\(\int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx\) [609]

Optimal result

Integrand size = 27, antiderivative size = 185 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {125 a^3 x}{128}-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cos ^7(c+d x)}{7 d}+\frac {125 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {125 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d} \]

[Out]

125/128*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+1/5*a^3*cos(d*x+c)^5/d-3/7*a^3
*cos(d*x+c)^7/d+125/128*a^3*cos(d*x+c)*sin(d*x+c)/d+125/192*a^3*cos(d*x+c)^3*sin(d*x+c)/d+25/48*a^3*cos(d*x+c)
^5*sin(d*x+c)/d-1/8*a^3*cos(d*x+c)^7*sin(d*x+c)/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 17, 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} \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {3 a^3 \cos ^7(c+d x)}{7 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {25 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {125 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {125 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {125 a^3 x}{128} \]

[In]

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

[Out]

(125*a^3*x)/128 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cos
[c + d*x]^5)/(5*d) - (3*a^3*Cos[c + d*x]^7)/(7*d) + (125*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (125*a^3*Cos
[c + d*x]^3*Sin[c + d*x])/(192*d) + (25*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a^3*Cos[c + d*x]^7*Sin[c +
d*x])/(8*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 ^6(c+d x)+a^3 \cos ^5(c+d x) \cot (c+d x)+3 a^3 \cos ^6(c+d x) \sin (c+d x)+a^3 \cos ^6(c+d x) \sin ^2(c+d x)\right ) \, dx \\ & = a^3 \int \cos ^5(c+d x) \cot (c+d x) \, dx+a^3 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin (c+d x) \, dx \\ & = \frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{2 d}-\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{8} a^3 \int \cos ^6(c+d x) \, dx+\frac {1}{2} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \frac {x^6}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {3 a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{8 d}+\frac {25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{48} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{8} \left (15 a^3\right ) \int \cos ^2(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \left (-1-x^2-x^4+\frac {1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cos ^7(c+d x)}{7 d}+\frac {15 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {125 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{64} \left (5 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{16} \left (15 a^3\right ) \int 1 \, dx-\frac {a^3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {15 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 {a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cos ^7(c+d x)}{7 d}+\frac {125 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {125 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{128} \left (5 a^3\right ) \int 1 \, dx \\ & = \frac {125 a^3 x}{128}-\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 {a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cos ^7(c+d x)}{7 d}+\frac {125 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {125 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.73 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.66 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (105000 c+105000 d x+122640 \cos (c+d x)+560 \cos (3 (c+d x))-3696 \cos (5 (c+d x))-720 \cos (7 (c+d x))-107520 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+107520 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+77280 \sin (2 (c+d x))+14280 \sin (4 (c+d x))+1120 \sin (6 (c+d x))-105 \sin (8 (c+d x))\right )}{107520 d} \]

[In]

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

[Out]

(a^3*(105000*c + 105000*d*x + 122640*Cos[c + d*x] + 560*Cos[3*(c + d*x)] - 3696*Cos[5*(c + d*x)] - 720*Cos[7*(
c + d*x)] - 107520*Log[Cos[(c + d*x)/2]] + 107520*Log[Sin[(c + d*x)/2]] + 77280*Sin[2*(c + d*x)] + 14280*Sin[4
*(c + d*x)] + 1120*Sin[6*(c + d*x)] - 105*Sin[8*(c + d*x)]))/(107520*d)

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.61

method	result	size
parallelrisch	\(\frac {a^{3} \left (105000 d x +122640 \cos \left (d x +c \right )-720 \cos \left (7 d x +7 c \right )+14280 \sin \left (4 d x +4 c \right )+77280 \sin \left (2 d x +2 c \right )-105 \sin \left (8 d x +8 c \right )+1120 \sin \left (6 d x +6 c \right )-3696 \cos \left (5 d x +5 c \right )+560 \cos \left (3 d x +3 c \right )+107520 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+118784\right )}{107520 d}\)	\(112\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+3 a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(177\)
default	\(\frac {a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+3 a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(177\)
risch	\(\frac {125 a^{3} x}{128}+\frac {73 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{128 d}+\frac {73 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{128 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}-\frac {a^{3} \sin \left (8 d x +8 c \right )}{1024 d}-\frac {3 a^{3} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{96 d}-\frac {11 a^{3} \cos \left (5 d x +5 c \right )}{320 d}+\frac {17 a^{3} \sin \left (4 d x +4 c \right )}{128 d}+\frac {a^{3} \cos \left (3 d x +3 c \right )}{192 d}+\frac {23 a^{3} \sin \left (2 d x +2 c \right )}{32 d}\)	\(200\)
norman	\(\frac {-\frac {259 a^{3} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {125 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {875 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {875 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {4375 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {875 a^{3} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {875 a^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {125 a^{3} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {125 a^{3} x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {1856 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 d}+\frac {125 a^{3} x}{128}+\frac {232 a^{3}}{105 d}-\frac {1861 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {1817 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {24 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {259 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {1216 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {232 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {1817 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {3805 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {3805 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {568 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {128 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {1861 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(450\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/107520*a^3*(105000*d*x+122640*cos(d*x+c)-720*cos(7*d*x+7*c)+14280*sin(4*d*x+4*c)+77280*sin(2*d*x+2*c)-105*si
n(8*d*x+8*c)+1120*sin(6*d*x+6*c)-3696*cos(5*d*x+5*c)+560*cos(3*d*x+3*c)+107520*ln(tan(1/2*d*x+1/2*c))+118784)/
d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.83 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {5760 \, a^{3} \cos \left (d x + c\right )^{7} - 2688 \, a^{3} \cos \left (d x + c\right )^{5} - 4480 \, a^{3} \cos \left (d x + c\right )^{3} - 13125 \, a^{3} d x - 13440 \, a^{3} \cos \left (d x + c\right ) + 6720 \, a^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 6720 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 35 \, {\left (48 \, a^{3} \cos \left (d x + c\right )^{7} - 200 \, a^{3} \cos \left (d x + c\right )^{5} - 250 \, a^{3} \cos \left (d x + c\right )^{3} - 375 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{13440 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/13440*(5760*a^3*cos(d*x + c)^7 - 2688*a^3*cos(d*x + c)^5 - 4480*a^3*cos(d*x + c)^3 - 13125*a^3*d*x - 13440*
a^3*cos(d*x + c) + 6720*a^3*log(1/2*cos(d*x + c) + 1/2) - 6720*a^3*log(-1/2*cos(d*x + c) + 1/2) + 35*(48*a^3*c
os(d*x + c)^7 - 200*a^3*cos(d*x + c)^5 - 250*a^3*cos(d*x + c)^3 - 375*a^3*cos(d*x + c))*sin(d*x + c))/d

Sympy [F(-1)]

Timed out. \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.92 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {46080 \, a^{3} \cos \left (d x + c\right )^{7} - 3584 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 35 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 1680 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3}}{107520 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/107520*(46080*a^3*cos(d*x + c)^7 - 3584*(6*cos(d*x + c)^5 + 10*cos(d*x + c)^3 + 30*cos(d*x + c) - 15*log(co
s(d*x + c) + 1) + 15*log(cos(d*x + c) - 1))*a^3 - 35*(64*sin(2*d*x + 2*c)^3 + 120*d*x + 120*c - 3*sin(8*d*x +
8*c) - 24*sin(4*d*x + 4*c))*a^3 + 1680*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d
*x + 2*c))*a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.50 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {13125 \, {\left (d x + c\right )} a^{3} + 13440 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (27195 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 65135 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 161280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 63595 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 286720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 133175 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 519680 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 133175 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 544768 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 63595 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 254464 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 65135 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 118784 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27195 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 14848 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8}}}{13440 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/13440*(13125*(d*x + c)*a^3 + 13440*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 2*(27195*a^3*tan(1/2*d*x + 1/2*c)^15
+ 65135*a^3*tan(1/2*d*x + 1/2*c)^13 - 161280*a^3*tan(1/2*d*x + 1/2*c)^12 + 63595*a^3*tan(1/2*d*x + 1/2*c)^11
- 286720*a^3*tan(1/2*d*x + 1/2*c)^10 + 133175*a^3*tan(1/2*d*x + 1/2*c)^9 - 519680*a^3*tan(1/2*d*x + 1/2*c)^8 -
133175*a^3*tan(1/2*d*x + 1/2*c)^7 - 544768*a^3*tan(1/2*d*x + 1/2*c)^6 - 63595*a^3*tan(1/2*d*x + 1/2*c)^5 - 25
4464*a^3*tan(1/2*d*x + 1/2*c)^4 - 65135*a^3*tan(1/2*d*x + 1/2*c)^3 - 118784*a^3*tan(1/2*d*x + 1/2*c)^2 - 27195
*a^3*tan(1/2*d*x + 1/2*c) - 14848*a^3)/(tan(1/2*d*x + 1/2*c)^2 + 1)^8)/d

Mupad [B] (verification not implemented)

Time = 13.07 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.32 \[ \int \cos ^5(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 {125\,a^3\,\mathrm {atan}\left (\frac {15625\,a^6}{4096\,\left (\frac {125\,a^6}{32}-\frac {15625\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4096}\right )}+\frac {125\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{32\,\left (\frac {125\,a^6}{32}-\frac {15625\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4096}\right )}\right )}{64\,d}+\frac {-\frac {259\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}-\frac {1861\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}+24\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {1817\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+\frac {128\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}-\frac {3805\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {232\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {3805\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{192}+\frac {1216\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}+\frac {1817\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+\frac {568\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}+\frac {1861\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {1856\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}+\frac {259\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {232\,a^3}{105}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^3)/sin(c + d*x),x)

[Out]

(a^3*log(tan(c/2 + (d*x)/2)))/d + (125*a^3*atan((15625*a^6)/(4096*((125*a^6)/32 - (15625*a^6*tan(c/2 + (d*x)/2
))/4096)) + (125*a^6*tan(c/2 + (d*x)/2))/(32*((125*a^6)/32 - (15625*a^6*tan(c/2 + (d*x)/2))/4096))))/(64*d) +
((1856*a^3*tan(c/2 + (d*x)/2)^2)/105 + (1861*a^3*tan(c/2 + (d*x)/2)^3)/192 + (568*a^3*tan(c/2 + (d*x)/2)^4)/15
+ (1817*a^3*tan(c/2 + (d*x)/2)^5)/192 + (1216*a^3*tan(c/2 + (d*x)/2)^6)/15 + (3805*a^3*tan(c/2 + (d*x)/2)^7)/
192 + (232*a^3*tan(c/2 + (d*x)/2)^8)/3 - (3805*a^3*tan(c/2 + (d*x)/2)^9)/192 + (128*a^3*tan(c/2 + (d*x)/2)^10)
/3 - (1817*a^3*tan(c/2 + (d*x)/2)^11)/192 + 24*a^3*tan(c/2 + (d*x)/2)^12 - (1861*a^3*tan(c/2 + (d*x)/2)^13)/19
2 - (259*a^3*tan(c/2 + (d*x)/2)^15)/64 + (232*a^3)/105 + (259*a^3*tan(c/2 + (d*x)/2))/64)/(d*(8*tan(c/2 + (d*x
)/2)^2 + 28*tan(c/2 + (d*x)/2)^4 + 56*tan(c/2 + (d*x)/2)^6 + 70*tan(c/2 + (d*x)/2)^8 + 56*tan(c/2 + (d*x)/2)^1
0 + 28*tan(c/2 + (d*x)/2)^12 + 8*tan(c/2 + (d*x)/2)^14 + tan(c/2 + (d*x)/2)^16 + 1))