\(\int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx\) [598]

Optimal result

Integrand size = 27, antiderivative size = 157 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=-2 a^2 x-\frac {25 a^2 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot (c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d} \]

[Out]

-2*a^2*x-25/16*a^2*arctanh(cos(d*x+c))/d+a^2*cos(d*x+c)/d-2*a^2*cot(d*x+c)/d+2/3*a^2*cot(d*x+c)^3/d-2/5*a^2*co
t(d*x+c)^5/d+7/16*a^2*cot(d*x+c)*csc(d*x+c)/d+7/24*a^2*cot(d*x+c)*csc(d*x+c)^3/d-1/6*a^2*cot(d*x+c)*csc(d*x+c)
^5/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2951, 3855, 3852, 8, 3853, 2718} \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {25 a^2 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot (c+d x)}{d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-2 a^2 x \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]

[Out]

-2*a^2*x - (25*a^2*ArcTanh[Cos[c + d*x]])/(16*d) + (a^2*Cos[c + d*x])/d - (2*a^2*Cot[c + d*x])/d + (2*a^2*Cot[
c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) + (7*a^2*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (7*a^2*Cot[c + d
*x]*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d)

Rule 8

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

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 (-2 a^8+2 a^8 \csc (c+d x)+6 a^8 \csc ^2(c+d x)-6 a^8 \csc ^4(c+d x)-2 a^8 \csc ^5(c+d x)+2 a^8 \csc ^6(c+d x)+a^8 \csc ^7(c+d x)-a^8 \sin (c+d x)\right ) \, dx}{a^6} \\ & = -2 a^2 x+a^2 \int \csc ^7(c+d x) \, dx-a^2 \int \sin (c+d x) \, dx+\left (2 a^2\right ) \int \csc (c+d x) \, dx-\left (2 a^2\right ) \int \csc ^5(c+d x) \, dx+\left (2 a^2\right ) \int \csc ^6(c+d x) \, dx+\left (6 a^2\right ) \int \csc ^2(c+d x) \, dx-\left (6 a^2\right ) \int \csc ^4(c+d x) \, dx \\ & = -2 a^2 x-\frac {2 a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{2 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{6} \left (5 a^2\right ) \int \csc ^5(c+d x) \, dx-\frac {1}{2} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac {\left (2 a^2\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^2\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac {\left (6 a^2\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d} \\ & = -2 a^2 x-\frac {2 a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot (c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{4 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{8} \left (5 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{4} \left (3 a^2\right ) \int \csc (c+d x) \, dx \\ & = -2 a^2 x-\frac {5 a^2 \text {arctanh}(\cos (c+d x))}{4 d}+\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot (c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{16} \left (5 a^2\right ) \int \csc (c+d x) \, dx \\ & = -2 a^2 x-\frac {25 a^2 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot (c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.11 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.72 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \left (-1920 \cot (c+d x)+\csc ^2\left (\frac {1}{2} (c+d x)\right ) (1472-210 \csc (c+d x))+\csc ^6\left (\frac {1}{2} (c+d x)\right ) (12+5 \csc (c+d x))-2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) (82+15 \csc (c+d x))+120 \csc (c+d x) \left (32 (c+d x)+25 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-25 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-2 (241+327 \cos (c+d x)+92 \cos (2 (c+d x))) \sec ^6\left (\frac {1}{2} (c+d x)\right )+840 \csc ^3(c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )+480 \csc ^5(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-320 \csc ^7(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x) (1+\sin (c+d x))^2}{1920 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]

[Out]

-1/1920*(a^2*(-1920*Cot[c + d*x] + Csc[(c + d*x)/2]^2*(1472 - 210*Csc[c + d*x]) + Csc[(c + d*x)/2]^6*(12 + 5*C
sc[c + d*x]) - 2*Csc[(c + d*x)/2]^4*(82 + 15*Csc[c + d*x]) + 120*Csc[c + d*x]*(32*(c + d*x) + 25*Log[Cos[(c +
d*x)/2]] - 25*Log[Sin[(c + d*x)/2]]) - 2*(241 + 327*Cos[c + d*x] + 92*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^6 + 8
40*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2 + 480*Csc[c + d*x]^5*Sin[(c + d*x)/2]^4 - 320*Csc[c + d*x]^7*Sin[(c + d*x
)/2]^6)*Sin[c + d*x]*(1 + Sin[c + d*x])^2)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.29

method	result	size
parallelrisch	\(-\frac {25 \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\left (-10+\cos \left (6 d x +6 c \right )-6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {96 d x}{5}-\frac {87}{8}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {192 d x}{25}+\frac {87}{20}\right ) \cos \left (4 d x +4 c \right )+\left (-\frac {32 d x}{25}-\frac {29}{40}\right ) \cos \left (6 d x +6 c \right )+\frac {64 d x}{5}+\frac {8 \cos \left (7 d x +7 c \right )}{25}+\frac {32 \sin \left (2 d x +2 c \right )}{5}-\frac {512 \sin \left (4 d x +4 c \right )}{125}+\frac {736 \sin \left (6 d x +6 c \right )}{375}-\frac {4 \cos \left (d x +c \right )}{5}+\frac {454 \cos \left (3 d x +3 c \right )}{75}-\frac {54 \cos \left (5 d x +5 c \right )}{25}+\frac {29}{4}\right ) \left (\csc ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{32768 d}\)	\(203\)
risch	\(-2 a^{2} x +\frac {a^{2} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {a^{2} \left (105 \,{\mathrm e}^{11 i \left (d x +c \right )}-595 \,{\mathrm e}^{9 i \left (d x +c \right )}+1440 i {\mathrm e}^{10 i \left (d x +c \right )}-150 \,{\mathrm e}^{7 i \left (d x +c \right )}-4320 i {\mathrm e}^{8 i \left (d x +c \right )}-150 \,{\mathrm e}^{5 i \left (d x +c \right )}+7360 i {\mathrm e}^{6 i \left (d x +c \right )}-595 \,{\mathrm e}^{3 i \left (d x +c \right )}-6720 i {\mathrm e}^{4 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}+2976 i {\mathrm e}^{2 i \left (d x +c \right )}-736 i\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {25 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}-\frac {25 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}\)	\(232\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(239\)
default	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(239\)
norman	\(\frac {-\frac {a^{2}}{384 d}-\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{80 d}+\frac {a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {29 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}+\frac {7 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d}-\frac {263 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}-\frac {59 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d}+\frac {59 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d}+\frac {263 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}-\frac {7 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d}-\frac {29 a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}-\frac {a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {a^{2} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {a^{2} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-2 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a^{2} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {163 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {163 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {25 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}\)	\(390\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-25/32768*sec(1/2*d*x+1/2*c)^6*((-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))
+(-96/5*d*x-87/8)*cos(2*d*x+2*c)+(192/25*d*x+87/20)*cos(4*d*x+4*c)+(-32/25*d*x-29/40)*cos(6*d*x+6*c)+64/5*d*x+
8/25*cos(7*d*x+7*c)+32/5*sin(2*d*x+2*c)-512/125*sin(4*d*x+4*c)+736/375*sin(6*d*x+6*c)-4/5*cos(d*x+c)+454/75*co
s(3*d*x+3*c)-54/25*cos(5*d*x+5*c)+29/4)*csc(1/2*d*x+1/2*c)^6*a^2/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (145) = 290\).

Time = 0.30 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.93 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {960 \, a^{2} d x \cos \left (d x + c\right )^{6} - 480 \, a^{2} \cos \left (d x + c\right )^{7} - 2880 \, a^{2} d x \cos \left (d x + c\right )^{4} + 1650 \, a^{2} \cos \left (d x + c\right )^{5} + 2880 \, a^{2} d x \cos \left (d x + c\right )^{2} - 2000 \, a^{2} \cos \left (d x + c\right )^{3} - 960 \, a^{2} d x + 750 \, a^{2} \cos \left (d x + c\right ) + 375 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 375 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 64 \, {\left (23 \, a^{2} \cos \left (d x + c\right )^{5} - 35 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/480*(960*a^2*d*x*cos(d*x + c)^6 - 480*a^2*cos(d*x + c)^7 - 2880*a^2*d*x*cos(d*x + c)^4 + 1650*a^2*cos(d*x +
c)^5 + 2880*a^2*d*x*cos(d*x + c)^2 - 2000*a^2*cos(d*x + c)^3 - 960*a^2*d*x + 750*a^2*cos(d*x + c) + 375*(a^2*
cos(d*x + c)^6 - 3*a^2*cos(d*x + c)^4 + 3*a^2*cos(d*x + c)^2 - a^2)*log(1/2*cos(d*x + c) + 1/2) - 375*(a^2*cos
(d*x + c)^6 - 3*a^2*cos(d*x + c)^4 + 3*a^2*cos(d*x + c)^2 - a^2)*log(-1/2*cos(d*x + c) + 1/2) - 64*(23*a^2*cos
(d*x + c)^5 - 35*a^2*cos(d*x + c)^3 + 15*a^2*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. \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**7*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.40 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {64 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{2} - 5 \, a^{2} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 30 \, a^{2} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/480*(64*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*a^2 - 5*a^2*(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*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) + 30*a^2*(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 = 0.42 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.65 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 280 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 255 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3840 \, {\left (d x + c\right )} a^{2} + 3000 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 2640 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {3840 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} - \frac {7350 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2640 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 255 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 280 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1920*(5*a^2*tan(1/2*d*x + 1/2*c)^6 + 24*a^2*tan(1/2*d*x + 1/2*c)^5 - 15*a^2*tan(1/2*d*x + 1/2*c)^4 - 280*a^2
*tan(1/2*d*x + 1/2*c)^3 - 255*a^2*tan(1/2*d*x + 1/2*c)^2 - 3840*(d*x + c)*a^2 + 3000*a^2*log(abs(tan(1/2*d*x +
1/2*c))) + 2640*a^2*tan(1/2*d*x + 1/2*c) + 3840*a^2/(tan(1/2*d*x + 1/2*c)^2 + 1) - (7350*a^2*tan(1/2*d*x + 1/
2*c)^6 + 2640*a^2*tan(1/2*d*x + 1/2*c)^5 - 255*a^2*tan(1/2*d*x + 1/2*c)^4 - 280*a^2*tan(1/2*d*x + 1/2*c)^3 - 1
5*a^2*tan(1/2*d*x + 1/2*c)^2 + 24*a^2*tan(1/2*d*x + 1/2*c) + 5*a^2)/tan(1/2*d*x + 1/2*c)^6)/d

Mupad [B] (verification not implemented)

Time = 12.38 (sec) , antiderivative size = 657, normalized size of antiderivative = 4.18 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5\,a^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-5\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+24\,a^2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-24\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-10\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-256\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-270\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+2360\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-255\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4095\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-2360\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+270\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+256\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+10\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+3000\,a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+3000\,a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+7680\,a^2\,\mathrm {atan}\left (\frac {32\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-25\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{25\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+32\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+7680\,a^2\,\mathrm {atan}\left (\frac {32\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-25\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{25\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+32\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^2)/sin(c + d*x)^7,x)

[Out]

(5*a^2*sin(c/2 + (d*x)/2)^14 - 5*a^2*cos(c/2 + (d*x)/2)^14 + 24*a^2*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^13 -
24*a^2*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2) - 10*a^2*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^12 - 256*a^2
*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^11 - 270*a^2*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10 + 2360*a^2*co
s(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^9 - 255*a^2*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 + 4095*a^2*cos(c/2
+ (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 - 2360*a^2*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^5 + 270*a^2*cos(c/2 + (d
*x)/2)^10*sin(c/2 + (d*x)/2)^4 + 256*a^2*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^3 + 10*a^2*cos(c/2 + (d*x)/2
)^12*sin(c/2 + (d*x)/2)^2 + 3000*a^2*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^6*sin(c/2 +
(d*x)/2)^8 + 3000*a^2*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 +
7680*a^2*atan((32*cos(c/2 + (d*x)/2) - 25*sin(c/2 + (d*x)/2))/(25*cos(c/2 + (d*x)/2) + 32*sin(c/2 + (d*x)/2)))
*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 + 7680*a^2*atan((32*cos(c/2 + (d*x)/2) - 25*sin(c/2 + (d*x)/2))/(25
*cos(c/2 + (d*x)/2) + 32*sin(c/2 + (d*x)/2)))*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6)/(1920*d*cos(c/2 + (d*
x)/2)^6*sin(c/2 + (d*x)/2)^6*(cos(c/2 + (d*x)/2)^2 + sin(c/2 + (d*x)/2)^2))