\(\int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx\) [615]

Optimal result

Integrand size = 27, antiderivative size = 182 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 x}{2}-\frac {85 a^3 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d} \]

[Out]

-1/2*a^3*x-85/16*a^3*arctanh(cos(d*x+c))/d+3*a^3*cos(d*x+c)/d-a^3*cot(d*x+c)/d+2/3*a^3*cot(d*x+c)^3/d-3/5*a^3*
cot(d*x+c)^5/d+43/16*a^3*cot(d*x+c)*csc(d*x+c)/d-5/24*a^3*cot(d*x+c)*csc(d*x+c)^3/d-1/6*a^3*cot(d*x+c)*csc(d*x
+c)^5/d+1/2*a^3*cos(d*x+c)*sin(d*x+c)/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2951, 3855, 3852, 8, 3853, 2718, 2715} \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {85 a^3 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {a^3 x}{2} \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x])^3,x]

[Out]

-1/2*(a^3*x) - (85*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (3*a^3*Cos[c + d*x])/d - (a^3*Cot[c + d*x])/d + (2*a^3*
Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]^5)/(5*d) + (43*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (5*a^3*Cot[
c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d) + (a^3*Cos[c + d*x]*Sin[c + d*x])/(2
*d)

Rule 8

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

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 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2951

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[1/a^p, Int[ExpandTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x]
)^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && ((GtQ[m,
0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

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

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 {\int \left (8 a^9 \csc (c+d x)+6 a^9 \csc ^2(c+d x)-6 a^9 \csc ^3(c+d x)-8 a^9 \csc ^4(c+d x)+3 a^9 \csc ^6(c+d x)+a^9 \csc ^7(c+d x)-3 a^9 \sin (c+d x)-a^9 \sin ^2(c+d x)\right ) \, dx}{a^6} \\ & = a^3 \int \csc ^7(c+d x) \, dx-a^3 \int \sin ^2(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^6(c+d x) \, dx-\left (3 a^3\right ) \int \sin (c+d x) \, dx+\left (6 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (6 a^3\right ) \int \csc ^3(c+d x) \, dx+\left (8 a^3\right ) \int \csc (c+d x) \, dx-\left (8 a^3\right ) \int \csc ^4(c+d x) \, dx \\ & = -\frac {8 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {1}{2} a^3 \int 1 \, dx+\frac {1}{6} \left (5 a^3\right ) \int \csc ^5(c+d x) \, dx-\left (3 a^3\right ) \int \csc (c+d x) \, dx-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (6 a^3\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac {\left (8 a^3\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d} \\ & = -\frac {a^3 x}{2}-\frac {5 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{8} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx \\ & = -\frac {a^3 x}{2}-\frac {5 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{16} \left (5 a^3\right ) \int \csc (c+d x) \, dx \\ & = -\frac {a^3 x}{2}-\frac {85 a^3 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.75 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.59 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (-960 (c+d x)+5760 \cos (c+d x)-2176 \cot \left (\frac {1}{2} (c+d x)\right )+1290 \csc ^2\left (\frac {1}{2} (c+d x)\right )-30 \csc ^4\left (\frac {1}{2} (c+d x)\right )-5 \csc ^6\left (\frac {1}{2} (c+d x)\right )-10200 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+10200 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-1290 \sec ^2\left (\frac {1}{2} (c+d x)\right )+30 \sec ^4\left (\frac {1}{2} (c+d x)\right )+5 \sec ^6\left (\frac {1}{2} (c+d x)\right )-3296 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+206 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-18 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+480 \sin (2 (c+d x))+2176 \tan \left (\frac {1}{2} (c+d x)\right )+36 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{1920 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(1 + Sin[c + d*x])^3*(-960*(c + d*x) + 5760*Cos[c + d*x] - 2176*Cot[(c + d*x)/2] + 1290*Csc[(c + d*x)/2]^
2 - 30*Csc[(c + d*x)/2]^4 - 5*Csc[(c + d*x)/2]^6 - 10200*Log[Cos[(c + d*x)/2]] + 10200*Log[Sin[(c + d*x)/2]] -
1290*Sec[(c + d*x)/2]^2 + 30*Sec[(c + d*x)/2]^4 + 5*Sec[(c + d*x)/2]^6 - 3296*Csc[c + d*x]^3*Sin[(c + d*x)/2]
^4 + 206*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 18*Csc[(c + d*x)/2]^6*Sin[c + d*x] + 480*Sin[2*(c + d*x)] + 2176*Ta
n[(c + d*x)/2] + 36*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(1920*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.27

method	result	size
parallelrisch	\(\frac {a^{3} \left (\csc ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-10200 \left (-10+\cos \left (6 d x +6 c \right )-6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+14400 \left (d x -\frac {543}{64}\right ) \cos \left (2 d x +2 c \right )+960 d x \cos \left (6 d x +6 c \right )-5760 d x \cos \left (4 d x +4 c \right )-9600 d x -8145 \cos \left (6 d x +6 c \right )-2880 \cos \left (7 d x +7 c \right )-8160 \sin \left (2 d x +2 c \right )-288 \sin \left (4 d x +4 c \right )-2912 \sin \left (6 d x +6 c \right )-240 \sin \left (8 d x +8 c \right )+21600 \cos \left (d x +c \right )-53680 \cos \left (3 d x +3 c \right )+48870 \cos \left (4 d x +4 c \right )+24720 \cos \left (5 d x +5 c \right )+81450\right )}{3932160 d}\)	\(231\)
risch	\(-\frac {a^{3} x}{2}-\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {3 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {3 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {a^{3} \left (645 \,{\mathrm e}^{11 i \left (d x +c \right )}-1735 \,{\mathrm e}^{9 i \left (d x +c \right )}+1440 i {\mathrm e}^{10 i \left (d x +c \right )}+450 \,{\mathrm e}^{7 i \left (d x +c \right )}-3360 i {\mathrm e}^{8 i \left (d x +c \right )}+450 \,{\mathrm e}^{5 i \left (d x +c \right )}+5440 i {\mathrm e}^{6 i \left (d x +c \right )}-1735 \,{\mathrm e}^{3 i \left (d x +c \right )}-4800 i {\mathrm e}^{4 i \left (d x +c \right )}+645 \,{\mathrm e}^{i \left (d x +c \right )}+1824 i {\mathrm e}^{2 i \left (d x +c \right )}-544 i\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {85 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}+\frac {85 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}\)	\(268\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(323\)
default	\(\frac {a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(323\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/3932160*a^3*csc(1/2*d*x+1/2*c)^6*sec(1/2*d*x+1/2*c)^6*(-10200*(-10+cos(6*d*x+6*c)-6*cos(4*d*x+4*c)+15*cos(2*
d*x+2*c))*ln(tan(1/2*d*x+1/2*c))+14400*(d*x-543/64)*cos(2*d*x+2*c)+960*d*x*cos(6*d*x+6*c)-5760*d*x*cos(4*d*x+4
*c)-9600*d*x-8145*cos(6*d*x+6*c)-2880*cos(7*d*x+7*c)-8160*sin(2*d*x+2*c)-288*sin(4*d*x+4*c)-2912*sin(6*d*x+6*c
)-240*sin(8*d*x+8*c)+21600*cos(d*x+c)-53680*cos(3*d*x+3*c)+48870*cos(4*d*x+4*c)+24720*cos(5*d*x+5*c)+81450)/d

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.74 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {240 \, a^{3} d x \cos \left (d x + c\right )^{6} - 1440 \, a^{3} \cos \left (d x + c\right )^{7} - 720 \, a^{3} d x \cos \left (d x + c\right )^{4} + 5610 \, a^{3} \cos \left (d x + c\right )^{5} + 720 \, a^{3} d x \cos \left (d x + c\right )^{2} - 6800 \, a^{3} \cos \left (d x + c\right )^{3} - 240 \, a^{3} d x + 2550 \, a^{3} \cos \left (d x + c\right ) + 1275 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 1275 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left (15 \, a^{3} \cos \left (d x + c\right )^{7} + 23 \, a^{3} \cos \left (d x + c\right )^{5} - 35 \, a^{3} \cos \left (d x + c\right )^{3} + 15 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/480*(240*a^3*d*x*cos(d*x + c)^6 - 1440*a^3*cos(d*x + c)^7 - 720*a^3*d*x*cos(d*x + c)^4 + 5610*a^3*cos(d*x +
c)^5 + 720*a^3*d*x*cos(d*x + c)^2 - 6800*a^3*cos(d*x + c)^3 - 240*a^3*d*x + 2550*a^3*cos(d*x + c) + 1275*(a^3
*cos(d*x + c)^6 - 3*a^3*cos(d*x + c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2) - 1275*(a^3*c
os(d*x + c)^6 - 3*a^3*cos(d*x + c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos(d*x + c) + 1/2) - 16*(15*a^3*c
os(d*x + c)^7 + 23*a^3*cos(d*x + c)^5 - 35*a^3*cos(d*x + c)^3 + 15*a^3*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x
+ c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**7*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.51 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {80 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{3} - 96 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} + 5 \, a^{3} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 90 \, a^{3} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/480*(80*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 10*tan(d*x + c)^2 - 2)/(tan(d*x + c)^5 + tan(d*x + c)^3))*a^3
- 96*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*a^3 + 5*a^3*(2*(33*cos(d*x +
c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) + 15*lo
g(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) - 90*a^3*(2*(9*cos(d*x + c)^3 - 7*cos(d*x + c))/(cos(d*x + c)^
4 - 2*cos(d*x + c)^2 + 1) - 16*cos(d*x + c) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.69 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 340 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1215 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 960 \, {\left (d x + c\right )} a^{3} + 10200 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 1800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {1920 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac {24990 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1215 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 340 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/1920*(5*a^3*tan(1/2*d*x + 1/2*c)^6 + 36*a^3*tan(1/2*d*x + 1/2*c)^5 + 45*a^3*tan(1/2*d*x + 1/2*c)^4 - 340*a^3
*tan(1/2*d*x + 1/2*c)^3 - 1215*a^3*tan(1/2*d*x + 1/2*c)^2 - 960*(d*x + c)*a^3 + 10200*a^3*log(abs(tan(1/2*d*x
+ 1/2*c))) + 1800*a^3*tan(1/2*d*x + 1/2*c) - 1920*(a^3*tan(1/2*d*x + 1/2*c)^3 - 6*a^3*tan(1/2*d*x + 1/2*c)^2 -
a^3*tan(1/2*d*x + 1/2*c) - 6*a^3)/(tan(1/2*d*x + 1/2*c)^2 + 1)^2 - (24990*a^3*tan(1/2*d*x + 1/2*c)^6 + 1800*a
^3*tan(1/2*d*x + 1/2*c)^5 - 1215*a^3*tan(1/2*d*x + 1/2*c)^4 - 340*a^3*tan(1/2*d*x + 1/2*c)^3 + 45*a^3*tan(1/2*
d*x + 1/2*c)^2 + 36*a^3*tan(1/2*d*x + 1/2*c) + 5*a^3)/tan(1/2*d*x + 1/2*c)^6)/d

Mupad [B] (verification not implemented)

Time = 10.43 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.18 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {17\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {81\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {85\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16\,d}+\frac {a^3\,\mathrm {atan}\left (\frac {a^6}{\frac {85\,a^6}{8}+a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {85\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,\left (\frac {85\,a^6}{8}+a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{d}-\frac {124\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-\frac {849\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2}+\frac {134\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}-\frac {927\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2}+\frac {578\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}-\frac {112\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-\frac {134\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{15}+\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6}+\frac {6\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+\frac {a^3}{6}}{d\,\left (64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\right )}+\frac {15\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^3)/sin(c + d*x)^7,x)

[Out]

(3*a^3*tan(c/2 + (d*x)/2)^4)/(128*d) - (17*a^3*tan(c/2 + (d*x)/2)^3)/(96*d) - (81*a^3*tan(c/2 + (d*x)/2)^2)/(1
28*d) + (3*a^3*tan(c/2 + (d*x)/2)^5)/(160*d) + (a^3*tan(c/2 + (d*x)/2)^6)/(384*d) + (85*a^3*log(tan(c/2 + (d*x
)/2)))/(16*d) + (a^3*atan(a^6/((85*a^6)/8 + a^6*tan(c/2 + (d*x)/2)) - (85*a^6*tan(c/2 + (d*x)/2))/(8*((85*a^6)
/8 + a^6*tan(c/2 + (d*x)/2)))))/d - ((11*a^3*tan(c/2 + (d*x)/2)^2)/6 - (134*a^3*tan(c/2 + (d*x)/2)^3)/15 - (11
2*a^3*tan(c/2 + (d*x)/2)^4)/3 + (578*a^3*tan(c/2 + (d*x)/2)^5)/15 - (927*a^3*tan(c/2 + (d*x)/2)^6)/2 + (134*a^
3*tan(c/2 + (d*x)/2)^7)/3 - (849*a^3*tan(c/2 + (d*x)/2)^8)/2 + 124*a^3*tan(c/2 + (d*x)/2)^9 + a^3/6 + (6*a^3*t
an(c/2 + (d*x)/2))/5)/(d*(64*tan(c/2 + (d*x)/2)^6 + 128*tan(c/2 + (d*x)/2)^8 + 64*tan(c/2 + (d*x)/2)^10)) + (1
5*a^3*tan(c/2 + (d*x)/2))/(16*d)