3.1329 \(\int \frac{\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=363 \[ \frac{2 b \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^7 d}+\frac{b \left (-35 a^2 b^2+23 a^4+15 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac{\left (-30 a^4 b^2+40 a^2 b^4+5 a^6-16 b^6\right ) \tanh ^{-1}(\cos (c+d x))}{16 a^7 d}-\frac{\left (-13 a^2 b^2+8 a^4+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac{\left (-22 a^2 b^2+15 a^4+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}-\frac{\left (-18 a^2 b^2+11 a^4+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}+\frac{b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{2 b d} \]

[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)

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Rubi [A]  time = 1.48389, antiderivative size = 363, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2896, 3055, 3001, 3770, 2660, 618, 204} \[ \frac{2 b \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^7 d}+\frac{b \left (-35 a^2 b^2+23 a^4+15 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac{\left (-30 a^4 b^2+40 a^2 b^4+5 a^6-16 b^6\right ) \tanh ^{-1}(\cos (c+d x))}{16 a^7 d}-\frac{\left (-13 a^2 b^2+8 a^4+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac{\left (-22 a^2 b^2+15 a^4+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}-\frac{\left (-18 a^2 b^2+11 a^4+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}+\frac{b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{2 b d} \]

Antiderivative was successfully verified.

[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 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 3055

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] + Dis
t[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] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3001

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 3770

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

Rule 2660

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

Rule 618

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

Rule 204

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

Rubi steps

\begin{align*} \int \frac{\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx &=-\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 ) \tanh ^{-1}(\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 ) \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 )}{a^7 d}\\ &=\frac{\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \tanh ^{-1}(\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 ) \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 )}{a^7 d}\\ &=\frac{2 b \left (a^2-b^2\right )^{5/2} \tan ^{-1}\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 ) \tanh ^{-1}(\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]  time = 1.50711, size = 356, normalized size = 0.98 \[ \frac{7680 b \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )+240 \left (30 a^4 b^2-40 a^2 b^4-5 a^6+16 b^6\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+240 \left (-30 a^4 b^2+40 a^2 b^4+5 a^6-16 b^6\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+2 a \cot (c+d x) \csc ^5(c+d x) \left (-2320 a^2 b^3 \sin (c+d x)+1240 a^2 b^3 \sin (3 (c+d x))-280 a^2 b^3 \sin (5 (c+d x))+20 \left (-42 a^3 b^2+7 a^5+24 a b^4\right ) \cos (2 (c+d x))-15 \left (-18 a^3 b^2+11 a^5+8 a b^4\right ) \cos (4 (c+d x))+570 a^3 b^2+1168 a^4 b \sin (c+d x)-568 a^4 b \sin (3 (c+d x))+184 a^4 b \sin (5 (c+d x))-295 a^5-360 a b^4+1200 b^5 \sin (c+d x)-600 b^5 \sin (3 (c+d x))+120 b^5 \sin (5 (c+d x))\right )}{3840 a^7 d} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.127, size = 780, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384/d/a*tan(1/2*d*x+1/2*c)^6-1/384/d/a/tan(1/2*d*x+1/2*c)^6+15/128/d/a*tan(1/2*d*x+1/2*c)^2-15/128/d/a/tan(1
/2*d*x+1/2*c)^2-5/2/d/a^5*ln(tan(1/2*d*x+1/2*c))*b^4-7/96/d/a^2*b/tan(1/2*d*x+1/2*c)^3-9/8/d*b^3/a^4/tan(1/2*d
*x+1/2*c)-5/16/d/a*ln(tan(1/2*d*x+1/2*c))-1/160/d/a^2*b*tan(1/2*d*x+1/2*c)^5+1/64/d/a^3*tan(1/2*d*x+1/2*c)^4*b
^2+1/8/d/a^5*tan(1/2*d*x+1/2*c)^2*b^4-1/24/d/a^4*tan(1/2*d*x+1/2*c)^3*b^3-1/64/d/a^3/tan(1/2*d*x+1/2*c)^4*b^2-
1/8/d/a^5/tan(1/2*d*x+1/2*c)^2*b^4+1/d/a^7*ln(tan(1/2*d*x+1/2*c))*b^6+1/160/d/a^2*b/tan(1/2*d*x+1/2*c)^5+1/2/d
*b^5/a^6/tan(1/2*d*x+1/2*c)-11/16/d/a^2*tan(1/2*d*x+1/2*c)*b+15/8/d/a^3*ln(tan(1/2*d*x+1/2*c))*b^2+11/16/d/a^2
*b/tan(1/2*d*x+1/2*c)-3/128/d/a*tan(1/2*d*x+1/2*c)^4+3/128/d/a/tan(1/2*d*x+1/2*c)^4-6/d/a^3*b^3/(a^2-b^2)^(1/2
)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-2/d*b^7/a^7/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2
*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-1/4/d/a^3*tan(1/2*d*x+1/2*c)^2*b^2+9/8/d/a^4*b^3*tan(1/2*d*x+1/2*c)+1/4/d/a^
3*b^2/tan(1/2*d*x+1/2*c)^2+2/d/a*b/(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))+1/
24/d/a^4*b^3/tan(1/2*d*x+1/2*c)^3-1/2/d/a^6*b^5*tan(1/2*d*x+1/2*c)+6/d*b^5/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))+7/96/d/a^2*tan(1/2*d*x+1/2*c)^3*b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 6.77411, size = 3362, normalized size = 9.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*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)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.29279, size = 846, normalized size = 2.33 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*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