\(\int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx\) [1329]

Optimal result

Integrand size = 27, antiderivative size = 363 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 b \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^7 d}+\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^7 d}+\frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d} \]

[Out]

2*b*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^7/d+1/16*(5*a^6-30*a^4*b^2+40*a^2*b^4-1
6*b^6)*arctanh(cos(d*x+c))/a^7/d+1/15*b*(23*a^4-35*a^2*b^2+15*b^4)*cot(d*x+c)/a^6/d-1/16*(11*a^4-18*a^2*b^2+8*
b^4)*cot(d*x+c)*csc(d*x+c)/a^5/d-1/2*cot(d*x+c)*csc(d*x+c)^2/b/d+1/30*(15*a^4-22*a^2*b^2+10*b^4)*cot(d*x+c)*cs
c(d*x+c)^2/a^4/b/d+1/3*a*cot(d*x+c)*csc(d*x+c)^3/b^2/d-1/24*(8*a^4-13*a^2*b^2+6*b^4)*cot(d*x+c)*csc(d*x+c)^3/a
^3/b^2/d+1/5*b*cot(d*x+c)*csc(d*x+c)^4/a^2/d-1/6*cot(d*x+c)*csc(d*x+c)^5/a/d

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2975, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {2 b \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^7 d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^7 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d} \]

[In]

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

[Out]

(2*b*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^7*d) + ((5*a^6 - 30*a^4*b^2 + 40*a
^2*b^4 - 16*b^6)*ArcTanh[Cos[c + d*x]])/(16*a^7*d) + (b*(23*a^4 - 35*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(15*a^6*d
) - ((11*a^4 - 18*a^2*b^2 + 8*b^4)*Cot[c + d*x]*Csc[c + d*x])/(16*a^5*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(2*b*
d) + ((15*a^4 - 22*a^2*b^2 + 10*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^4*b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^
3)/(3*b^2*d) - ((8*a^4 - 13*a^2*b^2 + 6*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^3*b^2*d) + (b*Cot[c + d*x]*Csc
[c + d*x]^4)/(5*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a*d)

Rule 210

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

Rule 632

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

Rule 2739

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

Rule 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 3080

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3134

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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*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] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\int \frac {\csc ^5(c+d x) \left (30 \left (8 a^4-13 a^2 b^2+6 b^4\right )-6 a b \left (5 a^2-b^2\right ) \sin (c+d x)-18 \left (10 a^4-15 a^2 b^2+8 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{180 a^2 b^2} \\ & = -\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\int \frac {\csc ^4(c+d x) \left (-72 b \left (15 a^4-22 a^2 b^2+10 b^4\right )-18 a b^2 \left (5 a^2+2 b^2\right ) \sin (c+d x)+90 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{720 a^3 b^2} \\ & = -\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\int \frac {\csc ^3(c+d x) \left (270 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right )-18 a b^3 \left (19 a^2-10 b^2\right ) \sin (c+d x)-144 b^2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2160 a^4 b^2} \\ & = -\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\int \frac {\csc ^2(c+d x) \left (-288 b^3 \left (23 a^4-35 a^2 b^2+15 b^4\right )-18 a b^2 \left (75 a^4-82 a^2 b^2+40 b^4\right ) \sin (c+d x)+270 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4320 a^5 b^2} \\ & = \frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\int \frac {\csc (c+d x) \left (-270 b^2 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right )+270 a b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4320 a^6 b^2} \\ & = \frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\left (b \left (a^2-b^2\right )^3\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^7}-\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \int \csc (c+d x) \, dx}{16 a^7} \\ & = \frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^7 d}+\frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\left (2 b \left (a^2-b^2\right )^3\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 )}{a^7 d} \\ & = \frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^7 d}+\frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\left (4 b \left (a^2-b^2\right )^3\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 )}{a^7 d} \\ & = \frac {2 b \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^7 d}+\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^7 d}+\frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.73 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.98 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {7680 b \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+240 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+240 \left (-5 a^6+30 a^4 b^2-40 a^2 b^4+16 b^6\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 a \cot (c+d x) \csc ^5(c+d x) \left (-295 a^5+570 a^3 b^2-360 a b^4+20 \left (7 a^5-42 a^3 b^2+24 a b^4\right ) \cos (2 (c+d x))-15 \left (11 a^5-18 a^3 b^2+8 a b^4\right ) \cos (4 (c+d x))+1168 a^4 b \sin (c+d x)-2320 a^2 b^3 \sin (c+d x)+1200 b^5 \sin (c+d x)-568 a^4 b \sin (3 (c+d x))+1240 a^2 b^3 \sin (3 (c+d x))-600 b^5 \sin (3 (c+d x))+184 a^4 b \sin (5 (c+d x))-280 a^2 b^3 \sin (5 (c+d x))+120 b^5 \sin (5 (c+d x))\right )}{3840 a^7 d} \]

[In]

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

[Out]

(7680*b*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 240*(5*a^6 - 30*a^4*b^2 + 40*a^2*
b^4 - 16*b^6)*Log[Cos[(c + d*x)/2]] + 240*(-5*a^6 + 30*a^4*b^2 - 40*a^2*b^4 + 16*b^6)*Log[Sin[(c + d*x)/2]] +
2*a*Cot[c + d*x]*Csc[c + d*x]^5*(-295*a^5 + 570*a^3*b^2 - 360*a*b^4 + 20*(7*a^5 - 42*a^3*b^2 + 24*a*b^4)*Cos[2
*(c + d*x)] - 15*(11*a^5 - 18*a^3*b^2 + 8*a*b^4)*Cos[4*(c + d*x)] + 1168*a^4*b*Sin[c + d*x] - 2320*a^2*b^3*Sin
[c + d*x] + 1200*b^5*Sin[c + d*x] - 568*a^4*b*Sin[3*(c + d*x)] + 1240*a^2*b^3*Sin[3*(c + d*x)] - 600*b^5*Sin[3
*(c + d*x)] + 184*a^4*b*Sin[5*(c + d*x)] - 280*a^2*b^3*Sin[5*(c + d*x)] + 120*b^5*Sin[5*(c + d*x)]))/(3840*a^7
*d)

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{6}-\frac {2 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{2}+\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b^{2}+\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4} b}{3}-\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{3}}{3}+\frac {15 a^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b^{2}+8 a \,b^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-44 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{3}-32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5}}{64 a^{6}}-\frac {1}{384 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {-6 a^{2}+4 b^{2}}{256 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {15 a^{4}-32 a^{2} b^{2}+16 b^{4}}{128 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-20 a^{6}+120 a^{4} b^{2}-160 a^{2} b^{4}+64 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{7}}+\frac {b}{160 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {b \left (7 a^{2}-4 b^{2}\right )}{96 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {b \left (11 a^{4}-18 a^{2} b^{2}+8 b^{4}\right )}{16 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{7} \sqrt {a^{2}-b^{2}}}}{d}\)	\(486\)
default	\(\frac {\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{6}-\frac {2 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{2}+\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b^{2}+\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4} b}{3}-\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{3}}{3}+\frac {15 a^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b^{2}+8 a \,b^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-44 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{3}-32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5}}{64 a^{6}}-\frac {1}{384 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {-6 a^{2}+4 b^{2}}{256 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {15 a^{4}-32 a^{2} b^{2}+16 b^{4}}{128 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-20 a^{6}+120 a^{4} b^{2}-160 a^{2} b^{4}+64 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{7}}+\frac {b}{160 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {b \left (7 a^{2}-4 b^{2}\right )}{96 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {b \left (11 a^{4}-18 a^{2} b^{2}+8 b^{4}\right )}{16 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{7} \sqrt {a^{2}-b^{2}}}}{d}\)	\(486\)
risch	\(\text {Expression too large to display}\)	\(1065\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^7/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/64/a^6*(1/6*tan(1/2*d*x+1/2*c)^6*a^5-2/5*b*tan(1/2*d*x+1/2*c)^5*a^4-3/2*tan(1/2*d*x+1/2*c)^4*a^5+tan(1/
2*d*x+1/2*c)^4*a^3*b^2+14/3*tan(1/2*d*x+1/2*c)^3*a^4*b-8/3*tan(1/2*d*x+1/2*c)^3*a^2*b^3+15/2*tan(1/2*d*x+1/2*c
)^2*a^5-16*tan(1/2*d*x+1/2*c)^2*a^3*b^2+8*a*b^4*tan(1/2*d*x+1/2*c)^2-44*a^4*b*tan(1/2*d*x+1/2*c)+72*tan(1/2*d*
x+1/2*c)*a^2*b^3-32*tan(1/2*d*x+1/2*c)*b^5)-1/384/a/tan(1/2*d*x+1/2*c)^6-1/256*(-6*a^2+4*b^2)/a^3/tan(1/2*d*x+
1/2*c)^4-1/128/a^5*(15*a^4-32*a^2*b^2+16*b^4)/tan(1/2*d*x+1/2*c)^2+1/64/a^7*(-20*a^6+120*a^4*b^2-160*a^2*b^4+6
4*b^6)*ln(tan(1/2*d*x+1/2*c))+1/160/a^2*b/tan(1/2*d*x+1/2*c)^5-1/96/a^4*b*(7*a^2-4*b^2)/tan(1/2*d*x+1/2*c)^3+1
/16*b*(11*a^4-18*a^2*b^2+8*b^4)/a^6/tan(1/2*d*x+1/2*c)+2*b*(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/a^7/(a^2-b^2)^(1/2)*a
rctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 689 vs. \(2 (340) = 680\).

Time = 0.98 (sec) , antiderivative size = 1462, normalized size of antiderivative = 4.03 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

[1/480*(30*(11*a^6 - 18*a^4*b^2 + 8*a^2*b^4)*cos(d*x + c)^5 - 80*(5*a^6 - 12*a^4*b^2 + 6*a^2*b^4)*cos(d*x + c)
^3 + 240*((a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c)^6 - a^4*b + 2*a^2*b^3 - b^5 - 3*(a^4*b - 2*a^2*b^3 + b^5)*cos
(d*x + c)^4 + 3*(a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c)^2)*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*(5*a^6 - 14*a^4*b^2 + 8*a^2*b^4)*cos(d*x + c) + 15*((5*a^6
- 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*cos(d*x + c)^6 - 5*a^6 + 30*a^4*b^2 - 40*a^2*b^4 + 16*b^6 - 3*(5*a^6 - 30
*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*cos(d*x + c)^4 + 3*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*cos(d*x + c)^2)*
log(1/2*cos(d*x + c) + 1/2) - 15*((5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*cos(d*x + c)^6 - 5*a^6 + 30*a^4*b
^2 - 40*a^2*b^4 + 16*b^6 - 3*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*cos(d*x + c)^4 + 3*(5*a^6 - 30*a^4*b^2
+ 40*a^2*b^4 - 16*b^6)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) - 32*((23*a^5*b - 35*a^3*b^3 + 15*a*b^5)*
cos(d*x + c)^5 - 5*(7*a^5*b - 13*a^3*b^3 + 6*a*b^5)*cos(d*x + c)^3 + 15*(a^5*b - 2*a^3*b^3 + a*b^5)*cos(d*x +
c))*sin(d*x + c))/(a^7*d*cos(d*x + c)^6 - 3*a^7*d*cos(d*x + c)^4 + 3*a^7*d*cos(d*x + c)^2 - a^7*d), 1/480*(30*
(11*a^6 - 18*a^4*b^2 + 8*a^2*b^4)*cos(d*x + c)^5 - 80*(5*a^6 - 12*a^4*b^2 + 6*a^2*b^4)*cos(d*x + c)^3 - 480*((
a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c)^6 - a^4*b + 2*a^2*b^3 - b^5 - 3*(a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c)^4
+ 3*(a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c)^2)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*c
os(d*x + c))) + 30*(5*a^6 - 14*a^4*b^2 + 8*a^2*b^4)*cos(d*x + c) + 15*((5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b
^6)*cos(d*x + c)^6 - 5*a^6 + 30*a^4*b^2 - 40*a^2*b^4 + 16*b^6 - 3*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*c
os(d*x + c)^4 + 3*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) - 15*
((5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*cos(d*x + c)^6 - 5*a^6 + 30*a^4*b^2 - 40*a^2*b^4 + 16*b^6 - 3*(5*a
^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*cos(d*x + c)^4 + 3*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*cos(d*x +
c)^2)*log(-1/2*cos(d*x + c) + 1/2) - 32*((23*a^5*b - 35*a^3*b^3 + 15*a*b^5)*cos(d*x + c)^5 - 5*(7*a^5*b - 13*
a^3*b^3 + 6*a*b^5)*cos(d*x + c)^3 + 15*(a^5*b - 2*a^3*b^3 + a*b^5)*cos(d*x + c))*sin(d*x + c))/(a^7*d*cos(d*x
+ c)^6 - 3*a^7*d*cos(d*x + c)^4 + 3*a^7*d*cos(d*x + c)^2 - a^7*d)]

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**7/(a+b*sin(d*x+c)),x)

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7/(a+b*sin(d*x+c)),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 = 0.46 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.73 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {5 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 30 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 80 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 480 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1320 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2160 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 960 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} - \frac {120 \, {\left (5 \, a^{6} - 30 \, a^{4} b^{2} + 40 \, a^{2} b^{4} - 16 \, b^{6}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{7}} + \frac {3840 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{7}} + \frac {1470 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 8820 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 11760 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4704 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1320 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2160 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 960 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 225 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 140 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{6}}{a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/1920*((5*a^5*tan(1/2*d*x + 1/2*c)^6 - 12*a^4*b*tan(1/2*d*x + 1/2*c)^5 - 45*a^5*tan(1/2*d*x + 1/2*c)^4 + 30*a
^3*b^2*tan(1/2*d*x + 1/2*c)^4 + 140*a^4*b*tan(1/2*d*x + 1/2*c)^3 - 80*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 225*a^5
*tan(1/2*d*x + 1/2*c)^2 - 480*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 + 240*a*b^4*tan(1/2*d*x + 1/2*c)^2 - 1320*a^4*b*t
an(1/2*d*x + 1/2*c) + 2160*a^2*b^3*tan(1/2*d*x + 1/2*c) - 960*b^5*tan(1/2*d*x + 1/2*c))/a^6 - 120*(5*a^6 - 30*
a^4*b^2 + 40*a^2*b^4 - 16*b^6)*log(abs(tan(1/2*d*x + 1/2*c)))/a^7 + 3840*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)
*(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)*a^7) + (1470*a^6*tan(1/2*d*x + 1/2*c)^6 - 8820*a^4*b^2*tan(1/2*d*x + 1/2*c)^6 + 11760*a^2*b^4*tan(1/2*d*x
+ 1/2*c)^6 - 4704*b^6*tan(1/2*d*x + 1/2*c)^6 + 1320*a^5*b*tan(1/2*d*x + 1/2*c)^5 - 2160*a^3*b^3*tan(1/2*d*x +
1/2*c)^5 + 960*a*b^5*tan(1/2*d*x + 1/2*c)^5 - 225*a^6*tan(1/2*d*x + 1/2*c)^4 + 480*a^4*b^2*tan(1/2*d*x + 1/2*
c)^4 - 240*a^2*b^4*tan(1/2*d*x + 1/2*c)^4 - 140*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 80*a^3*b^3*tan(1/2*d*x + 1/2*c)
^3 + 45*a^6*tan(1/2*d*x + 1/2*c)^2 - 30*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 + 12*a^5*b*tan(1/2*d*x + 1/2*c) - 5*a^6
)/(a^7*tan(1/2*d*x + 1/2*c)^6))/d

Mupad [B] (verification not implemented)

Time = 12.59 (sec) , antiderivative size = 1289, normalized size of antiderivative = 3.55 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^6/(sin(c + d*x)^7*(a + b*sin(c + d*x))),x)

[Out]

tan(c/2 + (d*x)/2)^6/(384*a*d) + (tan(c/2 + (d*x)/2)^3*(b/(96*a^2) + (2*b*(3/(32*a) - b^2/(16*a^3)))/(3*a)))/d
- (tan(c/2 + (d*x)/2)^4*(3/(128*a) - b^2/(64*a^3)))/d - (tan(c/2 + (d*x)/2)*(b/(32*a^2) - (2*b*(b^2/(16*a^3)
- 15/(64*a) + (2*b*(b/(32*a^2) + (2*b*(3/(32*a) - b^2/(16*a^3)))/a))/a))/a + (2*b*(3/(32*a) - b^2/(16*a^3)))/a
))/d - (tan(c/2 + (d*x)/2)^2*(b^2/(32*a^3) - 15/(128*a) + (b*(b/(32*a^2) + (2*b*(3/(32*a) - b^2/(16*a^3)))/a))
/a))/d - (tan(c/2 + (d*x)/2)^3*((14*a^4*b)/3 - (8*a^2*b^3)/3) + tan(c/2 + (d*x)/2)^4*(8*a*b^4 + (15*a^5)/2 - 1
6*a^3*b^2) - tan(c/2 + (d*x)/2)^5*(44*a^4*b + 32*b^5 - 72*a^2*b^3) + a^5/6 - tan(c/2 + (d*x)/2)^2*((3*a^5)/2 -
a^3*b^2) - (2*a^4*b*tan(c/2 + (d*x)/2))/5)/(64*a^6*d*tan(c/2 + (d*x)/2)^6) - (b*tan(c/2 + (d*x)/2)^5)/(160*a^
2*d) - (log(tan(c/2 + (d*x)/2))*(5*a^6 - 16*b^6 + 40*a^2*b^4 - 30*a^4*b^2))/(16*a^7*d) + (b*atan(((b*(-(a + b)
^5*(a - b)^5)^(1/2)*((tan(c/2 + (d*x)/2)*(5*a^13 + 64*a^5*b^8 - 192*a^7*b^6 + 196*a^9*b^4 - 72*a^11*b^2))/(8*a
^11) - (21*a^13*b - 32*a^7*b^7 + 88*a^9*b^5 - 78*a^11*b^3)/(8*a^12) + (b*(2*a^2*b - (tan(c/2 + (d*x)/2)*(48*a^
14 - 64*a^12*b^2))/(8*a^11))*(-(a + b)^5*(a - b)^5)^(1/2))/a^7)*1i)/a^7 - (b*(-(a + b)^5*(a - b)^5)^(1/2)*((21
*a^13*b - 32*a^7*b^7 + 88*a^9*b^5 - 78*a^11*b^3)/(8*a^12) - (tan(c/2 + (d*x)/2)*(5*a^13 + 64*a^5*b^8 - 192*a^7
*b^6 + 196*a^9*b^4 - 72*a^11*b^2))/(8*a^11) + (b*(2*a^2*b - (tan(c/2 + (d*x)/2)*(48*a^14 - 64*a^12*b^2))/(8*a^
11))*(-(a + b)^5*(a - b)^5)^(1/2))/a^7)*1i)/a^7)/((5*a^12*b + 16*b^13 - 88*a^2*b^11 + 198*a^4*b^9 - 231*a^6*b^
7 + 145*a^8*b^5 - 45*a^10*b^3)/(4*a^12) + (tan(c/2 + (d*x)/2)*(16*b^12 - 84*a^2*b^10 + 178*a^4*b^8 - 190*a^6*b
^6 + 102*a^8*b^4 - 22*a^10*b^2))/(4*a^11) + (b*(-(a + b)^5*(a - b)^5)^(1/2)*((tan(c/2 + (d*x)/2)*(5*a^13 + 64*
a^5*b^8 - 192*a^7*b^6 + 196*a^9*b^4 - 72*a^11*b^2))/(8*a^11) - (21*a^13*b - 32*a^7*b^7 + 88*a^9*b^5 - 78*a^11*
b^3)/(8*a^12) + (b*(2*a^2*b - (tan(c/2 + (d*x)/2)*(48*a^14 - 64*a^12*b^2))/(8*a^11))*(-(a + b)^5*(a - b)^5)^(1
/2))/a^7))/a^7 + (b*(-(a + b)^5*(a - b)^5)^(1/2)*((21*a^13*b - 32*a^7*b^7 + 88*a^9*b^5 - 78*a^11*b^3)/(8*a^12)
- (tan(c/2 + (d*x)/2)*(5*a^13 + 64*a^5*b^8 - 192*a^7*b^6 + 196*a^9*b^4 - 72*a^11*b^2))/(8*a^11) + (b*(2*a^2*b
- (tan(c/2 + (d*x)/2)*(48*a^14 - 64*a^12*b^2))/(8*a^11))*(-(a + b)^5*(a - b)^5)^(1/2))/a^7))/a^7))*(-(a + b)^
5*(a - b)^5)^(1/2)*2i)/(a^7*d)