\(\int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx\) [396]

Optimal result

Integrand size = 27, antiderivative size = 143 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {19 a^3 x}{16}-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {19 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d} \]

[Out]

19/16*a^3*x-a^3*arctanh(cos(d*x+c))/d+a^3*cos(d*x+c)/d+1/3*a^3*cos(d*x+c)^3/d-3/5*a^3*cos(d*x+c)^5/d+19/16*a^3
*cos(d*x+c)*sin(d*x+c)/d+19/24*a^3*cos(d*x+c)^3*sin(d*x+c)/d-1/6*a^3*cos(d*x+c)^5*sin(d*x+c)/d

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2952, 2715, 8, 2672, 308, 212, 2645, 30, 2648} \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {19 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {19 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {19 a^3 x}{16} \]

[In]

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

[Out]

(19*a^3*x)/16 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) - (3*a^3*Cos
[c + d*x]^5)/(5*d) + (19*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (19*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) -
(a^3*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 212

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

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2645

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 2648

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

Rule 2672

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2952

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (3 a^3 \cos ^4(c+d x)+a^3 \cos ^3(c+d x) \cot (c+d x)+3 a^3 \cos ^4(c+d x) \sin (c+d x)+a^3 \cos ^4(c+d x) \sin ^2(c+d x)\right ) \, dx \\ & = a^3 \int \cos ^3(c+d x) \cot (c+d x) \, dx+a^3 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \sin (c+d x) \, dx \\ & = \frac {3 a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{6} a^3 \int \cos ^4(c+d x) \, dx+\frac {1}{4} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^4 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {9 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{8} a^3 \int \cos ^2(c+d x) \, dx+\frac {1}{8} \left (9 a^3\right ) \int 1 \, dx-\frac {a^3 \text {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {9 a^3 x}{8}+\frac {a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {19 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{16} a^3 \int 1 \, dx-\frac {a^3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {19 a^3 x}{16}-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {19 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.89 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.71 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (1140 c+1140 d x+840 \cos (c+d x)-100 \cos (3 (c+d x))-36 \cos (5 (c+d x))-960 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+960 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+735 \sin (2 (c+d x))+75 \sin (4 (c+d x))-5 \sin (6 (c+d x))\right )}{960 d} \]

[In]

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

[Out]

(a^3*(1140*c + 1140*d*x + 840*Cos[c + d*x] - 100*Cos[3*(c + d*x)] - 36*Cos[5*(c + d*x)] - 960*Log[Cos[(c + d*x
)/2]] + 960*Log[Sin[(c + d*x)/2]] + 735*Sin[2*(c + d*x)] + 75*Sin[4*(c + d*x)] - 5*Sin[6*(c + d*x)]))/(960*d)

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.62

method	result	size
parallelrisch	\(-\frac {\left (-228 d x +\sin \left (6 d x +6 c \right )-192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-168 \cos \left (d x +c \right )+20 \cos \left (3 d x +3 c \right )+\frac {36 \cos \left (5 d x +5 c \right )}{5}-147 \sin \left (2 d x +2 c \right )-15 \sin \left (4 d x +4 c \right )-\frac {704}{5}\right ) a^{3}}{192 d}\)	\(88\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+3 a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(147\)
default	\(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+3 a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(147\)
risch	\(\frac {19 a^{3} x}{16}+\frac {7 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 d}+\frac {7 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}-\frac {3 a^{3} \cos \left (5 d x +5 c \right )}{80 d}+\frac {5 a^{3} \sin \left (4 d x +4 c \right )}{64 d}-\frac {5 a^{3} \cos \left (3 d x +3 c \right )}{48 d}+\frac {49 a^{3} \sin \left (2 d x +2 c \right )}{64 d}\)	\(166\)
norman	\(\frac {\frac {19 a^{3} x}{16}+\frac {22 a^{3}}{15 d}+\frac {29 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {173 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {7 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {7 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {173 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {29 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {57 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {285 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {95 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {285 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {57 a^{3} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {19 a^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {2 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {10 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {12 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {54 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {44 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(359\)

[In]

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

[Out]

-1/192*(-228*d*x+sin(6*d*x+6*c)-192*ln(tan(1/2*d*x+1/2*c))-168*cos(d*x+c)+20*cos(3*d*x+3*c)+36/5*cos(5*d*x+5*c
)-147*sin(2*d*x+2*c)-15*sin(4*d*x+4*c)-704/5)*a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.90 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {144 \, a^{3} \cos \left (d x + c\right )^{5} - 80 \, a^{3} \cos \left (d x + c\right )^{3} - 285 \, a^{3} d x - 240 \, a^{3} \cos \left (d x + c\right ) + 120 \, a^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 120 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 5 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{5} - 38 \, a^{3} \cos \left (d x + c\right )^{3} - 57 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]

[In]

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

[Out]

-1/240*(144*a^3*cos(d*x + c)^5 - 80*a^3*cos(d*x + c)^3 - 285*a^3*d*x - 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) + 5*(8*a^3*cos(d*x + c)^5 - 38*a^3*cos(d*x + c)^3 -
57*a^3*cos(d*x + c))*sin(d*x + c))/d

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.94 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {576 \, a^{3} \cos \left (d x + c\right )^{5} - 160 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3}}{960 \, d} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.60 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {285 \, {\left (d x + c\right )} a^{3} + 240 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (435 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 240 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 865 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1200 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 210 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1760 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 210 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1440 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 865 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1296 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 435 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 176 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]

[In]

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

[Out]

1/240*(285*(d*x + c)*a^3 + 240*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 2*(435*a^3*tan(1/2*d*x + 1/2*c)^11 + 240*a
^3*tan(1/2*d*x + 1/2*c)^10 + 865*a^3*tan(1/2*d*x + 1/2*c)^9 - 1200*a^3*tan(1/2*d*x + 1/2*c)^8 - 210*a^3*tan(1/
2*d*x + 1/2*c)^7 - 1760*a^3*tan(1/2*d*x + 1/2*c)^6 + 210*a^3*tan(1/2*d*x + 1/2*c)^5 - 1440*a^3*tan(1/2*d*x + 1
/2*c)^4 - 865*a^3*tan(1/2*d*x + 1/2*c)^3 - 1296*a^3*tan(1/2*d*x + 1/2*c)^2 - 435*a^3*tan(1/2*d*x + 1/2*c) - 17
6*a^3)/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d

Mupad [B] (verification not implemented)

Time = 12.39 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.48 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {19\,a^3\,\mathrm {atan}\left (\frac {361\,a^6}{64\,\left (\frac {19\,a^6}{4}-\frac {361\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}+\frac {19\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {19\,a^6}{4}-\frac {361\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}\right )}{8\,d}+\frac {-\frac {29\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}-2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {173\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+10\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {44\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {173\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {54\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {29\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {22\,a^3}{15}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

[In]

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

[Out]

(a^3*log(tan(c/2 + (d*x)/2)))/d + (19*a^3*atan((361*a^6)/(64*((19*a^6)/4 - (361*a^6*tan(c/2 + (d*x)/2))/64)) +
(19*a^6*tan(c/2 + (d*x)/2))/(4*((19*a^6)/4 - (361*a^6*tan(c/2 + (d*x)/2))/64))))/(8*d) + ((54*a^3*tan(c/2 + (
d*x)/2)^2)/5 + (173*a^3*tan(c/2 + (d*x)/2)^3)/24 + 12*a^3*tan(c/2 + (d*x)/2)^4 - (7*a^3*tan(c/2 + (d*x)/2)^5)/
4 + (44*a^3*tan(c/2 + (d*x)/2)^6)/3 + (7*a^3*tan(c/2 + (d*x)/2)^7)/4 + 10*a^3*tan(c/2 + (d*x)/2)^8 - (173*a^3*
tan(c/2 + (d*x)/2)^9)/24 - 2*a^3*tan(c/2 + (d*x)/2)^10 - (29*a^3*tan(c/2 + (d*x)/2)^11)/8 + (22*a^3)/15 + (29*
a^3*tan(c/2 + (d*x)/2))/8)/(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
*tan(c/2 + (d*x)/2)^8 + 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1))