\(\int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1266]

Optimal result

Integrand size = 29, antiderivative size = 536 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right ) x}{16 b^9}+\frac {a \sqrt {a^2-b^2} \left (56 a^4-47 a^2 b^2+6 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^9 d}-\frac {a \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{30 b^8 d}+\frac {\left (224 a^4-244 a^2 b^2+43 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^7 d}-\frac {\left (280 a^4-291 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a b^6 d}+\frac {\left (168 a^4-169 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^7(c+d x)}{6 b d (a+b \sin (c+d x))^2}-\frac {\left (112 a^4-110 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))} \]

[Out]

-1/16*(448*a^6-600*a^4*b^2+180*a^2*b^4-5*b^6)*x/b^9-1/30*a*(840*a^4-985*a^2*b^2+213*b^4)*cos(d*x+c)/b^8/d+1/16
*(224*a^4-244*a^2*b^2+43*b^4)*cos(d*x+c)*sin(d*x+c)/b^7/d-1/30*(280*a^4-291*a^2*b^2+45*b^4)*cos(d*x+c)*sin(d*x
+c)^2/a/b^6/d+1/24*(168*a^4-169*a^2*b^2+24*b^4)*cos(d*x+c)*sin(d*x+c)^3/a^2/b^5/d+1/4*cos(d*x+c)*sin(d*x+c)^4/
a/d/(a+b*sin(d*x+c))^2-1/10*b*cos(d*x+c)*sin(d*x+c)^5/a^2/d/(a+b*sin(d*x+c))^2-1/60*(56*a^4-60*a^2*b^2+9*b^4)*
cos(d*x+c)*sin(d*x+c)^5/a^2/b^3/d/(a+b*sin(d*x+c))^2-4/15*a*cos(d*x+c)*sin(d*x+c)^6/b^2/d/(a+b*sin(d*x+c))^2+1
/6*cos(d*x+c)*sin(d*x+c)^7/b/d/(a+b*sin(d*x+c))^2-1/20*(112*a^4-110*a^2*b^2+15*b^4)*cos(d*x+c)*sin(d*x+c)^4/a^
2/b^4/d/(a+b*sin(d*x+c))+a*(56*a^4-47*a^2*b^2+6*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))*(a^2-b^2
)^(1/2)/b^9/d

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2975, 3126, 3128, 3102, 2814, 2739, 632, 210} \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}+\frac {a \sqrt {a^2-b^2} \left (56 a^4-47 a^2 b^2+6 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^9 d}-\frac {\left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}-\frac {a \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{30 b^8 d}+\frac {\left (224 a^4-244 a^2 b^2+43 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^7 d}-\frac {\left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{30 a b^6 d}+\frac {\left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^5 d}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {x \left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right )}{16 b^9}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2} \]

[In]

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

[Out]

-1/16*((448*a^6 - 600*a^4*b^2 + 180*a^2*b^4 - 5*b^6)*x)/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 210

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

Rule 632

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

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2975

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*S
in[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[Cos
[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 3102

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 3126

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 3128

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*} \text {integral}& = \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 ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^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 ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^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 ) \arctan \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 [A] (verified)

Time = 11.02 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.18 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {3840 a \left (a^2-b^2\right )^{5/2} \left (56 a^4-47 a^2 b^2+6 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+\frac {\left (a^2-b^2\right )^2 \left (-107520 a^8 c+90240 a^6 b^2 c+28800 a^4 b^4 c-20400 a^2 b^6 c+600 b^8 c-107520 a^8 d x+90240 a^6 b^2 d x+28800 a^4 b^4 d x-20400 a^2 b^6 d x+600 b^8 d x-80 a b \left (1344 a^6-1464 a^4 b^2+202 a^2 b^4+33 b^6\right ) \cos (c+d x)+120 b^2 \left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right ) (c+d x) \cos (2 (c+d x))+8960 a^5 b^3 \cos (3 (c+d x))-10880 a^3 b^5 \cos (3 (c+d x))+2436 a b^7 \cos (3 (c+d x))-224 a^3 b^5 \cos (5 (c+d x))+188 a b^7 \cos (5 (c+d x))+16 a b^7 \cos (7 (c+d x))-215040 a^7 b c \sin (c+d x)+288000 a^5 b^3 c \sin (c+d x)-86400 a^3 b^5 c \sin (c+d x)+2400 a b^7 c \sin (c+d x)-215040 a^7 b d x \sin (c+d x)+288000 a^5 b^3 d x \sin (c+d x)-86400 a^3 b^5 d x \sin (c+d x)+2400 a b^7 d x \sin (c+d x)-80640 a^6 b^2 \sin (2 (c+d x))+99040 a^4 b^4 \sin (2 (c+d x))-24600 a^2 b^6 \sin (2 (c+d x))+405 b^8 \sin (2 (c+d x))-1120 a^4 b^4 \sin (4 (c+d x))+1164 a^2 b^6 \sin (4 (c+d x))-140 b^8 \sin (4 (c+d x))+56 a^2 b^6 \sin (6 (c+d x))-35 b^8 \sin (6 (c+d x))-5 b^8 \sin (8 (c+d x))\right )}{(a+b \sin (c+d x))^2}}{3840 (a-b)^2 b^9 (a+b)^2 d} \]

[In]

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

[Out]

(3840*a*(a^2 - b^2)^(5/2)*(56*a^4 - 47*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + ((a
^2 - b^2)^2*(-107520*a^8*c + 90240*a^6*b^2*c + 28800*a^4*b^4*c - 20400*a^2*b^6*c + 600*b^8*c - 107520*a^8*d*x
+ 90240*a^6*b^2*d*x + 28800*a^4*b^4*d*x - 20400*a^2*b^6*d*x + 600*b^8*d*x - 80*a*b*(1344*a^6 - 1464*a^4*b^2 +
202*a^2*b^4 + 33*b^6)*Cos[c + d*x] + 120*b^2*(448*a^6 - 600*a^4*b^2 + 180*a^2*b^4 - 5*b^6)*(c + d*x)*Cos[2*(c
+ d*x)] + 8960*a^5*b^3*Cos[3*(c + d*x)] - 10880*a^3*b^5*Cos[3*(c + d*x)] + 2436*a*b^7*Cos[3*(c + d*x)] - 224*a
^3*b^5*Cos[5*(c + d*x)] + 188*a*b^7*Cos[5*(c + d*x)] + 16*a*b^7*Cos[7*(c + d*x)] - 215040*a^7*b*c*Sin[c + d*x]
+ 288000*a^5*b^3*c*Sin[c + d*x] - 86400*a^3*b^5*c*Sin[c + d*x] + 2400*a*b^7*c*Sin[c + d*x] - 215040*a^7*b*d*x
*Sin[c + d*x] + 288000*a^5*b^3*d*x*Sin[c + d*x] - 86400*a^3*b^5*d*x*Sin[c + d*x] + 2400*a*b^7*d*x*Sin[c + d*x]
- 80640*a^6*b^2*Sin[2*(c + d*x)] + 99040*a^4*b^4*Sin[2*(c + d*x)] - 24600*a^2*b^6*Sin[2*(c + d*x)] + 405*b^8*
Sin[2*(c + d*x)] - 1120*a^4*b^4*Sin[4*(c + d*x)] + 1164*a^2*b^6*Sin[4*(c + d*x)] - 140*b^8*Sin[4*(c + d*x)] +
56*a^2*b^6*Sin[6*(c + d*x)] - 35*b^8*Sin[6*(c + d*x)] - 5*b^8*Sin[8*(c + d*x)]))/(a + b*Sin[c + d*x])^2)/(3840
*(a - b)^2*b^9*(a + b)^2*d)

Maple [A] (verified)

Time = 5.79 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.30

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {\left (\frac {15}{2} a^{4} b^{2}-\frac {27}{4} a^{2} b^{4}+\frac {11}{16} b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 a^{5} b -30 a^{3} b^{3}+9 a \,b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {45}{2} a^{4} b^{2}-\frac {57}{4} a^{2} b^{4}-\frac {5}{48} b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (105 a^{5} b -130 a^{3} b^{3}+27 a \,b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{4} b^{2}-\frac {15}{2} a^{2} b^{4}+\frac {15}{8} b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (210 a^{5} b -\frac {700}{3} a^{3} b^{3}+46 a \,b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-15 a^{4} b^{2}+\frac {15}{2} a^{2} b^{4}-\frac {15}{8} b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (210 a^{5} b -220 a^{3} b^{3}+42 a \,b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {45}{2} a^{4} b^{2}+\frac {57}{4} a^{2} b^{4}+\frac {5}{48} b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (105 a^{5} b -110 a^{3} b^{3}+\frac {93}{5} a \,b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {15}{2} a^{4} b^{2}+\frac {27}{4} a^{2} b^{4}-\frac {11}{16} b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+21 a^{5} b -\frac {70 a^{3} b^{3}}{3}+\frac {23 a \,b^{5}}{5}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {\left (448 a^{6}-600 a^{4} b^{2}+180 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}\right )}{b^{9}}+\frac {2 a \left (\frac {-\frac {a \,b^{2} \left (13 a^{4}-17 a^{2} b^{2}+4 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b \left (14 a^{6}+9 a^{4} b^{2}-33 a^{2} b^{4}+10 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b^{2} a \left (43 a^{4}-59 a^{2} b^{2}+16 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-7 a^{6} b +\frac {19 a^{4} b^{3}}{2}-\frac {5 a^{2} b^{5}}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (56 a^{6}-103 a^{4} b^{2}+53 a^{2} b^{4}-6 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{9}}}{d}\)	\(696\)
default	\(\frac {-\frac {2 \left (\frac {\left (\frac {15}{2} a^{4} b^{2}-\frac {27}{4} a^{2} b^{4}+\frac {11}{16} b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 a^{5} b -30 a^{3} b^{3}+9 a \,b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {45}{2} a^{4} b^{2}-\frac {57}{4} a^{2} b^{4}-\frac {5}{48} b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (105 a^{5} b -130 a^{3} b^{3}+27 a \,b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{4} b^{2}-\frac {15}{2} a^{2} b^{4}+\frac {15}{8} b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (210 a^{5} b -\frac {700}{3} a^{3} b^{3}+46 a \,b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-15 a^{4} b^{2}+\frac {15}{2} a^{2} b^{4}-\frac {15}{8} b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (210 a^{5} b -220 a^{3} b^{3}+42 a \,b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {45}{2} a^{4} b^{2}+\frac {57}{4} a^{2} b^{4}+\frac {5}{48} b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (105 a^{5} b -110 a^{3} b^{3}+\frac {93}{5} a \,b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {15}{2} a^{4} b^{2}+\frac {27}{4} a^{2} b^{4}-\frac {11}{16} b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+21 a^{5} b -\frac {70 a^{3} b^{3}}{3}+\frac {23 a \,b^{5}}{5}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {\left (448 a^{6}-600 a^{4} b^{2}+180 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}\right )}{b^{9}}+\frac {2 a \left (\frac {-\frac {a \,b^{2} \left (13 a^{4}-17 a^{2} b^{2}+4 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b \left (14 a^{6}+9 a^{4} b^{2}-33 a^{2} b^{4}+10 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b^{2} a \left (43 a^{4}-59 a^{2} b^{2}+16 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-7 a^{6} b +\frac {19 a^{4} b^{3}}{2}-\frac {5 a^{2} b^{5}}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (56 a^{6}-103 a^{4} b^{2}+53 a^{2} b^{4}-6 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{9}}}{d}\)	\(696\)
risch	\(\frac {\sin \left (6 d x +6 c \right )}{192 b^{3} d}+\frac {5 x}{16 b^{3}}-\frac {28 x \,a^{6}}{b^{9}}+\frac {75 x \,a^{4}}{2 b^{7}}-\frac {45 x \,a^{2}}{4 b^{5}}-\frac {3 a \cos \left (5 d x +5 c \right )}{80 d \,b^{4}}-\frac {3 \sin \left (4 d x +4 c \right ) a^{2}}{16 b^{5} d}-\frac {15 i {\mathrm e}^{2 i \left (d x +c \right )}}{128 b^{3} d}+\frac {5 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{12 b^{6} d}-\frac {7 a \,{\mathrm e}^{3 i \left (d x +c \right )}}{32 b^{4} d}-\frac {21 a^{5} {\mathrm e}^{i \left (d x +c \right )}}{2 b^{8} d}+\frac {45 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{4 b^{6} d}-\frac {21 a^{5} {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{8} d}+\frac {45 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{4 b^{6} d}-\frac {33 a \,{\mathrm e}^{-i \left (d x +c \right )}}{16 b^{4} d}-\frac {33 a \,{\mathrm e}^{i \left (d x +c \right )}}{16 b^{4} d}+\frac {5 a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{12 b^{6} d}-\frac {7 a \,{\mathrm e}^{-3 i \left (d x +c \right )}}{32 b^{4} d}+\frac {15 i {\mathrm e}^{-2 i \left (d x +c \right )}}{128 b^{3} d}+\frac {3 \sin \left (4 d x +4 c \right )}{64 b^{3} d}+\frac {3 i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{5}}-\frac {28 i \sqrt {a^{2}-b^{2}}\, a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{9}}+\frac {47 i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{2 d \,b^{7}}+\frac {28 i \sqrt {a^{2}-b^{2}}\, a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{9}}-\frac {47 i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{2 d \,b^{7}}-\frac {3 i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{5}}+\frac {i a^{2} \left (-16 i a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}+23 i a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-7 i a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+44 i a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-61 i a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}+17 i a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}+30 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-27 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+6 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-15 a^{4} b^{2}+21 a^{2} b^{4}-6 b^{6}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} d \,b^{9}}-\frac {15 i {\mathrm e}^{2 i \left (d x +c \right )} a^{4}}{8 b^{7} d}+\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} a^{2}}{2 b^{5} d}+\frac {15 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4}}{8 b^{7} d}-\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{2}}{2 b^{5} d}\)	\(967\)

[In]

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

[Out]

1/d*(-2/b^9*(((15/2*a^4*b^2-27/4*a^2*b^4+11/16*b^6)*tan(1/2*d*x+1/2*c)^11+(21*a^5*b-30*a^3*b^3+9*a*b^5)*tan(1/
2*d*x+1/2*c)^10+(45/2*a^4*b^2-57/4*a^2*b^4-5/48*b^6)*tan(1/2*d*x+1/2*c)^9+(105*a^5*b-130*a^3*b^3+27*a*b^5)*tan
(1/2*d*x+1/2*c)^8+(15*a^4*b^2-15/2*a^2*b^4+15/8*b^6)*tan(1/2*d*x+1/2*c)^7+(210*a^5*b-700/3*a^3*b^3+46*a*b^5)*t
an(1/2*d*x+1/2*c)^6+(-15*a^4*b^2+15/2*a^2*b^4-15/8*b^6)*tan(1/2*d*x+1/2*c)^5+(210*a^5*b-220*a^3*b^3+42*a*b^5)*
tan(1/2*d*x+1/2*c)^4+(-45/2*a^4*b^2+57/4*a^2*b^4+5/48*b^6)*tan(1/2*d*x+1/2*c)^3+(105*a^5*b-110*a^3*b^3+93/5*a*
b^5)*tan(1/2*d*x+1/2*c)^2+(-15/2*a^4*b^2+27/4*a^2*b^4-11/16*b^6)*tan(1/2*d*x+1/2*c)+21*a^5*b-70/3*a^3*b^3+23/5
*a*b^5)/(1+tan(1/2*d*x+1/2*c)^2)^6+1/16*(448*a^6-600*a^4*b^2+180*a^2*b^4-5*b^6)*arctan(tan(1/2*d*x+1/2*c)))+2*
a/b^9*((-1/2*a*b^2*(13*a^4-17*a^2*b^2+4*b^4)*tan(1/2*d*x+1/2*c)^3-1/2*b*(14*a^6+9*a^4*b^2-33*a^2*b^4+10*b^6)*t
an(1/2*d*x+1/2*c)^2-1/2*b^2*a*(43*a^4-59*a^2*b^2+16*b^4)*tan(1/2*d*x+1/2*c)-7*a^6*b+19/2*a^4*b^3-5/2*a^2*b^5)/
(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+1/2*(56*a^6-103*a^4*b^2+53*a^2*b^4-6*b^6)/(a^2-b^2)^(1/2)*
arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))))

Fricas [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 1128, normalized size of antiderivative = 2.10 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[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)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 968, normalized size of antiderivative = 1.81 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[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

Mupad [B] (verification not implemented)

Time = 48.95 (sec) , antiderivative size = 4362, normalized size of antiderivative = 8.14 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- ((840*a^7 + 213*a^3*b^4 - 985*a^5*b^2)/(15*b^8) + (tan(c/2 + (d*x)/2)^14*(31*a*b^6 + 112*a^7 - 138*a^3*b^4 +
18*a^5*b^2))/(2*b^8) + (tan(c/2 + (d*x)/2)^12*(410*a*b^6 + 1176*a^7 - 1533*a^3*b^4 + 189*a^5*b^2))/(3*b^8) +
(tan(c/2 + (d*x)/2)^10*(2281*a*b^6 + 7056*a^7 - 8766*a^3*b^4 + 686*a^5*b^2))/(6*b^8) + (tan(c/2 + (d*x)/2)^2*(
1239*a*b^6 + 11760*a^7 - 2402*a^3*b^4 - 9310*a^5*b^2))/(30*b^8) + (tan(c/2 + (d*x)/2)^4*(3062*a*b^6 + 17640*a^
7 - 10011*a^3*b^4 - 8365*a^5*b^2))/(15*b^8) + (tan(c/2 + (d*x)/2)^6*(14155*a*b^6 + 58800*a^7 - 50514*a^3*b^4 -
12950*a^5*b^2))/(30*b^8) + (tan(c/2 + (d*x)/2)*(23520*a^6 + 6171*a^2*b^4 - 27860*a^4*b^2))/(120*b^7) + (tan(c
/2 + (d*x)/2)^15*(224*a^6 + 43*a^2*b^4 - 244*a^4*b^2))/(8*b^7) + (tan(c/2 + (d*x)/2)^13*(8736*a^6 + 132*b^6 +
1453*a^2*b^4 - 9516*a^4*b^2))/(24*b^7) + (tan(c/2 + (d*x)/2)^11*(38304*a^6 - 20*b^6 + 8033*a^2*b^4 - 43068*a^4
*b^2))/(24*b^7) + (tan(c/2 + (d*x)/2)^9*(84000*a^6 + 360*b^6 + 20341*a^2*b^4 - 97324*a^4*b^2))/(24*b^7) + (tan
(c/2 + (d*x)/2)^7*(104160*a^6 - 360*b^6 + 27371*a^2*b^4 - 123316*a^4*b^2))/(24*b^7) + (tan(c/2 + (d*x)/2)^3*(1
44480*a^6 - 660*b^6 + 40447*a^2*b^4 - 173060*a^4*b^2))/(120*b^7) + (tan(c/2 + (d*x)/2)^5*(372960*a^6 + 100*b^6
+ 102971*a^2*b^4 - 446580*a^4*b^2))/(120*b^7) + (tan(c/2 + (d*x)/2)^8*(7*a^2 + 8*b^2)*(213*a*b^4 + 840*a^5 -
985*a^3*b^2))/(3*b^8))/(d*(tan(c/2 + (d*x)/2)^2*(8*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^14*(8*a^2 + 4*b^2) + tan(
c/2 + (d*x)/2)^4*(28*a^2 + 24*b^2) + tan(c/2 + (d*x)/2)^12*(28*a^2 + 24*b^2) + tan(c/2 + (d*x)/2)^6*(56*a^2 +
60*b^2) + tan(c/2 + (d*x)/2)^10*(56*a^2 + 60*b^2) + tan(c/2 + (d*x)/2)^8*(70*a^2 + 80*b^2) + a^2*tan(c/2 + (d*
x)/2)^16 + a^2 + 28*a*b*tan(c/2 + (d*x)/2)^3 + 84*a*b*tan(c/2 + (d*x)/2)^5 + 140*a*b*tan(c/2 + (d*x)/2)^7 + 14
0*a*b*tan(c/2 + (d*x)/2)^9 + 84*a*b*tan(c/2 + (d*x)/2)^11 + 28*a*b*tan(c/2 + (d*x)/2)^13 + 4*a*b*tan(c/2 + (d*
x)/2)^15 + 4*a*b*tan(c/2 + (d*x)/2))) - (atan((((((25*a^2*b^20)/8 - 225*a^4*b^18 + 4800*a^6*b^16 - 27560*a^8*b
^14 + 65160*a^10*b^12 - 67200*a^12*b^10 + 25088*a^14*b^8)/b^23 - (((10*a*b^24 - 274*a^3*b^22 + 712*a^5*b^20 -
448*a^7*b^18)/b^23 - ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^28 - 512*a^3*b^26))/(8*b^24))*(a^6*448i - b^6
*5i + a^2*b^4*180i - a^4*b^2*600i))/(16*b^9) + (tan(c/2 + (d*x)/2)*(1536*a^2*b^24 - 13568*a^4*b^22 + 26368*a^6
*b^20 - 14336*a^8*b^18))/(8*b^24))*(a^6*448i - b^6*5i + a^2*b^4*180i - a^4*b^2*600i))/(16*b^9) + (tan(c/2 + (d
*x)/2)*(50*a*b^22 - 5929*a^3*b^20 + 119304*a^5*b^18 - 738240*a^7*b^16 + 2004800*a^9*b^14 - 2655360*a^11*b^12 +
1677312*a^13*b^10 - 401408*a^15*b^8))/(8*b^24))*(a^6*448i - b^6*5i + a^2*b^4*180i - a^4*b^2*600i)*1i)/(16*b^9
) + ((((25*a^2*b^20)/8 - 225*a^4*b^18 + 4800*a^6*b^16 - 27560*a^8*b^14 + 65160*a^10*b^12 - 67200*a^12*b^10 + 2
5088*a^14*b^8)/b^23 + (((10*a*b^24 - 274*a^3*b^22 + 712*a^5*b^20 - 448*a^7*b^18)/b^23 + ((32*a^2*b^3 + (tan(c/
2 + (d*x)/2)*(768*a*b^28 - 512*a^3*b^26))/(8*b^24))*(a^6*448i - b^6*5i + a^2*b^4*180i - a^4*b^2*600i))/(16*b^9
) + (tan(c/2 + (d*x)/2)*(1536*a^2*b^24 - 13568*a^4*b^22 + 26368*a^6*b^20 - 14336*a^8*b^18))/(8*b^24))*(a^6*448
i - b^6*5i + a^2*b^4*180i - a^4*b^2*600i))/(16*b^9) + (tan(c/2 + (d*x)/2)*(50*a*b^22 - 5929*a^3*b^20 + 119304*
a^5*b^18 - 738240*a^7*b^16 + 2004800*a^9*b^14 - 2655360*a^11*b^12 + 1677312*a^13*b^10 - 401408*a^15*b^8))/(8*b
^24))*(a^6*448i - b^6*5i + a^2*b^4*180i - a^4*b^2*600i)*1i)/(16*b^9))/((702464*a^19 + (645*a^3*b^16)/4 - (6515
5*a^5*b^14)/8 + (922065*a^7*b^12)/8 - 740668*a^9*b^10 + 2522213*a^11*b^8 - 4837780*a^13*b^6 + 5244512*a^15*b^4
- 2998016*a^17*b^2)/b^23 + ((((25*a^2*b^20)/8 - 225*a^4*b^18 + 4800*a^6*b^16 - 27560*a^8*b^14 + 65160*a^10*b^
12 - 67200*a^12*b^10 + 25088*a^14*b^8)/b^23 - (((10*a*b^24 - 274*a^3*b^22 + 712*a^5*b^20 - 448*a^7*b^18)/b^23
- ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^28 - 512*a^3*b^26))/(8*b^24))*(a^6*448i - b^6*5i + a^2*b^4*180i
- a^4*b^2*600i))/(16*b^9) + (tan(c/2 + (d*x)/2)*(1536*a^2*b^24 - 13568*a^4*b^22 + 26368*a^6*b^20 - 14336*a^8*b
^18))/(8*b^24))*(a^6*448i - b^6*5i + a^2*b^4*180i - a^4*b^2*600i))/(16*b^9) + (tan(c/2 + (d*x)/2)*(50*a*b^22 -
5929*a^3*b^20 + 119304*a^5*b^18 - 738240*a^7*b^16 + 2004800*a^9*b^14 - 2655360*a^11*b^12 + 1677312*a^13*b^10
- 401408*a^15*b^8))/(8*b^24))*(a^6*448i - b^6*5i + a^2*b^4*180i - a^4*b^2*600i))/(16*b^9) - ((((25*a^2*b^20)/8
- 225*a^4*b^18 + 4800*a^6*b^16 - 27560*a^8*b^14 + 65160*a^10*b^12 - 67200*a^12*b^10 + 25088*a^14*b^8)/b^23 +
(((10*a*b^24 - 274*a^3*b^22 + 712*a^5*b^20 - 448*a^7*b^18)/b^23 + ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^
28 - 512*a^3*b^26))/(8*b^24))*(a^6*448i - b^6*5i + a^2*b^4*180i - a^4*b^2*600i))/(16*b^9) + (tan(c/2 + (d*x)/2
)*(1536*a^2*b^24 - 13568*a^4*b^22 + 26368*a^6*b^20 - 14336*a^8*b^18))/(8*b^24))*(a^6*448i - b^6*5i + a^2*b^4*1
80i - a^4*b^2*600i))/(16*b^9) + (tan(c/2 + (d*x)/2)*(50*a*b^22 - 5929*a^3*b^20 + 119304*a^5*b^18 - 738240*a^7*
b^16 + 2004800*a^9*b^14 - 2655360*a^11*b^12 + 1677312*a^13*b^10 - 401408*a^15*b^8))/(8*b^24))*(a^6*448i - b^6*
5i + a^2*b^4*180i - a^4*b^2*600i))/(16*b^9) + (tan(c/2 + (d*x)/2)*(11239424*a^20 - 150*a^2*b^18 + 12125*a^4*b^
16 - 328375*a^6*b^14 + 3544880*a^8*b^12 - 18869120*a^10*b^10 + 55713280*a^12*b^8 - 95735744*a^14*b^6 + 9520179
2*a^16*b^4 - 50778112*a^18*b^2))/(4*b^24)))*(a^6*448i - b^6*5i + a^2*b^4*180i - a^4*b^2*600i)*1i)/(8*b^9*d) -
(a*atan(((a*(-(a + b)*(a - b))^(1/2)*(56*a^4 + 6*b^4 - 47*a^2*b^2)*(((25*a^2*b^20)/8 - 225*a^4*b^18 + 4800*a^6
*b^16 - 27560*a^8*b^14 + 65160*a^10*b^12 - 67200*a^12*b^10 + 25088*a^14*b^8)/b^23 + (tan(c/2 + (d*x)/2)*(50*a*
b^22 - 5929*a^3*b^20 + 119304*a^5*b^18 - 738240*a^7*b^16 + 2004800*a^9*b^14 - 2655360*a^11*b^12 + 1677312*a^13
*b^10 - 401408*a^15*b^8))/(8*b^24) - (a*(-(a + b)*(a - b))^(1/2)*((10*a*b^24 - 274*a^3*b^22 + 712*a^5*b^20 - 4
48*a^7*b^18)/b^23 + (tan(c/2 + (d*x)/2)*(1536*a^2*b^24 - 13568*a^4*b^22 + 26368*a^6*b^20 - 14336*a^8*b^18))/(8
*b^24) - (a*(-(a + b)*(a - b))^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^28 - 512*a^3*b^26))/(8*b^24))*
(56*a^4 + 6*b^4 - 47*a^2*b^2))/(2*b^9))*(56*a^4 + 6*b^4 - 47*a^2*b^2))/(2*b^9))*1i)/(2*b^9) + (a*(-(a + b)*(a
- b))^(1/2)*(56*a^4 + 6*b^4 - 47*a^2*b^2)*(((25*a^2*b^20)/8 - 225*a^4*b^18 + 4800*a^6*b^16 - 27560*a^8*b^14 +
65160*a^10*b^12 - 67200*a^12*b^10 + 25088*a^14*b^8)/b^23 + (tan(c/2 + (d*x)/2)*(50*a*b^22 - 5929*a^3*b^20 + 11
9304*a^5*b^18 - 738240*a^7*b^16 + 2004800*a^9*b^14 - 2655360*a^11*b^12 + 1677312*a^13*b^10 - 401408*a^15*b^8))
/(8*b^24) + (a*(-(a + b)*(a - b))^(1/2)*((10*a*b^24 - 274*a^3*b^22 + 712*a^5*b^20 - 448*a^7*b^18)/b^23 + (tan(
c/2 + (d*x)/2)*(1536*a^2*b^24 - 13568*a^4*b^22 + 26368*a^6*b^20 - 14336*a^8*b^18))/(8*b^24) + (a*(-(a + b)*(a
- b))^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^28 - 512*a^3*b^26))/(8*b^24))*(56*a^4 + 6*b^4 - 47*a^2*
b^2))/(2*b^9))*(56*a^4 + 6*b^4 - 47*a^2*b^2))/(2*b^9))*1i)/(2*b^9))/((702464*a^19 + (645*a^3*b^16)/4 - (65155*
a^5*b^14)/8 + (922065*a^7*b^12)/8 - 740668*a^9*b^10 + 2522213*a^11*b^8 - 4837780*a^13*b^6 + 5244512*a^15*b^4 -
2998016*a^17*b^2)/b^23 + (tan(c/2 + (d*x)/2)*(11239424*a^20 - 150*a^2*b^18 + 12125*a^4*b^16 - 328375*a^6*b^14
+ 3544880*a^8*b^12 - 18869120*a^10*b^10 + 55713280*a^12*b^8 - 95735744*a^14*b^6 + 95201792*a^16*b^4 - 5077811
2*a^18*b^2))/(4*b^24) + (a*(-(a + b)*(a - b))^(1/2)*(56*a^4 + 6*b^4 - 47*a^2*b^2)*(((25*a^2*b^20)/8 - 225*a^4*
b^18 + 4800*a^6*b^16 - 27560*a^8*b^14 + 65160*a^10*b^12 - 67200*a^12*b^10 + 25088*a^14*b^8)/b^23 + (tan(c/2 +
(d*x)/2)*(50*a*b^22 - 5929*a^3*b^20 + 119304*a^5*b^18 - 738240*a^7*b^16 + 2004800*a^9*b^14 - 2655360*a^11*b^12
+ 1677312*a^13*b^10 - 401408*a^15*b^8))/(8*b^24) - (a*(-(a + b)*(a - b))^(1/2)*((10*a*b^24 - 274*a^3*b^22 + 7
12*a^5*b^20 - 448*a^7*b^18)/b^23 + (tan(c/2 + (d*x)/2)*(1536*a^2*b^24 - 13568*a^4*b^22 + 26368*a^6*b^20 - 1433
6*a^8*b^18))/(8*b^24) - (a*(-(a + b)*(a - b))^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^28 - 512*a^3*b^
26))/(8*b^24))*(56*a^4 + 6*b^4 - 47*a^2*b^2))/(2*b^9))*(56*a^4 + 6*b^4 - 47*a^2*b^2))/(2*b^9)))/(2*b^9) - (a*(
-(a + b)*(a - b))^(1/2)*(56*a^4 + 6*b^4 - 47*a^2*b^2)*(((25*a^2*b^20)/8 - 225*a^4*b^18 + 4800*a^6*b^16 - 27560
*a^8*b^14 + 65160*a^10*b^12 - 67200*a^12*b^10 + 25088*a^14*b^8)/b^23 + (tan(c/2 + (d*x)/2)*(50*a*b^22 - 5929*a
^3*b^20 + 119304*a^5*b^18 - 738240*a^7*b^16 + 2004800*a^9*b^14 - 2655360*a^11*b^12 + 1677312*a^13*b^10 - 40140
8*a^15*b^8))/(8*b^24) + (a*(-(a + b)*(a - b))^(1/2)*((10*a*b^24 - 274*a^3*b^22 + 712*a^5*b^20 - 448*a^7*b^18)/
b^23 + (tan(c/2 + (d*x)/2)*(1536*a^2*b^24 - 13568*a^4*b^22 + 26368*a^6*b^20 - 14336*a^8*b^18))/(8*b^24) + (a*(
-(a + b)*(a - b))^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^28 - 512*a^3*b^26))/(8*b^24))*(56*a^4 + 6*b
^4 - 47*a^2*b^2))/(2*b^9))*(56*a^4 + 6*b^4 - 47*a^2*b^2))/(2*b^9)))/(2*b^9)))*(-(a + b)*(a - b))^(1/2)*(56*a^4
+ 6*b^4 - 47*a^2*b^2)*1i)/(b^9*d)