3.1273 \(\int \frac{\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=355 \[ -\frac{15 b \left (a^2-2 b^2\right ) \sqrt{a^2-b^2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^7 d}+\frac{\left (-25 a^2 b^2+a^4+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}-\frac{15 \left (-8 a^2 b^2+a^4+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}+\frac{15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac{\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2} \]

[Out]

(-15*b*(a^2 - 2*b^2)*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^7*d) - (15*(a^4 - 8*
a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^7*d) + ((a^4 - 25*a^2*b^2 + 30*b^4)*Cot[c + d*x])/(2*a^6*b*d) + (
15*(3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^5*d) - Cot[c + d*x]/(2*b*d*(a + b*Sin[c + d*x])^2) - ((4*a^
2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(4*a^3*d*(a + b*Sin[c + d*x])^2) + (b*Cot[c + d*x]*Csc[c + d*x]^2)/(2*a^
2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d*(a + b*Sin[c + d*x])^2) - ((7*a^2 - 10*b^2)
*Cot[c + d*x]*Csc[c + d*x])/(2*a^4*d*(a + b*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.68169, antiderivative size = 355, 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} \[ -\frac{15 b \left (a^2-2 b^2\right ) \sqrt{a^2-b^2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^7 d}+\frac{\left (-25 a^2 b^2+a^4+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}-\frac{15 \left (-8 a^2 b^2+a^4+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}+\frac{15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac{\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + b*Sin[c + d*x])^3,x]

[Out]

(-15*b*(a^2 - 2*b^2)*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^7*d) - (15*(a^4 - 8*
a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^7*d) + ((a^4 - 25*a^2*b^2 + 30*b^4)*Cot[c + d*x])/(2*a^6*b*d) + (
15*(3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^5*d) - Cot[c + d*x]/(2*b*d*(a + b*Sin[c + d*x])^2) - ((4*a^
2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(4*a^3*d*(a + b*Sin[c + d*x])^2) + (b*Cot[c + d*x]*Csc[c + d*x]^2)/(2*a^
2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d*(a + b*Sin[c + d*x])^2) - ((7*a^2 - 10*b^2)
*Cot[c + d*x]*Csc[c + d*x])/(2*a^4*d*(a + b*Sin[c + d*x]))

Rule 2896

Int[cos[(e_.) + (f_.)*(x_)]^6*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1))/(a*d*f*(n + 1)), x] +
(Dist[1/(a^2*b^2*d^2*(n + 1)*(n + 2)*(m + n + 5)*(m + n + 6)), Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*
x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2*n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m +
n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e +
f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4)*(m + n + 5)*(m + n + 6) - a^2*b^2*(
n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))*Sin[e + f*x]^2, x], x], x] - Simp[(b*(m + n + 2)*Cos[e + f*x]*(d*
Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1))/(a^2*d^2*f*(n + 1)*(n + 2)), x] - Simp[(a*(n + 5)*Cos[e +
f*x]*(d*Sin[e + f*x])^(n + 3)*(a + b*Sin[e + f*x])^(m + 1))/(b^2*d^3*f*(m + n + 5)*(m + n + 6)), x] + Simp[(Co
s[e + f*x]*(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^(m + 1))/(b*d^4*f*(m + n + 6)), x]) /; FreeQ[{a, b, d
, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0]
&& NeQ[m + n + 6, 0] &&  !IGtQ[m, 0]

Rule 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{\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=-\frac{\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac{\int \frac{\csc ^3(c+d x) \left (-18 b^2 \left (9 a^2-10 b^2\right )-18 a b \left (2 a^2-b^2\right ) \sin (c+d x)+36 b^2 \left (3 a^2-4 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{72 a^2 b^2}\\ &=-\frac{\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac{\int \frac{\csc ^3(c+d x) \left (-36 b^2 \left (17 a^4-37 a^2 b^2+20 b^4\right )-72 a b \left (a^2-b^2\right )^2 \sin (c+d x)+108 b^2 \left (4 a^4-9 a^2 b^2+5 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{144 a^3 b^2 \left (a^2-b^2\right )}\\ &=-\frac{\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^3(c+d x) \left (-540 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2-36 a b \left (2 a^2-5 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)+144 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right )^2 \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{144 a^4 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^2(c+d x) \left (-144 b \left (a^2-b^2\right )^2 \left (a^4-25 a^2 b^2+30 b^4\right )+36 a b^2 \left (11 a^2-20 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-540 b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{288 a^5 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}+\frac{15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac{\int \frac{\csc (c+d x) \left (540 b^2 \left (a^2-b^2\right )^2 \left (a^4-8 a^2 b^2+8 b^4\right )-540 a b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{288 a^6 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}+\frac{15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac{\left (15 b \left (a^2-2 b^2\right ) \left (a^2-b^2\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{2 a^7}+\frac{\left (15 \left (a^4-8 a^2 b^2+8 b^4\right )\right ) \int \csc (c+d x) \, dx}{8 a^7}\\ &=-\frac{15 \left (a^4-8 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}+\frac{\left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}+\frac{15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac{\left (15 b \left (a^2-2 b^2\right ) \left (a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^7 d}\\ &=-\frac{15 \left (a^4-8 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}+\frac{\left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}+\frac{15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac{\left (30 b \left (a^2-2 b^2\right ) \left (a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^7 d}\\ &=-\frac{15 b \left (a^2-2 b^2\right ) \sqrt{a^2-b^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{15 \left (a^4-8 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}+\frac{\left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}+\frac{15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac{\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 1.33923, size = 363, normalized size = 1.02 \[ \frac{-\frac{1920 b \left (-3 a^2 b^2+a^4+2 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}+240 \left (-8 a^2 b^2+a^4+8 b^4\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-240 \left (-8 a^2 b^2+a^4+8 b^4\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{2 a \cot (c+d x) \csc ^5(c+d x) \left (-260 a^2 b^3 \sin (c+d x)+170 a^2 b^3 \sin (3 (c+d x))-50 a^2 b^3 \sin (5 (c+d x))+\left (660 a^3 b^2-68 a^5-720 a b^4\right ) \cos (2 (c+d x))+\left (-155 a^3 b^2+8 a^5+180 a b^4\right ) \cos (4 (c+d x))-505 a^3 b^2-176 a^4 b \sin (c+d x)+66 a^4 b \sin (3 (c+d x))+2 a^4 b \sin (5 (c+d x))+44 a^5+540 a b^4+600 b^5 \sin (c+d x)-300 b^5 \sin (3 (c+d x))+60 b^5 \sin (5 (c+d x))\right )}{(a \csc (c+d x)+b)^2}}{128 a^7 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + b*Sin[c + d*x])^3,x]

[Out]

((-1920*b*(a^4 - 3*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 240*(a
^4 - 8*a^2*b^2 + 8*b^4)*Log[Cos[(c + d*x)/2]] + 240*(a^4 - 8*a^2*b^2 + 8*b^4)*Log[Sin[(c + d*x)/2]] + (2*a*Cot
[c + d*x]*Csc[c + d*x]^5*(44*a^5 - 505*a^3*b^2 + 540*a*b^4 + (-68*a^5 + 660*a^3*b^2 - 720*a*b^4)*Cos[2*(c + d*
x)] + (8*a^5 - 155*a^3*b^2 + 180*a*b^4)*Cos[4*(c + d*x)] - 176*a^4*b*Sin[c + d*x] - 260*a^2*b^3*Sin[c + d*x] +
600*b^5*Sin[c + d*x] + 66*a^4*b*Sin[3*(c + d*x)] + 170*a^2*b^3*Sin[3*(c + d*x)] - 300*b^5*Sin[3*(c + d*x)] +
2*a^4*b*Sin[5*(c + d*x)] - 50*a^2*b^3*Sin[5*(c + d*x)] + 60*b^5*Sin[5*(c + d*x)]))/(b + a*Csc[c + d*x])^2)/(12
8*a^7*d)

________________________________________________________________________________________

Maple [B]  time = 0.228, size = 1070, normalized size = 3. \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)^5/(a+b*sin(d*x+c))^3,x)

[Out]

1/64/d/a^3*tan(1/2*d*x+1/2*c)^4-1/64/d/a^3/tan(1/2*d*x+1/2*c)^4+2/d/a/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/
2*c)*b+a)^2+15/8/d/a^3*ln(tan(1/2*d*x+1/2*c))+12/d/a^6/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan
(1/2*d*x+1/2*c)^3*b^5-30/d/a^7*b^5/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-15
/d/a^4*b^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^3-15/d/a^5*b^4/(tan(1/2*d*x+
1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2+3/d/a^2*b/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/
2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^3-9/d/a^3*b^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1
/2*c)^2+5/d/a^2*b/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)-15/d/a^3*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))+27/8/d/a^4*tan(1/2*d*x+1/2*c)*b-13/d/a^3*b^2/(t
an(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2-15/d/a^5*ln(tan(1/2*d*x+1/2*c))*b^2-27/8/d*b/a^4/tan(1/2*d*x
+1/2*c)+32/d/a^6/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)*b^5+22/d/a^7/(tan(1/2*
d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2*b^6+2/d/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+
1/2*c)*b+a)^2/a*tan(1/2*d*x+1/2*c)^2+1/4/d/a^3/tan(1/2*d*x+1/2*c)^2-1/4/d/a^3*tan(1/2*d*x+1/2*c)^2-37/d/a^4*b^
3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)+45/d/a^5*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))+1/8/d/a^4*b/tan(1/2*d*x+1/2*c)^3+5/d*b^3/a^6/tan(1/2*d*x+1/2*
c)-1/8/d/a^4*tan(1/2*d*x+1/2*c)^3*b+3/4/d/a^5*tan(1/2*d*x+1/2*c)^2*b^2-5/d/a^6*b^3*tan(1/2*d*x+1/2*c)+11/d/a^5
/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*b^4-3/4/d/a^5/tan(1/2*d*x+1/2*c)^2*b^2+15/d/a^7*ln(tan(1/
2*d*x+1/2*c))*b^4

________________________________________________________________________________________

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)^5/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 3.54943, size = 4537, normalized size = 12.78 \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)^5/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[-1/16*(2*(8*a^6 - 155*a^4*b^2 + 180*a^2*b^4)*cos(d*x + c)^5 - 10*(5*a^6 - 64*a^4*b^2 + 72*a^2*b^4)*cos(d*x +
c)^3 + 60*((a^2*b^3 - 2*b^5)*cos(d*x + c)^6 - a^4*b + a^2*b^3 + 2*b^5 - (a^4*b + a^2*b^3 - 6*b^5)*cos(d*x + c)
^4 + (2*a^4*b - a^2*b^3 - 6*b^5)*cos(d*x + c)^2 - 2*(a^3*b^2 - 2*a*b^4 + (a^3*b^2 - 2*a*b^4)*cos(d*x + c)^4 -
2*(a^3*b^2 - 2*a*b^4)*cos(d*x + c)^2)*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*(a^6 - 11*a^4*b^2 + 12*a^2*b^4)*cos(d*x + c) + 15*((a^4*b^2 - 8*
a^2*b^4 + 8*b^6)*cos(d*x + c)^6 - a^6 + 7*a^4*b^2 - 8*b^6 - (a^6 - 5*a^4*b^2 - 16*a^2*b^4 + 24*b^6)*cos(d*x +
c)^4 + (2*a^6 - 13*a^4*b^2 - 8*a^2*b^4 + 24*b^6)*cos(d*x + c)^2 - 2*(a^5*b - 8*a^3*b^3 + 8*a*b^5 + (a^5*b - 8*
a^3*b^3 + 8*a*b^5)*cos(d*x + c)^4 - 2*(a^5*b - 8*a^3*b^3 + 8*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(
d*x + c) + 1/2) - 15*((a^4*b^2 - 8*a^2*b^4 + 8*b^6)*cos(d*x + c)^6 - a^6 + 7*a^4*b^2 - 8*b^6 - (a^6 - 5*a^4*b^
2 - 16*a^2*b^4 + 24*b^6)*cos(d*x + c)^4 + (2*a^6 - 13*a^4*b^2 - 8*a^2*b^4 + 24*b^6)*cos(d*x + c)^2 - 2*(a^5*b
- 8*a^3*b^3 + 8*a*b^5 + (a^5*b - 8*a^3*b^3 + 8*a*b^5)*cos(d*x + c)^4 - 2*(a^5*b - 8*a^3*b^3 + 8*a*b^5)*cos(d*x
+ c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 4*(2*(a^5*b - 25*a^3*b^3 + 30*a*b^5)*cos(d*x + c)^5 + 5*
(3*a^5*b + 16*a^3*b^3 - 24*a*b^5)*cos(d*x + c)^3 - 15*(a^5*b + 2*a^3*b^3 - 4*a*b^5)*cos(d*x + c))*sin(d*x + c)
)/(a^7*b^2*d*cos(d*x + c)^6 - (a^9 + 3*a^7*b^2)*d*cos(d*x + c)^4 + (2*a^9 + 3*a^7*b^2)*d*cos(d*x + c)^2 - (a^9
+ a^7*b^2)*d - 2*(a^8*b*d*cos(d*x + c)^4 - 2*a^8*b*d*cos(d*x + c)^2 + a^8*b*d)*sin(d*x + c)), -1/16*(2*(8*a^6
- 155*a^4*b^2 + 180*a^2*b^4)*cos(d*x + c)^5 - 10*(5*a^6 - 64*a^4*b^2 + 72*a^2*b^4)*cos(d*x + c)^3 - 120*((a^2
*b^3 - 2*b^5)*cos(d*x + c)^6 - a^4*b + a^2*b^3 + 2*b^5 - (a^4*b + a^2*b^3 - 6*b^5)*cos(d*x + c)^4 + (2*a^4*b -
a^2*b^3 - 6*b^5)*cos(d*x + c)^2 - 2*(a^3*b^2 - 2*a*b^4 + (a^3*b^2 - 2*a*b^4)*cos(d*x + c)^4 - 2*(a^3*b^2 - 2*
a*b^4)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c
))) + 30*(a^6 - 11*a^4*b^2 + 12*a^2*b^4)*cos(d*x + c) + 15*((a^4*b^2 - 8*a^2*b^4 + 8*b^6)*cos(d*x + c)^6 - a^6
+ 7*a^4*b^2 - 8*b^6 - (a^6 - 5*a^4*b^2 - 16*a^2*b^4 + 24*b^6)*cos(d*x + c)^4 + (2*a^6 - 13*a^4*b^2 - 8*a^2*b^
4 + 24*b^6)*cos(d*x + c)^2 - 2*(a^5*b - 8*a^3*b^3 + 8*a*b^5 + (a^5*b - 8*a^3*b^3 + 8*a*b^5)*cos(d*x + c)^4 - 2
*(a^5*b - 8*a^3*b^3 + 8*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 15*((a^4*b^2 - 8*a^
2*b^4 + 8*b^6)*cos(d*x + c)^6 - a^6 + 7*a^4*b^2 - 8*b^6 - (a^6 - 5*a^4*b^2 - 16*a^2*b^4 + 24*b^6)*cos(d*x + c)
^4 + (2*a^6 - 13*a^4*b^2 - 8*a^2*b^4 + 24*b^6)*cos(d*x + c)^2 - 2*(a^5*b - 8*a^3*b^3 + 8*a*b^5 + (a^5*b - 8*a^
3*b^3 + 8*a*b^5)*cos(d*x + c)^4 - 2*(a^5*b - 8*a^3*b^3 + 8*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d
*x + c) + 1/2) + 4*(2*(a^5*b - 25*a^3*b^3 + 30*a*b^5)*cos(d*x + c)^5 + 5*(3*a^5*b + 16*a^3*b^3 - 24*a*b^5)*cos
(d*x + c)^3 - 15*(a^5*b + 2*a^3*b^3 - 4*a*b^5)*cos(d*x + c))*sin(d*x + c))/(a^7*b^2*d*cos(d*x + c)^6 - (a^9 +
3*a^7*b^2)*d*cos(d*x + c)^4 + (2*a^9 + 3*a^7*b^2)*d*cos(d*x + c)^2 - (a^9 + a^7*b^2)*d - 2*(a^8*b*d*cos(d*x +
c)^4 - 2*a^8*b*d*cos(d*x + c)^2 + a^8*b*d)*sin(d*x + c))]

________________________________________________________________________________________

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)**5/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.35957, size = 814, normalized size = 2.29 \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)^5/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/64*(120*(a^4 - 8*a^2*b^2 + 8*b^4)*log(abs(tan(1/2*d*x + 1/2*c)))/a^7 - 960*(a^4*b - 3*a^2*b^3 + 2*b^5)*(pi*f
loor(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) + 64*(3*a^5*b*tan(1/2*d*x + 1/2*c)^3 - 15*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 12*a*b^5*tan(1/2*d*x + 1/2*c)^3
+ 2*a^6*tan(1/2*d*x + 1/2*c)^2 - 9*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 - 15*a^2*b^4*tan(1/2*d*x + 1/2*c)^2 + 22*b^
6*tan(1/2*d*x + 1/2*c)^2 + 5*a^5*b*tan(1/2*d*x + 1/2*c) - 37*a^3*b^3*tan(1/2*d*x + 1/2*c) + 32*a*b^5*tan(1/2*d
*x + 1/2*c) + 2*a^6 - 13*a^4*b^2 + 11*a^2*b^4)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2*a^
7) - (250*a^4*tan(1/2*d*x + 1/2*c)^4 - 2000*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 2000*b^4*tan(1/2*d*x + 1/2*c)^4 +
216*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 320*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 16*a^4*tan(1/2*d*x + 1/2*c)^2 + 48*a^2*
b^2*tan(1/2*d*x + 1/2*c)^2 - 8*a^3*b*tan(1/2*d*x + 1/2*c) + a^4)/(a^7*tan(1/2*d*x + 1/2*c)^4) + (a^9*tan(1/2*d
*x + 1/2*c)^4 - 8*a^8*b*tan(1/2*d*x + 1/2*c)^3 - 16*a^9*tan(1/2*d*x + 1/2*c)^2 + 48*a^7*b^2*tan(1/2*d*x + 1/2*
c)^2 + 216*a^8*b*tan(1/2*d*x + 1/2*c) - 320*a^6*b^3*tan(1/2*d*x + 1/2*c))/a^12)/d