1104*a*b^4)/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*b^8))/d