3.1328 \(\int \frac{\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=241 \[ -\frac{2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^6 d}-\frac{\left (-35 a^2 b^2+23 a^4+15 b^4\right ) \cot (c+d x)}{15 a^5 d}+\frac{b \left (-20 a^2 b^2+15 a^4+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}+\frac{\left (11 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac{b \left (4 b^2-9 a^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d} \]

[Out]

(-2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*d) + (b*(15*a^4 - 20*a^2*b^2 + 8*
b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) - ((23*a^4 - 35*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(15*a^5*d) + (b*(-9*a^2
+ 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*d) + ((11*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*a^3*d) + (b
*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d)

________________________________________________________________________________________

Rubi [A]  time = 1.11981, antiderivative size = 307, normalized size of antiderivative = 1.27, number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2726, 3055, 3001, 3770, 2660, 618, 204} \[ -\frac{2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^6 d}-\frac{\left (-35 a^2 b^2+23 a^4+15 b^4\right ) \cot (c+d x)}{15 a^5 d}+\frac{b \left (-20 a^2 b^2+15 a^4+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac{\left (-22 a^2 b^2+15 a^4+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^2 d}+\frac{\left (-9 a^2 b^2+8 a^4+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac{\cot (c+d x) \csc (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*d) + (b*(15*a^4 - 20*a^2*b^2 + 8*
b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) - ((23*a^4 - 35*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(15*a^5*d) - (Cot[c + d*
x]*Csc[c + d*x])/(b*d) + ((8*a^4 - 9*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*b*d) + (a*Cot[c + d*x]
*Csc[c + d*x]^2)/(2*b^2*d) - ((15*a^4 - 22*a^2*b^2 + 10*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^3*b^2*d) + (b*
Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d)

Rule 2726

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^6, x_Symbol] :> -Simp[(Cos[e + f*x]*(a
+ b*Sin[e + f*x])^(m + 1))/(5*a*f*Sin[e + f*x]^5), x] + (Dist[1/(20*a^2*b^2*m*(m - 1)), Int[((a + b*Sin[e + f
*x])^m*Simp[60*a^4 - 44*a^2*b^2*(m - 1)*m + b^4*m*(m - 1)*(m - 3)*(m - 4) + a*b*m*(20*a^2 - b^2*m*(m - 1))*Sin
[e + f*x] - (40*a^4 + b^4*m*(m - 1)*(m - 2)*(m - 4) - 20*a^2*b^2*(m - 1)*(2*m + 1))*Sin[e + f*x]^2, x])/Sin[e
+ f*x]^4, x], x] + Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*m*Sin[e + f*x]^2), x] + Simp[(a*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*m*(m - 1)*Sin[e + f*x]^3), x] - Simp[(b*(m - 4)*Cos[e + f*x]*(a +
b*Sin[e + f*x])^(m + 1))/(20*a^2*f*Sin[e + f*x]^4), x]) /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] &&
NeQ[m, 1] && IntegerQ[2*m]

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)}{a+b \sin (c+d x)} \, dx &=-\frac{\cot (c+d x) \csc (c+d x)}{b d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac{\int \frac{\csc ^4(c+d x) \left (4 \left (15 a^4-22 a^2 b^2+10 b^4\right )-2 a b \left (10 a^2-b^2\right ) \sin (c+d x)-10 \left (4 a^4-4 a^2 b^2+3 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{40 a^2 b^2}\\ &=-\frac{\cot (c+d x) \csc (c+d x)}{b d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 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^3 b^2 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac{\int \frac{\csc ^3(c+d x) \left (-30 b \left (8 a^4-9 a^2 b^2+4 b^4\right )-2 a b^2 \left (28 a^2+5 b^2\right ) \sin (c+d x)+8 b \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}{120 a^3 b^2}\\ &=-\frac{\cot (c+d x) \csc (c+d x)}{b d}+\frac{\left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 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^3 b^2 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac{\int \frac{\csc ^2(c+d x) \left (16 b^2 \left (23 a^4-35 a^2 b^2+15 b^4\right )-2 a b^3 \left (41 a^2-20 b^2\right ) \sin (c+d x)-30 b^2 \left (8 a^4-9 a^2 b^2+4 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{240 a^4 b^2}\\ &=-\frac{\left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac{\cot (c+d x) \csc (c+d x)}{b d}+\frac{\left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 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^3 b^2 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac{\int \frac{\csc (c+d x) \left (-30 b^3 \left (15 a^4-20 a^2 b^2+8 b^4\right )-30 a b^2 \left (8 a^4-9 a^2 b^2+4 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{240 a^5 b^2}\\ &=-\frac{\left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac{\cot (c+d x) \csc (c+d x)}{b d}+\frac{\left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 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^3 b^2 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac{\left (a^2-b^2\right )^3 \int \frac{1}{a+b \sin (c+d x)} \, dx}{a^6}-\frac{\left (b \left (15 a^4-20 a^2 b^2+8 b^4\right )\right ) \int \csc (c+d x) \, dx}{8 a^6}\\ &=\frac{b \left (15 a^4-20 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac{\left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac{\cot (c+d x) \csc (c+d x)}{b d}+\frac{\left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 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^3 b^2 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac{\left (2 \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^6 d}\\ &=\frac{b \left (15 a^4-20 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac{\left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac{\cot (c+d x) \csc (c+d x)}{b d}+\frac{\left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 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^3 b^2 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac{\left (4 \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^6 d}\\ &=-\frac{2 \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^6 d}+\frac{b \left (15 a^4-20 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac{\left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac{\cot (c+d x) \csc (c+d x)}{b d}+\frac{\left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 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^3 b^2 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}\\ \end{align*}

Mathematica [B]  time = 1.36323, size = 504, normalized size = 2.09 \[ \frac{-1920 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )-1120 a^3 b^2 \tan \left (\frac{1}{2} (c+d x)\right )-32 \left (-35 a^3 b^2+23 a^5+15 a b^4\right ) \cot \left (\frac{1}{2} (c+d x)\right )+120 a^2 b^3 \csc ^2\left (\frac{1}{2} (c+d x)\right )-120 a^2 b^3 \sec ^2\left (\frac{1}{2} (c+d x)\right )+2400 a^2 b^3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2400 a^2 b^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+320 a^3 b^2 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-20 a^3 b^2 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )+15 a^4 b \csc ^4\left (\frac{1}{2} (c+d x)\right )-270 a^4 b \csc ^2\left (\frac{1}{2} (c+d x)\right )-15 a^4 b \sec ^4\left (\frac{1}{2} (c+d x)\right )+270 a^4 b \sec ^2\left (\frac{1}{2} (c+d x)\right )-1800 a^4 b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+1800 a^4 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+736 a^5 \tan \left (\frac{1}{2} (c+d x)\right )-656 a^5 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-3 a^5 \sin (c+d x) \csc ^6\left (\frac{1}{2} (c+d x)\right )+41 a^5 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )+6 a^5 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )+480 a b^4 \tan \left (\frac{1}{2} (c+d x)\right )-960 b^5 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+960 b^5 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{960 a^6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1920*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 32*(23*a^5 - 35*a^3*b^2 + 15*a*b^4
)*Cot[(c + d*x)/2] - 270*a^4*b*Csc[(c + d*x)/2]^2 + 120*a^2*b^3*Csc[(c + d*x)/2]^2 + 15*a^4*b*Csc[(c + d*x)/2]
^4 + 1800*a^4*b*Log[Cos[(c + d*x)/2]] - 2400*a^2*b^3*Log[Cos[(c + d*x)/2]] + 960*b^5*Log[Cos[(c + d*x)/2]] - 1
800*a^4*b*Log[Sin[(c + d*x)/2]] + 2400*a^2*b^3*Log[Sin[(c + d*x)/2]] - 960*b^5*Log[Sin[(c + d*x)/2]] + 270*a^4
*b*Sec[(c + d*x)/2]^2 - 120*a^2*b^3*Sec[(c + d*x)/2]^2 - 15*a^4*b*Sec[(c + d*x)/2]^4 - 656*a^5*Csc[c + d*x]^3*
Sin[(c + d*x)/2]^4 + 320*a^3*b^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 41*a^5*Csc[(c + d*x)/2]^4*Sin[c + d*x] -
20*a^3*b^2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 3*a^5*Csc[(c + d*x)/2]^6*Sin[c + d*x] + 736*a^5*Tan[(c + d*x)/2]
- 1120*a^3*b^2*Tan[(c + d*x)/2] + 480*a*b^4*Tan[(c + d*x)/2] + 6*a^5*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(960
*a^6*d)

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Maple [B]  time = 0.126, size = 629, normalized size = 2.6 \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)^6/(a+b*sin(d*x+c)),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 4.37732, size = 2566, normalized size = 10.65 \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)^6/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

[-1/240*(16*(23*a^5 - 35*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5 - 80*(7*a^5 - 13*a^3*b^2 + 6*a*b^4)*cos(d*x + c)^3
- 120*((a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*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)*s
in(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))*sin(d*x
+ c) - 15*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*
a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 15*(15*a^4*b - 20*a^2*b^3 + 8*b^5
+ (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(-1/2*
cos(d*x + c) + 1/2)*sin(d*x + c) + 240*(a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c) - 30*((9*a^4*b - 4*a^2*b^3)*cos(
d*x + c)^3 - (7*a^4*b - 4*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/((a^6*d*cos(d*x + c)^4 - 2*a^6*d*cos(d*x + c)^2
+ a^6*d)*sin(d*x + c)), -1/240*(16*(23*a^5 - 35*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5 - 80*(7*a^5 - 13*a^3*b^2 +
6*a*b^4)*cos(d*x + c)^3 - 240*((a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^
2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c)))*sin(
d*x + c) - 15*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b -
20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 15*(15*a^4*b - 20*a^2*b^3 + 8*b
^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(-1
/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 240*(a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c) - 30*((9*a^4*b - 4*a^2*b^3)*c
os(d*x + c)^3 - (7*a^4*b - 4*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/((a^6*d*cos(d*x + c)^4 - 2*a^6*d*cos(d*x + c
)^2 + a^6*d)*sin(d*x + c))]

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

[Out]

Timed out

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Giac [B]  time = 1.26008, size = 662, normalized size = 2.75 \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)^6/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/960*((6*a^4*tan(1/2*d*x + 1/2*c)^5 - 15*a^3*b*tan(1/2*d*x + 1/2*c)^4 - 70*a^4*tan(1/2*d*x + 1/2*c)^3 + 40*a^
2*b^2*tan(1/2*d*x + 1/2*c)^3 + 240*a^3*b*tan(1/2*d*x + 1/2*c)^2 - 120*a*b^3*tan(1/2*d*x + 1/2*c)^2 + 660*a^4*t
an(1/2*d*x + 1/2*c) - 1080*a^2*b^2*tan(1/2*d*x + 1/2*c) + 480*b^4*tan(1/2*d*x + 1/2*c))/a^5 - 120*(15*a^4*b -
20*a^2*b^3 + 8*b^5)*log(abs(tan(1/2*d*x + 1/2*c)))/a^6 - 1920*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - 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)*a^6) + (
4110*a^4*b*tan(1/2*d*x + 1/2*c)^5 - 5480*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 + 2192*b^5*tan(1/2*d*x + 1/2*c)^5 - 66
0*a^5*tan(1/2*d*x + 1/2*c)^4 + 1080*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 - 480*a*b^4*tan(1/2*d*x + 1/2*c)^4 - 240*a^
4*b*tan(1/2*d*x + 1/2*c)^3 + 120*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 70*a^5*tan(1/2*d*x + 1/2*c)^2 - 40*a^3*b^2*t
an(1/2*d*x + 1/2*c)^2 + 15*a^4*b*tan(1/2*d*x + 1/2*c) - 6*a^5)/(a^6*tan(1/2*d*x + 1/2*c)^5))/d