3.1257 \(\int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=471 \[ \frac {2 a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^8 d}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{6 a^2 d (a+b \sin (c+d x))}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^4 d}-\frac {a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{15 a b^5 d}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {x \left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right )}{16 b^8}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^6(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))} \]

[Out]

-1/16*(112*a^6-200*a^4*b^2+90*a^2*b^4-5*b^6)*x/b^8+2*a*(7*a^2-2*b^2)*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*d*x+1
/2*c))/(a^2-b^2)^(1/2))/b^8/d-1/15*a*(105*a^4-170*a^2*b^2+61*b^4)*cos(d*x+c)/b^7/d+1/16*(56*a^4-86*a^2*b^2+27*
b^4)*cos(d*x+c)*sin(d*x+c)/b^6/d-1/15*(35*a^4-52*a^2*b^2+15*b^4)*cos(d*x+c)*sin(d*x+c)^2/a/b^5/d+1/24*(42*a^4-
61*a^2*b^2+16*b^4)*cos(d*x+c)*sin(d*x+c)^3/a^2/b^4/d+1/3*cos(d*x+c)*sin(d*x+c)^3/a/d/(a+b*sin(d*x+c))-1/6*b*co
s(d*x+c)*sin(d*x+c)^4/a^2/d/(a+b*sin(d*x+c))-1/10*(14*a^4-20*a^2*b^2+5*b^4)*cos(d*x+c)*sin(d*x+c)^4/a^2/b^3/d/
(a+b*sin(d*x+c))-7/30*a*cos(d*x+c)*sin(d*x+c)^5/b^2/d/(a+b*sin(d*x+c))+1/6*cos(d*x+c)*sin(d*x+c)^6/b/d/(a+b*si
n(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 1.55, antiderivative size = 471, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ -\frac {a \left (-170 a^2 b^2+105 a^4+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {2 a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^8 d}-\frac {\left (-20 a^2 b^2+14 a^4+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}+\frac {\left (-61 a^2 b^2+42 a^4+16 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^4 d}-\frac {\left (-52 a^2 b^2+35 a^4+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{15 a b^5 d}+\frac {\left (-86 a^2 b^2+56 a^4+27 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^6 d}-\frac {x \left (-200 a^4 b^2+90 a^2 b^4+112 a^6-5 b^6\right )}{16 b^8}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^6(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((112*a^6 - 200*a^4*b^2 + 90*a^2*b^4 - 5*b^6)*x)/(16*b^8) + (2*a*(7*a^2 - 2*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b
+ a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^8*d) - (a*(105*a^4 - 170*a^2*b^2 + 61*b^4)*Cos[c + d*x])/(15*b^7*d)
+ ((56*a^4 - 86*a^2*b^2 + 27*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^6*d) - ((35*a^4 - 52*a^2*b^2 + 15*b^4)*Cos
[c + d*x]*Sin[c + d*x]^2)/(15*a*b^5*d) + ((42*a^4 - 61*a^2*b^2 + 16*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^2*
b^4*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*d*(a + b*Sin[c + d*x])) - (b*Cos[c + d*x]*Sin[c + d*x]^4)/(6*a^2*d
*(a + b*Sin[c + d*x])) - ((14*a^4 - 20*a^2*b^2 + 5*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(10*a^2*b^3*d*(a + b*Sin[
c + d*x])) - (7*a*Cos[c + d*x]*Sin[c + d*x]^5)/(30*b^2*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*Sin[c + d*x]^6)
/(6*b*d*(a + b*Sin[c + d*x]))

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])

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 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 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 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 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 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])))

Rubi steps

\begin {align*} \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}+\frac {\int \frac {\sin ^4(c+d x) \left (60 \left (7 a^4-10 a^2 b^2+3 b^4\right )-12 a b \left (2 a^2-5 b^2\right ) \sin (c+d x)-12 \left (42 a^4-65 a^2 b^2+20 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{360 a^2 b^2}\\ &=\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin ^3(c+d x) \left (-144 \left (14 a^6-34 a^4 b^2+25 a^2 b^4-5 b^6\right )+12 a b \left (7 a^4-17 a^2 b^2+10 b^4\right ) \sin (c+d x)+60 \left (42 a^6-103 a^4 b^2+77 a^2 b^4-16 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{360 a^2 b^3 \left (a^2-b^2\right )}\\ &=\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin ^2(c+d x) \left (180 a \left (42 a^6-103 a^4 b^2+77 a^2 b^4-16 b^6\right )-36 a^2 b \left (14 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x)-288 a \left (35 a^6-87 a^4 b^2+67 a^2 b^4-15 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1440 a^2 b^4 \left (a^2-b^2\right )}\\ &=-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin (c+d x) \left (-576 a^2 \left (35 a^6-87 a^4 b^2+67 a^2 b^4-15 b^6\right )+36 a^3 b \left (70 a^4-153 a^2 b^2+83 b^4\right ) \sin (c+d x)+540 a^2 \left (56 a^6-142 a^4 b^2+113 a^2 b^4-27 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4320 a^2 b^5 \left (a^2-b^2\right )}\\ &=\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac {\int \frac {540 a^3 \left (56 a^6-142 a^4 b^2+113 a^2 b^4-27 b^6\right )-36 a^2 b \left (280 a^6-654 a^4 b^2+449 a^2 b^4-75 b^6\right ) \sin (c+d x)-576 a^3 \left (105 a^6-275 a^4 b^2+231 a^2 b^4-61 b^6\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{8640 a^2 b^6 \left (a^2-b^2\right )}\\ &=-\frac {a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac {\int \frac {540 a^3 b \left (56 a^6-142 a^4 b^2+113 a^2 b^4-27 b^6\right )+540 a^2 \left (112 a^8-312 a^6 b^2+290 a^4 b^4-95 a^2 b^6+5 b^8\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{8640 a^2 b^7 \left (a^2-b^2\right )}\\ &=-\frac {\left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac {a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}+\frac {\left (a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^8}\\ &=-\frac {\left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac {a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}+\frac {\left (2 a \left (7 a^2-2 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^8 d}\\ &=-\frac {\left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac {a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac {\left (4 a \left (7 a^2-2 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^8 d}\\ &=-\frac {\left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right ) x}{16 b^8}+\frac {2 a \left (7 a^2-2 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^8 d}-\frac {a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 8.17, size = 462, normalized size = 0.98 \[ \frac {3840 a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )-\frac {13440 a^7 c+13440 a^7 d x+13440 a^6 b c \sin (c+d x)+13440 a^6 b d x \sin (c+d x)+3360 a^5 b^2 \sin (2 (c+d x))-24000 a^5 b^2 c-24000 a^5 b^2 d x-24000 a^4 b^3 c \sin (c+d x)-24000 a^4 b^3 d x \sin (c+d x)-5440 a^3 b^4 \sin (2 (c+d x))-140 a^3 b^4 \sin (4 (c+d x))+10800 a^3 b^4 c+10800 a^3 b^4 d x+10800 a^2 b^5 c \sin (c+d x)+10800 a^2 b^5 d x \sin (c+d x)-42 a^2 b^5 \cos (5 (c+d x))+10 \left (56 a^4 b^3-79 a^2 b^5+18 b^7\right ) \cos (3 (c+d x))+15 b \left (896 a^6-1488 a^4 b^2+576 a^2 b^4-15 b^6\right ) \cos (c+d x)+1910 a b^6 \sin (2 (c+d x))+166 a b^6 \sin (4 (c+d x))+14 a b^6 \sin (6 (c+d x))-600 a b^6 c-600 a b^6 d x-600 b^7 c \sin (c+d x)-600 b^7 d x \sin (c+d x)+40 b^7 \cos (5 (c+d x))+5 b^7 \cos (7 (c+d x))}{a+b \sin (c+d x)}}{1920 b^8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3840*a*(7*a^2 - 2*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - (13440*a^7*c - 24
000*a^5*b^2*c + 10800*a^3*b^4*c - 600*a*b^6*c + 13440*a^7*d*x - 24000*a^5*b^2*d*x + 10800*a^3*b^4*d*x - 600*a*
b^6*d*x + 15*b*(896*a^6 - 1488*a^4*b^2 + 576*a^2*b^4 - 15*b^6)*Cos[c + d*x] + 10*(56*a^4*b^3 - 79*a^2*b^5 + 18
*b^7)*Cos[3*(c + d*x)] - 42*a^2*b^5*Cos[5*(c + d*x)] + 40*b^7*Cos[5*(c + d*x)] + 5*b^7*Cos[7*(c + d*x)] + 1344
0*a^6*b*c*Sin[c + d*x] - 24000*a^4*b^3*c*Sin[c + d*x] + 10800*a^2*b^5*c*Sin[c + d*x] - 600*b^7*c*Sin[c + d*x]
+ 13440*a^6*b*d*x*Sin[c + d*x] - 24000*a^4*b^3*d*x*Sin[c + d*x] + 10800*a^2*b^5*d*x*Sin[c + d*x] - 600*b^7*d*x
*Sin[c + d*x] + 3360*a^5*b^2*Sin[2*(c + d*x)] - 5440*a^3*b^4*Sin[2*(c + d*x)] + 1910*a*b^6*Sin[2*(c + d*x)] -
140*a^3*b^4*Sin[4*(c + d*x)] + 166*a*b^6*Sin[4*(c + d*x)] + 14*a*b^6*Sin[6*(c + d*x)])/(a + b*Sin[c + d*x]))/(
1920*b^8*d)

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fricas [A]  time = 1.01, size = 814, normalized size = 1.73 \[ \left [-\frac {40 \, b^{7} \cos \left (d x + c\right )^{7} - 2 \, {\left (42 \, a^{2} b^{5} - 5 \, b^{7}\right )} \cos \left (d x + c\right )^{5} + 5 \, {\left (56 \, a^{4} b^{3} - 58 \, a^{2} b^{5} + 5 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (112 \, a^{7} - 200 \, a^{5} b^{2} + 90 \, a^{3} b^{4} - 5 \, a b^{6}\right )} d x - 120 \, {\left (7 \, a^{6} - 9 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + {\left (7 \, a^{5} b - 9 \, a^{3} b^{3} + 2 \, a b^{5}\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 ) + 15 \, {\left (112 \, a^{6} b - 200 \, a^{4} b^{3} + 90 \, a^{2} b^{5} - 5 \, b^{7}\right )} \cos \left (d x + c\right ) + {\left (56 \, a b^{6} \cos \left (d x + c\right )^{5} - 10 \, {\left (14 \, a^{3} b^{4} - 11 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (112 \, a^{6} b - 200 \, a^{4} b^{3} + 90 \, a^{2} b^{5} - 5 \, b^{7}\right )} d x + 15 \, {\left (56 \, a^{5} b^{2} - 86 \, a^{3} b^{4} + 27 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (b^{9} d \sin \left (d x + c\right ) + a b^{8} d\right )}}, -\frac {40 \, b^{7} \cos \left (d x + c\right )^{7} - 2 \, {\left (42 \, a^{2} b^{5} - 5 \, b^{7}\right )} \cos \left (d x + c\right )^{5} + 5 \, {\left (56 \, a^{4} b^{3} - 58 \, a^{2} b^{5} + 5 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (112 \, a^{7} - 200 \, a^{5} b^{2} + 90 \, a^{3} b^{4} - 5 \, a b^{6}\right )} d x + 240 \, {\left (7 \, a^{6} - 9 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + {\left (7 \, a^{5} b - 9 \, a^{3} b^{3} + 2 \, a b^{5}\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 ) + 15 \, {\left (112 \, a^{6} b - 200 \, a^{4} b^{3} + 90 \, a^{2} b^{5} - 5 \, b^{7}\right )} \cos \left (d x + c\right ) + {\left (56 \, a b^{6} \cos \left (d x + c\right )^{5} - 10 \, {\left (14 \, a^{3} b^{4} - 11 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (112 \, a^{6} b - 200 \, a^{4} b^{3} + 90 \, a^{2} b^{5} - 5 \, b^{7}\right )} d x + 15 \, {\left (56 \, a^{5} b^{2} - 86 \, a^{3} b^{4} + 27 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (b^{9} d \sin \left (d x + c\right ) + a b^{8} d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/240*(40*b^7*cos(d*x + c)^7 - 2*(42*a^2*b^5 - 5*b^7)*cos(d*x + c)^5 + 5*(56*a^4*b^3 - 58*a^2*b^5 + 5*b^7)*c
os(d*x + c)^3 + 15*(112*a^7 - 200*a^5*b^2 + 90*a^3*b^4 - 5*a*b^6)*d*x - 120*(7*a^6 - 9*a^4*b^2 + 2*a^2*b^4 + (
7*a^5*b - 9*a^3*b^3 + 2*a*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)) + 15*(112*a^6*b - 200*a^4*b^3 + 90*a^2*b^5 - 5*b^7)*cos(d*x + c) + (56*a*b^6
*cos(d*x + c)^5 - 10*(14*a^3*b^4 - 11*a*b^6)*cos(d*x + c)^3 + 15*(112*a^6*b - 200*a^4*b^3 + 90*a^2*b^5 - 5*b^7
)*d*x + 15*(56*a^5*b^2 - 86*a^3*b^4 + 27*a*b^6)*cos(d*x + c))*sin(d*x + c))/(b^9*d*sin(d*x + c) + a*b^8*d), -1
/240*(40*b^7*cos(d*x + c)^7 - 2*(42*a^2*b^5 - 5*b^7)*cos(d*x + c)^5 + 5*(56*a^4*b^3 - 58*a^2*b^5 + 5*b^7)*cos(
d*x + c)^3 + 15*(112*a^7 - 200*a^5*b^2 + 90*a^3*b^4 - 5*a*b^6)*d*x + 240*(7*a^6 - 9*a^4*b^2 + 2*a^2*b^4 + (7*a
^5*b - 9*a^3*b^3 + 2*a*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))) + 15*(112*a^6*b - 200*a^4*b^3 + 90*a^2*b^5 - 5*b^7)*cos(d*x + c) + (56*a*b^6*cos(d*x + c)^5 - 10*(14*
a^3*b^4 - 11*a*b^6)*cos(d*x + c)^3 + 15*(112*a^6*b - 200*a^4*b^3 + 90*a^2*b^5 - 5*b^7)*d*x + 15*(56*a^5*b^2 -
86*a^3*b^4 + 27*a*b^6)*cos(d*x + c))*sin(d*x + c))/(b^9*d*sin(d*x + c) + a*b^8*d)]

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giac [A]  time = 0.24, size = 835, normalized size = 1.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/240*(15*(112*a^6 - 200*a^4*b^2 + 90*a^2*b^4 - 5*b^6)*(d*x + c)/b^8 - 480*(7*a^7 - 16*a^5*b^2 + 11*a^3*b^4 -
2*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)))/(sq
rt(a^2 - b^2)*b^8) + 480*(a^5*b*tan(1/2*d*x + 1/2*c) - 2*a^3*b^3*tan(1/2*d*x + 1/2*c) + a*b^5*tan(1/2*d*x + 1/
2*c) + a^6 - 2*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)*b^7) + 2*(600*a^4
*b*tan(1/2*d*x + 1/2*c)^11 - 810*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 + 1440*a^5*
tan(1/2*d*x + 1/2*c)^10 - 2880*a^3*b^2*tan(1/2*d*x + 1/2*c)^10 + 1440*a*b^4*tan(1/2*d*x + 1/2*c)^10 + 1800*a^4
*b*tan(1/2*d*x + 1/2*c)^9 - 1710*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 + 7200*a^5*tan
(1/2*d*x + 1/2*c)^8 - 12480*a^3*b^2*tan(1/2*d*x + 1/2*c)^8 + 4320*a*b^4*tan(1/2*d*x + 1/2*c)^8 + 1200*a^4*b*ta
n(1/2*d*x + 1/2*c)^7 - 900*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 + 14400*a^5*tan(1/2
*d*x + 1/2*c)^6 - 22400*a^3*b^2*tan(1/2*d*x + 1/2*c)^6 + 7360*a*b^4*tan(1/2*d*x + 1/2*c)^6 - 1200*a^4*b*tan(1/
2*d*x + 1/2*c)^5 + 900*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 + 14400*a^5*tan(1/2*d*x
+ 1/2*c)^4 - 21120*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 + 6720*a*b^4*tan(1/2*d*x + 1/2*c)^4 - 1800*a^4*b*tan(1/2*d*
x + 1/2*c)^3 + 1710*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 25*b^5*tan(1/2*d*x + 1/2*c)^3 + 7200*a^5*tan(1/2*d*x + 1/
2*c)^2 - 10560*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 + 2976*a*b^4*tan(1/2*d*x + 1/2*c)^2 - 600*a^4*b*tan(1/2*d*x + 1/
2*c) + 810*a^2*b^3*tan(1/2*d*x + 1/2*c) - 165*b^5*tan(1/2*d*x + 1/2*c) + 1440*a^5 - 2240*a^3*b^2 + 736*a*b^4)/
((tan(1/2*d*x + 1/2*c)^2 + 1)^6*b^7))/d

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maple [B]  time = 0.54, size = 1817, normalized size = 3.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5*a^4-12/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^6*a^5-11/8/d/b^
2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11+5/24/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9-
15/4/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7+15/4/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1
/2*c)^5-5/24/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3+11/8/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1
/2*d*x+1/2*c)+56/3/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*a^3-92/15/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*a-14/d/b^8*arct
an(tan(1/2*d*x+1/2*c))*a^6+25/d/b^6*arctan(tan(1/2*d*x+1/2*c))*a^4-45/4/d/b^4*arctan(tan(1/2*d*x+1/2*c))*a^2+4
/d/b^5/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)*a^4-2/d/b^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2
*c)*b+a)*a^2-2/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)+5/8/d/b^2*arctan(tan(1/2*d*x+1/2*c)
)-60/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8*a^5+104/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*
x+1/2*c)^8*a^3-36/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8*a-5/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*t
an(1/2*d*x+1/2*c)^11*a^4-120/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^4*a^5+176/d/b^5/(1+tan(1/2*d*
x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^4*a^3-56/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^4*a-32/d/b^6*a^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))+22/d/b^4*a^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))-4/d/b^2*a/(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))+4/d/b^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)*a^3*tan(1/2*d*x+1/2*c)+14/d*a
^7/b^8/(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))+27/4/d/b^4/(1+tan(1/2*d*x+1/2*
c)^2)^6*tan(1/2*d*x+1/2*c)^11*a^2-15/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9*a^4+57/4/d/b^4/(1+t
an(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9*a^2-10/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7*a^4+1
5/2/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7*a^2-120/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x
+1/2*c)^6*a^5+560/3/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^6*a^3-184/3/d/b^3/(1+tan(1/2*d*x+1/2*c
)^2)^6*tan(1/2*d*x+1/2*c)^6*a-15/2/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5*a^2+15/d/b^6/(1+tan(1
/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3*a^4-57/4/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3*a^2-60/
d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^2*a^5+88/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*
c)^2*a^3-124/5/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^2*a+5/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(
1/2*d*x+1/2*c)*a^4-27/4/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)*a^2-12/d/b^7/(1+tan(1/2*d*x+1/2*c)
^2)^6*tan(1/2*d*x+1/2*c)^10*a^5+24/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^10*a^3-12/d/b^3/(1+tan(
1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^10*a-2/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)*tan(
1/2*d*x+1/2*c)-2/d/b^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)*a*tan(1/2*d*x+1/2*c)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+b*sin(d*x+c))^2,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 details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 16.38, size = 4067, normalized size = 8.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*(105*a^6 + 61*a^2*b^4 - 170*a^4*b^2))/(15*b^7) + (tan(c/2 + (d*x)/2)^13*(27*a*b^4 + 56*a^5 - 86*a^3*b^2)
)/(8*b^6) + (8*tan(c/2 + (d*x)/2)^7*(61*a*b^4 + 105*a^5 - 170*a^3*b^2))/(3*b^6) + (tan(c/2 + (d*x)/2)^11*(223*
a*b^4 + 336*a^5 - 558*a^3*b^2))/(6*b^6) + (tan(c/2 + (d*x)/2)^3*(1813*a*b^4 + 3360*a^5 - 5370*a^3*b^2))/(30*b^
6) + (tan(c/2 + (d*x)/2)^9*(2533*a*b^4 + 4200*a^5 - 6954*a^3*b^2))/(24*b^6) + (tan(c/2 + (d*x)/2)^5*(3323*a*b^
4 + 5880*a^5 - 9366*a^3*b^2))/(24*b^6) + (tan(c/2 + (d*x)/2)^12*(56*a^6 + 11*b^6 + 2*a^2*b^4 - 72*a^4*b^2))/(4
*b^7) + (tan(c/2 + (d*x)/2)^10*(1008*a^6 - 5*b^6 + 378*a^2*b^4 - 1464*a^4*b^2))/(12*b^7) + (tan(c/2 + (d*x)/2)
^8*(1260*a^6 + 45*b^6 + 674*a^2*b^4 - 1984*a^4*b^2))/(6*b^7) + (tan(c/2 + (d*x)/2)^6*(1680*a^6 - 45*b^6 + 1034
*a^2*b^4 - 2776*a^4*b^2))/(6*b^7) + (tan(c/2 + (d*x)/2)^2*(5040*a^6 - 165*b^6 + 3386*a^2*b^4 - 8440*a^4*b^2))/
(60*b^7) + (tan(c/2 + (d*x)/2)^4*(12600*a^6 + 25*b^6 + 8358*a^2*b^4 - 21240*a^4*b^2))/(60*b^7) + (tan(c/2 + (d
*x)/2)*(1547*a*b^4 + 2520*a^5 - 4150*a^3*b^2))/(120*b^6))/(d*(a + 2*b*tan(c/2 + (d*x)/2) + 7*a*tan(c/2 + (d*x)
/2)^2 + 21*a*tan(c/2 + (d*x)/2)^4 + 35*a*tan(c/2 + (d*x)/2)^6 + 35*a*tan(c/2 + (d*x)/2)^8 + 21*a*tan(c/2 + (d*
x)/2)^10 + 7*a*tan(c/2 + (d*x)/2)^12 + a*tan(c/2 + (d*x)/2)^14 + 12*b*tan(c/2 + (d*x)/2)^3 + 30*b*tan(c/2 + (d
*x)/2)^5 + 40*b*tan(c/2 + (d*x)/2)^7 + 30*b*tan(c/2 + (d*x)/2)^9 + 12*b*tan(c/2 + (d*x)/2)^11 + 2*b*tan(c/2 +
(d*x)/2)^13)) - (atan((((((25*a^2*b^19)/8 - (225*a^4*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b
^11 - 5600*a^12*b^9 + 1568*a^14*b^7)/b^20 - (((10*a*b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 -
((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a
^4*b^2*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(1024*a^2*b^22 - 5632*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/
(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a
^3*b^19 + 32364*a^5*b^17 - 131700*a^7*b^15 + 254720*a^9*b^13 - 254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15
*b^7))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i)*1i)/(16*b^8) + ((((25*a^2*b^19)/8 - (225*a^4
*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^11 - 5600*a^12*b^9 + 1568*a^14*b^7)/b^20 + (((10*a*
b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 + ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512
*a^3*b^23))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(1024*a
^2*b^22 - 5632*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2
*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3*b^19 + 32364*a^5*b^17 - 131700*a^7*b^15 + 254720*
a^9*b^13 - 254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15*b^7))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i -
a^4*b^2*200i)*1i)/(16*b^8))/((10976*a^19 + (135*a^3*b^16)/2 - (7205*a^5*b^14)/4 + 15115*a^7*b^12 - (244853*a^9
*b^10)/4 + 138577*a^11*b^8 - 184965*a^13*b^6 + 144788*a^15*b^4 - 61544*a^17*b^2)/b^20 + (tan(c/2 + (d*x)/2)*(1
75616*a^20 - 100*a^2*b^18 + 4150*a^4*b^16 - 61000*a^6*b^14 + 399830*a^8*b^12 - 1393080*a^10*b^10 + 2831960*a^1
2*b^8 - 3480576*a^14*b^6 + 2551808*a^16*b^4 - 1028608*a^18*b^2))/(4*b^21) + ((((25*a^2*b^19)/8 - (225*a^4*b^17
)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^11 - 5600*a^12*b^9 + 1568*a^14*b^7)/b^20 - (((10*a*b^22
- 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 - ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*
b^23))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(1024*a^2*b^
22 - 5632*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i
))/(16*b^8) + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3*b^19 + 32364*a^5*b^17 - 131700*a^7*b^15 + 254720*a^9*b
^13 - 254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15*b^7))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b
^2*200i))/(16*b^8) - ((((25*a^2*b^19)/8 - (225*a^4*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^1
1 - 5600*a^12*b^9 + 1568*a^14*b^7)/b^20 + (((10*a*b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 + ((
32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4
*b^2*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(1024*a^2*b^22 - 5632*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/(8
*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3
*b^19 + 32364*a^5*b^17 - 131700*a^7*b^15 + 254720*a^9*b^13 - 254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15*b
^7))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i))/(16*b^8)))*(a^6*112i - b^6*5i + a^2*b^4*90i -
a^4*b^2*200i)*1i)/(8*b^8*d) - (a*atan(((a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*(((25*a^2*b^19)/8 - (2
25*a^4*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^11 - 5600*a^12*b^9 + 1568*a^14*b^7)/b^20 + (t
an(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3*b^19 + 32364*a^5*b^17 - 131700*a^7*b^15 + 254720*a^9*b^13 - 254720*a^1
1*b^11 + 127232*a^13*b^9 - 25088*a^15*b^7))/(8*b^21) + (a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((10*a*
b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 + (tan(c/2 + (d*x)/2)*(1024*a^2*b^22 - 5632*a^4*b^20 +
8192*a^6*b^18 - 3584*a^8*b^16))/(8*b^21) + (a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*(32*a^2*b^3 + (tan
(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21)))/b^8))/b^8)*1i)/b^8 + (a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a
- b)^3)^(1/2)*(((25*a^2*b^19)/8 - (225*a^4*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^11 - 560
0*a^12*b^9 + 1568*a^14*b^7)/b^20 + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3*b^19 + 32364*a^5*b^17 - 131700*a^
7*b^15 + 254720*a^9*b^13 - 254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15*b^7))/(8*b^21) - (a*(7*a^2 - 2*b^2)
*(-(a + b)^3*(a - b)^3)^(1/2)*((10*a*b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 + (tan(c/2 + (d*x
)/2)*(1024*a^2*b^22 - 5632*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/(8*b^21) - (a*(7*a^2 - 2*b^2)*(-(a + b)^
3*(a - b)^3)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21)))/b^8))/b^8)*1i)/b^
8)/((10976*a^19 + (135*a^3*b^16)/2 - (7205*a^5*b^14)/4 + 15115*a^7*b^12 - (244853*a^9*b^10)/4 + 138577*a^11*b^
8 - 184965*a^13*b^6 + 144788*a^15*b^4 - 61544*a^17*b^2)/b^20 + (tan(c/2 + (d*x)/2)*(175616*a^20 - 100*a^2*b^18
+ 4150*a^4*b^16 - 61000*a^6*b^14 + 399830*a^8*b^12 - 1393080*a^10*b^10 + 2831960*a^12*b^8 - 3480576*a^14*b^6
+ 2551808*a^16*b^4 - 1028608*a^18*b^2))/(4*b^21) - (a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*(((25*a^2*b
^19)/8 - (225*a^4*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^11 - 5600*a^12*b^9 + 1568*a^14*b^7
)/b^20 + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3*b^19 + 32364*a^5*b^17 - 131700*a^7*b^15 + 254720*a^9*b^13 -
254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15*b^7))/(8*b^21) + (a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1
/2)*((10*a*b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 + (tan(c/2 + (d*x)/2)*(1024*a^2*b^22 - 5632
*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/(8*b^21) + (a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*(32*a^2
*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21)))/b^8))/b^8))/b^8 + (a*(7*a^2 - 2*b^2)*(-(a +
b)^3*(a - b)^3)^(1/2)*(((25*a^2*b^19)/8 - (225*a^4*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^
11 - 5600*a^12*b^9 + 1568*a^14*b^7)/b^20 + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3*b^19 + 32364*a^5*b^17 - 1
31700*a^7*b^15 + 254720*a^9*b^13 - 254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15*b^7))/(8*b^21) - (a*(7*a^2
- 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((10*a*b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 + (tan(c/
2 + (d*x)/2)*(1024*a^2*b^22 - 5632*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/(8*b^21) - (a*(7*a^2 - 2*b^2)*(-
(a + b)^3*(a - b)^3)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21)))/b^8))/b^8
))/b^8))*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*2i)/(b^8*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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