\(\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^3 \, dx\) [1118]

Optimal result

Integrand size = 27, antiderivative size = 250 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {1}{16} b \left (18 a^2+b^2\right ) x-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {a \left (2 a^4-43 a^2 b^2+36 b^4\right ) \cos (c+d x)}{60 b^2 d}-\frac {\left (4 a^4-84 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin (c+d x)}{240 b d}-\frac {a \left (2 a^2-39 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{120 b^2 d}-\frac {\left (2 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 b^2 d}+\frac {a \cos (c+d x) (a+b \sin (c+d x))^4}{15 b^2 d}-\frac {\cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^4}{6 b d} \]

[Out]

1/16*b*(18*a^2+b^2)*x-a^3*arctanh(cos(d*x+c))/d-1/60*a*(2*a^4-43*a^2*b^2+36*b^4)*cos(d*x+c)/b^2/d-1/240*(4*a^4
-84*a^2*b^2+15*b^4)*cos(d*x+c)*sin(d*x+c)/b/d-1/120*a*(2*a^2-39*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^2/b^2/d-1/120
*(2*a^2-35*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^3/b^2/d+1/15*a*cos(d*x+c)*(a+b*sin(d*x+c))^4/b^2/d-1/6*cos(d*x+c)*
sin(d*x+c)*(a+b*sin(d*x+c))^4/b/d

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2974, 3128, 3112, 3102, 2814, 3855} \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {\left (2 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 b^2 d}-\frac {a \left (2 a^2-39 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{120 b^2 d}+\frac {1}{16} b x \left (18 a^2+b^2\right )-\frac {a \left (2 a^4-43 a^2 b^2+36 b^4\right ) \cos (c+d x)}{60 b^2 d}-\frac {\left (4 a^4-84 a^2 b^2+15 b^4\right ) \sin (c+d x) \cos (c+d x)}{240 b d}+\frac {a \cos (c+d x) (a+b \sin (c+d x))^4}{15 b^2 d}-\frac {\sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{6 b d} \]

[In]

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

[Out]

(b*(18*a^2 + b^2)*x)/16 - (a^3*ArcTanh[Cos[c + d*x]])/d - (a*(2*a^4 - 43*a^2*b^2 + 36*b^4)*Cos[c + d*x])/(60*b
^2*d) - ((4*a^4 - 84*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Sin[c + d*x])/(240*b*d) - (a*(2*a^2 - 39*b^2)*Cos[c + d*x]
*(a + b*Sin[c + d*x])^2)/(120*b^2*d) - ((2*a^2 - 35*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(120*b^2*d) + (a
*Cos[c + d*x]*(a + b*Sin[c + d*x])^4)/(15*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^4)/(6*b*d)

Rule 2814

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

Rule 2974

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

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3112

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a +
b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*
c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e
+ f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

Rule 3128

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d
^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

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 {a \cos (c+d x) (a+b \sin (c+d x))^4}{15 b^2 d}-\frac {\cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^4}{6 b d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x))^3 \left (-30 b^2+3 a b \sin (c+d x)-\left (2 a^2-35 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{30 b^2} \\ & = -\frac {\left (2 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 b^2 d}+\frac {a \cos (c+d x) (a+b \sin (c+d x))^4}{15 b^2 d}-\frac {\cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^4}{6 b d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x))^2 \left (-120 a b^2+3 b \left (2 a^2-5 b^2\right ) \sin (c+d x)-3 a \left (2 a^2-39 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{120 b^2} \\ & = -\frac {a \left (2 a^2-39 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{120 b^2 d}-\frac {\left (2 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 b^2 d}+\frac {a \cos (c+d x) (a+b \sin (c+d x))^4}{15 b^2 d}-\frac {\cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^4}{6 b d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x)) \left (-360 a^2 b^2+3 a b \left (2 a^2-57 b^2\right ) \sin (c+d x)-3 \left (4 a^4-84 a^2 b^2+15 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{360 b^2} \\ & = -\frac {\left (4 a^4-84 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin (c+d x)}{240 b d}-\frac {a \left (2 a^2-39 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{120 b^2 d}-\frac {\left (2 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 b^2 d}+\frac {a \cos (c+d x) (a+b \sin (c+d x))^4}{15 b^2 d}-\frac {\cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^4}{6 b d}-\frac {\int \csc (c+d x) \left (-720 a^3 b^2-45 b^3 \left (18 a^2+b^2\right ) \sin (c+d x)-12 a \left (2 a^4-43 a^2 b^2+36 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{720 b^2} \\ & = -\frac {a \left (2 a^4-43 a^2 b^2+36 b^4\right ) \cos (c+d x)}{60 b^2 d}-\frac {\left (4 a^4-84 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin (c+d x)}{240 b d}-\frac {a \left (2 a^2-39 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{120 b^2 d}-\frac {\left (2 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 b^2 d}+\frac {a \cos (c+d x) (a+b \sin (c+d x))^4}{15 b^2 d}-\frac {\cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^4}{6 b d}-\frac {\int \csc (c+d x) \left (-720 a^3 b^2-45 b^3 \left (18 a^2+b^2\right ) \sin (c+d x)\right ) \, dx}{720 b^2} \\ & = \frac {1}{16} b \left (18 a^2+b^2\right ) x-\frac {a \left (2 a^4-43 a^2 b^2+36 b^4\right ) \cos (c+d x)}{60 b^2 d}-\frac {\left (4 a^4-84 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin (c+d x)}{240 b d}-\frac {a \left (2 a^2-39 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{120 b^2 d}-\frac {\left (2 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 b^2 d}+\frac {a \cos (c+d x) (a+b \sin (c+d x))^4}{15 b^2 d}-\frac {\cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^4}{6 b d}+a^3 \int \csc (c+d x) \, dx \\ & = \frac {1}{16} b \left (18 a^2+b^2\right ) x-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {a \left (2 a^4-43 a^2 b^2+36 b^4\right ) \cos (c+d x)}{60 b^2 d}-\frac {\left (4 a^4-84 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin (c+d x)}{240 b d}-\frac {a \left (2 a^2-39 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{120 b^2 d}-\frac {\left (2 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 b^2 d}+\frac {a \cos (c+d x) (a+b \sin (c+d x))^4}{15 b^2 d}-\frac {\cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^4}{6 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.76 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {1080 a^2 b c+60 b^3 c+1080 a^2 b d x+60 b^3 d x+120 a \left (10 a^2-3 b^2\right ) \cos (c+d x)+20 \left (4 a^3-9 a b^2\right ) \cos (3 (c+d x))-36 a b^2 \cos (5 (c+d x))-960 a^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+960 a^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+720 a^2 b \sin (2 (c+d x))+15 b^3 \sin (2 (c+d x))+90 a^2 b \sin (4 (c+d x))-15 b^3 \sin (4 (c+d x))-5 b^3 \sin (6 (c+d x))}{960 d} \]

[In]

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

[Out]

(1080*a^2*b*c + 60*b^3*c + 1080*a^2*b*d*x + 60*b^3*d*x + 120*a*(10*a^2 - 3*b^2)*Cos[c + d*x] + 20*(4*a^3 - 9*a
*b^2)*Cos[3*(c + d*x)] - 36*a*b^2*Cos[5*(c + d*x)] - 960*a^3*Log[Cos[(c + d*x)/2]] + 960*a^3*Log[Sin[(c + d*x)
/2]] + 720*a^2*b*Sin[2*(c + d*x)] + 15*b^3*Sin[2*(c + d*x)] + 90*a^2*b*Sin[4*(c + d*x)] - 15*b^3*Sin[4*(c + d*
x)] - 5*b^3*Sin[6*(c + d*x)])/(960*d)

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.60

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{2} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {3 a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+b^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )}{d}\)	\(149\)
default	\(\frac {a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{2} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {3 a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+b^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )}{d}\)	\(149\)
parallelrisch	\(\frac {1080 a^{2} b d x +60 b^{3} d x -5 b^{3} \sin \left (6 d x +6 c \right )-36 a \,b^{2} \cos \left (5 d x +5 c \right )+90 a^{2} b \sin \left (4 d x +4 c \right )-15 b^{3} \sin \left (4 d x +4 c \right )+720 a^{2} b \sin \left (2 d x +2 c \right )+15 b^{3} \sin \left (2 d x +2 c \right )+80 a^{3} \cos \left (3 d x +3 c \right )-180 a \,b^{2} \cos \left (3 d x +3 c \right )+1200 \cos \left (d x +c \right ) a^{3}-360 \cos \left (d x +c \right ) a \,b^{2}+960 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1280 a^{3}-576 a \,b^{2}}{960 d}\)	\(187\)
risch	\(\frac {9 a^{2} b x}{8}+\frac {b^{3} x}{16}+\frac {5 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {3 \,{\mathrm e}^{i \left (d x +c \right )} a \,b^{2}}{16 d}+\frac {5 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )} a \,b^{2}}{16 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}-\frac {b^{3} \sin \left (6 d x +6 c \right )}{192 d}-\frac {3 \cos \left (5 d x +5 c \right ) a \,b^{2}}{80 d}+\frac {3 b \sin \left (4 d x +4 c \right ) a^{2}}{32 d}-\frac {b^{3} \sin \left (4 d x +4 c \right )}{64 d}+\frac {a^{3} \cos \left (3 d x +3 c \right )}{12 d}-\frac {3 \cos \left (3 d x +3 c \right ) a \,b^{2}}{16 d}+\frac {3 a^{2} b \sin \left (2 d x +2 c \right )}{4 d}+\frac {b^{3} \sin \left (2 d x +2 c \right )}{64 d}\)	\(264\)
norman	\(\frac {\left (\frac {9}{8} a^{2} b +\frac {1}{16} b^{3}\right ) x +\left (\frac {9}{8} a^{2} b +\frac {1}{16} b^{3}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {27}{4} a^{2} b +\frac {3}{8} b^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {27}{4} a^{2} b +\frac {3}{8} b^{3}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {45}{2} a^{2} b +\frac {5}{4} b^{3}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {135}{8} a^{2} b +\frac {15}{16} b^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {135}{8} a^{2} b +\frac {15}{16} b^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (16 a^{3}-6 a \,b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {40 a^{3}-18 a \,b^{2}}{15 d}+\frac {2 \left (2 a^{3}-3 a \,b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {3 \left (8 a^{3}-4 a \,b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {6 \left (10 a^{3}-a \,b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {4 \left (20 a^{3}-9 a \,b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {b \left (6 a^{2}-13 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {b \left (6 a^{2}-13 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {b \left (30 a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {b \left (30 a^{2}-b^{2}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {b \left (126 a^{2}+47 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {b \left (126 a^{2}+47 b^{2}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(522\)

[In]

int(cos(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^3*(1/3*cos(d*x+c)^3+cos(d*x+c)+ln(csc(d*x+c)-cot(d*x+c)))+3*a^2*b*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*si
n(d*x+c)+3/8*d*x+3/8*c)-3/5*a*b^2*cos(d*x+c)^5+b^3*(-1/6*sin(d*x+c)*cos(d*x+c)^5+1/24*(cos(d*x+c)^3+3/2*cos(d*
x+c))*sin(d*x+c)+1/16*d*x+1/16*c))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.60 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {144 \, a b^{2} \cos \left (d x + c\right )^{5} - 80 \, a^{3} \cos \left (d x + c\right )^{3} - 240 \, a^{3} \cos \left (d x + c\right ) + 120 \, a^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 120 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 15 \, {\left (18 \, a^{2} b + b^{3}\right )} d x + 5 \, {\left (8 \, b^{3} \cos \left (d x + c\right )^{5} - 2 \, {\left (18 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (18 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/240*(144*a*b^2*cos(d*x + c)^5 - 80*a^3*cos(d*x + c)^3 - 240*a^3*cos(d*x + c) + 120*a^3*log(1/2*cos(d*x + c)
+ 1/2) - 120*a^3*log(-1/2*cos(d*x + c) + 1/2) - 15*(18*a^2*b + b^3)*d*x + 5*(8*b^3*cos(d*x + c)^5 - 2*(18*a^2
*b + b^3)*cos(d*x + c)^3 - 3*(18*a^2*b + b^3)*cos(d*x + c))*sin(d*x + c))/d

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)*(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.55 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {576 \, a b^{2} \cos \left (d x + c\right )^{5} - 160 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} b - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{3}}{960 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/960*(576*a*b^2*cos(d*x + c)^5 - 160*(2*cos(d*x + c)^3 + 6*cos(d*x + c) - 3*log(cos(d*x + c) + 1) + 3*log(co
s(d*x + c) - 1))*a^3 - 90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^2*b - 5*(4*sin(2*d*x + 2*c
)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*b^3)/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.71 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {240 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 15 \, {\left (18 \, a^{2} b + b^{3}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (450 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 15 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 480 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 720 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 630 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 235 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 720 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 180 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 390 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3200 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1440 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 180 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 390 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1440 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 630 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 235 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1440 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 144 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 450 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 320 \, a^{3} + 144 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/240*(240*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 15*(18*a^2*b + b^3)*(d*x + c) - 2*(450*a^2*b*tan(1/2*d*x + 1/2
*c)^11 - 15*b^3*tan(1/2*d*x + 1/2*c)^11 - 480*a^3*tan(1/2*d*x + 1/2*c)^10 + 720*a*b^2*tan(1/2*d*x + 1/2*c)^10
+ 630*a^2*b*tan(1/2*d*x + 1/2*c)^9 + 235*b^3*tan(1/2*d*x + 1/2*c)^9 - 1920*a^3*tan(1/2*d*x + 1/2*c)^8 + 720*a*
b^2*tan(1/2*d*x + 1/2*c)^8 + 180*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 390*b^3*tan(1/2*d*x + 1/2*c)^7 - 3200*a^3*tan(
1/2*d*x + 1/2*c)^6 + 1440*a*b^2*tan(1/2*d*x + 1/2*c)^6 - 180*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 390*b^3*tan(1/2*d*
x + 1/2*c)^5 - 2880*a^3*tan(1/2*d*x + 1/2*c)^4 + 1440*a*b^2*tan(1/2*d*x + 1/2*c)^4 - 630*a^2*b*tan(1/2*d*x + 1
/2*c)^3 - 235*b^3*tan(1/2*d*x + 1/2*c)^3 - 1440*a^3*tan(1/2*d*x + 1/2*c)^2 + 144*a*b^2*tan(1/2*d*x + 1/2*c)^2
- 450*a^2*b*tan(1/2*d*x + 1/2*c) + 15*b^3*tan(1/2*d*x + 1/2*c) - 320*a^3 + 144*a*b^2)/(tan(1/2*d*x + 1/2*c)^2
+ 1)^6)/d

Mupad [B] (verification not implemented)

Time = 12.23 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.76 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {\frac {6\,a\,b^2}{5}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {15\,a^2\,b}{4}-\frac {b^3}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (6\,a\,b^2-4\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {6\,a\,b^2}{5}-12\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (6\,a\,b^2-16\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (12\,a\,b^2-24\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (12\,a\,b^2-\frac {80\,a^3}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,a^2\,b}{2}-\frac {13\,b^3}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {3\,a^2\,b}{2}-\frac {13\,b^3}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {15\,a^2\,b}{4}-\frac {b^3}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {21\,a^2\,b}{4}+\frac {47\,b^3}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {21\,a^2\,b}{4}+\frac {47\,b^3}{24}\right )-\frac {8\,a^3}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {b\,\mathrm {atan}\left (\frac {\frac {b\,\left (18\,a^2+b^2\right )\,\left (\frac {9\,a^2\,b}{4}+\frac {b^3}{8}+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (18\,a^2+b^2\right )\,3{}\mathrm {i}}{8}\right )}{16}+\frac {b\,\left (18\,a^2+b^2\right )\,\left (\frac {9\,a^2\,b}{4}+\frac {b^3}{8}+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (18\,a^2+b^2\right )\,3{}\mathrm {i}}{8}\right )}{16}}{2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {81\,a^4\,b^2}{16}+\frac {9\,a^2\,b^4}{16}+\frac {b^6}{64}\right )+\frac {9\,a^5\,b}{2}+\frac {a^3\,b^3}{4}-\frac {b\,\left (18\,a^2+b^2\right )\,\left (\frac {9\,a^2\,b}{4}+\frac {b^3}{8}+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (18\,a^2+b^2\right )\,3{}\mathrm {i}}{8}\right )\,1{}\mathrm {i}}{16}+\frac {b\,\left (18\,a^2+b^2\right )\,\left (\frac {9\,a^2\,b}{4}+\frac {b^3}{8}+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (18\,a^2+b^2\right )\,3{}\mathrm {i}}{8}\right )\,1{}\mathrm {i}}{16}}\right )\,\left (18\,a^2+b^2\right )}{8\,d} \]

[In]

int((cos(c + d*x)^4*(a + b*sin(c + d*x))^3)/sin(c + d*x),x)

[Out]

(a^3*log(tan(c/2 + (d*x)/2)))/d - ((6*a*b^2)/5 - tan(c/2 + (d*x)/2)*((15*a^2*b)/4 - b^3/8) + tan(c/2 + (d*x)/2
)^10*(6*a*b^2 - 4*a^3) + tan(c/2 + (d*x)/2)^2*((6*a*b^2)/5 - 12*a^3) + tan(c/2 + (d*x)/2)^8*(6*a*b^2 - 16*a^3)
+ tan(c/2 + (d*x)/2)^4*(12*a*b^2 - 24*a^3) + tan(c/2 + (d*x)/2)^6*(12*a*b^2 - (80*a^3)/3) - tan(c/2 + (d*x)/2
)^5*((3*a^2*b)/2 - (13*b^3)/4) + tan(c/2 + (d*x)/2)^7*((3*a^2*b)/2 - (13*b^3)/4) + tan(c/2 + (d*x)/2)^11*((15*
a^2*b)/4 - b^3/8) - tan(c/2 + (d*x)/2)^3*((21*a^2*b)/4 + (47*b^3)/24) + tan(c/2 + (d*x)/2)^9*((21*a^2*b)/4 + (
47*b^3)/24) - (8*a^3)/3)/(d*(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c/2 + (d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*t
an(c/2 + (d*x)/2)^8 + 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (b*atan(((b*(18*a^2 + b^2)*((9*a
^2*b)/4 + b^3/8 + 2*a^3*tan(c/2 + (d*x)/2) - (b*tan(c/2 + (d*x)/2)*(18*a^2 + b^2)*3i)/8))/16 + (b*(18*a^2 + b^
2)*((9*a^2*b)/4 + b^3/8 + 2*a^3*tan(c/2 + (d*x)/2) + (b*tan(c/2 + (d*x)/2)*(18*a^2 + b^2)*3i)/8))/16)/(2*tan(c
/2 + (d*x)/2)*(b^6/64 + (9*a^2*b^4)/16 + (81*a^4*b^2)/16) + (9*a^5*b)/2 + (a^3*b^3)/4 - (b*(18*a^2 + b^2)*((9*
a^2*b)/4 + b^3/8 + 2*a^3*tan(c/2 + (d*x)/2) - (b*tan(c/2 + (d*x)/2)*(18*a^2 + b^2)*3i)/8)*1i)/16 + (b*(18*a^2
+ b^2)*((9*a^2*b)/4 + b^3/8 + 2*a^3*tan(c/2 + (d*x)/2) + (b*tan(c/2 + (d*x)/2)*(18*a^2 + b^2)*3i)/8)*1i)/16))*
(18*a^2 + b^2))/(8*d)