\(\int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx\) [586]

Optimal result

Integrand size = 27, antiderivative size = 160 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 a \text {arctanh}(\cos (c+d x))}{256 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{256 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d} \]

[Out]

3/256*a*arctanh(cos(d*x+c))/d-1/7*a*cot(d*x+c)^7/d-1/9*a*cot(d*x+c)^9/d+3/256*a*cot(d*x+c)*csc(d*x+c)/d+1/128*
a*cot(d*x+c)*csc(d*x+c)^3/d-1/32*a*cot(d*x+c)*csc(d*x+c)^5/d+1/16*a*cot(d*x+c)^3*csc(d*x+c)^5/d-1/10*a*cot(d*x
+c)^5*csc(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2917, 2691, 3853, 3855, 2687, 14} \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 a \text {arctanh}(\cos (c+d x))}{256 d}-\frac {a \cot ^9(c+d x)}{9 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{256 d} \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

(3*a*ArcTanh[Cos[c + d*x]])/(256*d) - (a*Cot[c + d*x]^7)/(7*d) - (a*Cot[c + d*x]^9)/(9*d) + (3*a*Cot[c + d*x]*
Csc[c + d*x])/(256*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) - (a*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a*
Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2917

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

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}& = a \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+a \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx \\ & = -\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{2} a \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac {a \text {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {1}{16} (3 a) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {a \text {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{32} a \int \csc ^5(c+d x) \, dx \\ & = -\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{128} (3 a) \int \csc ^3(c+d x) \, dx \\ & = -\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{256 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{256} (3 a) \int \csc (c+d x) \, dx \\ & = \frac {3 a \text {arctanh}(\cos (c+d x))}{256 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{256 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(341\) vs. \(2(160)=320\).

Time = 0.18 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.13 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {2 a \cot (c+d x)}{63 d}+\frac {3 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {3 a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{2048 d}+\frac {3 a \csc ^8\left (\frac {1}{2} (c+d x)\right )}{4096 d}-\frac {a \csc ^{10}\left (\frac {1}{2} (c+d x)\right )}{10240 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{63 d}-\frac {5 a \cot (c+d x) \csc ^4(c+d x)}{21 d}+\frac {19 a \cot (c+d x) \csc ^6(c+d x)}{63 d}-\frac {a \cot (c+d x) \csc ^8(c+d x)}{9 d}+\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{256 d}-\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{256 d}-\frac {3 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {3 a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{2048 d}-\frac {3 a \sec ^8\left (\frac {1}{2} (c+d x)\right )}{4096 d}+\frac {a \sec ^{10}\left (\frac {1}{2} (c+d x)\right )}{10240 d} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

(2*a*Cot[c + d*x])/(63*d) + (3*a*Csc[(c + d*x)/2]^2)/(1024*d) - (a*Csc[(c + d*x)/2]^4)/(1024*d) - (3*a*Csc[(c
+ d*x)/2]^6)/(2048*d) + (3*a*Csc[(c + d*x)/2]^8)/(4096*d) - (a*Csc[(c + d*x)/2]^10)/(10240*d) + (a*Cot[c + d*x
]*Csc[c + d*x]^2)/(63*d) - (5*a*Cot[c + d*x]*Csc[c + d*x]^4)/(21*d) + (19*a*Cot[c + d*x]*Csc[c + d*x]^6)/(63*d
) - (a*Cot[c + d*x]*Csc[c + d*x]^8)/(9*d) + (3*a*Log[Cos[(c + d*x)/2]])/(256*d) - (3*a*Log[Sin[(c + d*x)/2]])/
(256*d) - (3*a*Sec[(c + d*x)/2]^2)/(1024*d) + (a*Sec[(c + d*x)/2]^4)/(1024*d) + (3*a*Sec[(c + d*x)/2]^6)/(2048
*d) - (3*a*Sec[(c + d*x)/2]^8)/(4096*d) + (a*Sec[(c + d*x)/2]^10)/(10240*d)

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\)	\(184\)
default	\(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\)	\(184\)
parallelrisch	\(-\frac {\left (120 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cot ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {20 \left (\cot ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {5 \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {60 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-5 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {160 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+10 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-\left (-120+\frac {160 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+20 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {60 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {20 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{10240 d}\)	\(246\)
risch	\(-\frac {a \left (945 \,{\mathrm e}^{19 i \left (d x +c \right )}-9135 \,{\mathrm e}^{17 i \left (d x +c \right )}-218484 \,{\mathrm e}^{15 i \left (d x +c \right )}+107520 i {\mathrm e}^{14 i \left (d x +c \right )}-653940 \,{\mathrm e}^{13 i \left (d x +c \right )}+161280 i {\mathrm e}^{16 i \left (d x +c \right )}-1183770 \,{\mathrm e}^{11 i \left (d x +c \right )}-414720 i {\mathrm e}^{6 i \left (d x +c \right )}-1183770 \,{\mathrm e}^{9 i \left (d x +c \right )}+537600 i {\mathrm e}^{12 i \left (d x +c \right )}-653940 \,{\mathrm e}^{7 i \left (d x +c \right )}-218484 \,{\mathrm e}^{5 i \left (d x +c \right )}-322560 i {\mathrm e}^{10 i \left (d x +c \right )}-9135 \,{\mathrm e}^{3 i \left (d x +c \right )}-46080 i {\mathrm e}^{4 i \left (d x +c \right )}+945 \,{\mathrm e}^{i \left (d x +c \right )}-25600 i {\mathrm e}^{2 i \left (d x +c \right )}+2560 i\right )}{40320 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}\)	\(254\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(a*(-1/9/sin(d*x+c)^9*cos(d*x+c)^7-2/63/sin(d*x+c)^7*cos(d*x+c)^7)+a*(-1/10/sin(d*x+c)^10*cos(d*x+c)^7-3/8
0/sin(d*x+c)^8*cos(d*x+c)^7-1/160/sin(d*x+c)^6*cos(d*x+c)^7+1/640/sin(d*x+c)^4*cos(d*x+c)^7-3/1280/sin(d*x+c)^
2*cos(d*x+c)^7-3/1280*cos(d*x+c)^5-1/256*cos(d*x+c)^3-3/256*cos(d*x+c)-3/256*ln(csc(d*x+c)-cot(d*x+c))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (144) = 288\).

Time = 0.28 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.81 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {1890 \, a \cos \left (d x + c\right )^{9} - 8820 \, a \cos \left (d x + c\right )^{7} - 16128 \, a \cos \left (d x + c\right )^{5} + 8820 \, a \cos \left (d x + c\right )^{3} - 1890 \, a \cos \left (d x + c\right ) - 945 \, {\left (a \cos \left (d x + c\right )^{10} - 5 \, a \cos \left (d x + c\right )^{8} + 10 \, a \cos \left (d x + c\right )^{6} - 10 \, a \cos \left (d x + c\right )^{4} + 5 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 945 \, {\left (a \cos \left (d x + c\right )^{10} - 5 \, a \cos \left (d x + c\right )^{8} + 10 \, a \cos \left (d x + c\right )^{6} - 10 \, a \cos \left (d x + c\right )^{4} + 5 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2560 \, {\left (2 \, a \cos \left (d x + c\right )^{9} - 9 \, a \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{161280 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/161280*(1890*a*cos(d*x + c)^9 - 8820*a*cos(d*x + c)^7 - 16128*a*cos(d*x + c)^5 + 8820*a*cos(d*x + c)^3 - 18
90*a*cos(d*x + c) - 945*(a*cos(d*x + c)^10 - 5*a*cos(d*x + c)^8 + 10*a*cos(d*x + c)^6 - 10*a*cos(d*x + c)^4 +
5*a*cos(d*x + c)^2 - a)*log(1/2*cos(d*x + c) + 1/2) + 945*(a*cos(d*x + c)^10 - 5*a*cos(d*x + c)^8 + 10*a*cos(d
*x + c)^6 - 10*a*cos(d*x + c)^4 + 5*a*cos(d*x + c)^2 - a)*log(-1/2*cos(d*x + c) + 1/2) + 2560*(2*a*cos(d*x + c
)^9 - 9*a*cos(d*x + c)^7)*sin(d*x + c))/(d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*c
os(d*x + c)^4 + 5*d*cos(d*x + c)^2 - d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**11*(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.99 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {63 \, a {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {2560 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/161280*(63*a*(2*(15*cos(d*x + c)^9 - 70*cos(d*x + c)^7 - 128*cos(d*x + c)^5 + 70*cos(d*x + c)^3 - 15*cos(d*
x + c))/(cos(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*cos(d*x + c)^2 - 1) -
15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 2560*(9*tan(d*x + c)^2 + 7)*a/tan(d*x + c)^9)/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.78 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {126 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 280 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1080 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 630 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2520 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 6720 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1260 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15120 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 15120 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {44286 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 15120 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1260 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 6720 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2520 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 630 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1080 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 280 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 126 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{1290240 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/1290240*(126*a*tan(1/2*d*x + 1/2*c)^10 + 280*a*tan(1/2*d*x + 1/2*c)^9 - 315*a*tan(1/2*d*x + 1/2*c)^8 - 1080*
a*tan(1/2*d*x + 1/2*c)^7 - 630*a*tan(1/2*d*x + 1/2*c)^6 + 2520*a*tan(1/2*d*x + 1/2*c)^4 + 6720*a*tan(1/2*d*x +
1/2*c)^3 + 1260*a*tan(1/2*d*x + 1/2*c)^2 - 15120*a*log(abs(tan(1/2*d*x + 1/2*c))) - 15120*a*tan(1/2*d*x + 1/2
*c) + (44286*a*tan(1/2*d*x + 1/2*c)^10 + 15120*a*tan(1/2*d*x + 1/2*c)^9 - 1260*a*tan(1/2*d*x + 1/2*c)^8 - 6720
*a*tan(1/2*d*x + 1/2*c)^7 - 2520*a*tan(1/2*d*x + 1/2*c)^6 + 630*a*tan(1/2*d*x + 1/2*c)^4 + 1080*a*tan(1/2*d*x
+ 1/2*c)^3 + 315*a*tan(1/2*d*x + 1/2*c)^2 - 280*a*tan(1/2*d*x + 1/2*c) - 126*a)/tan(1/2*d*x + 1/2*c)^10)/d

Mupad [B] (verification not implemented)

Time = 10.48 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.99 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3\,a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}+\frac {3\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x)))/sin(c + d*x)^11,x)

[Out]

(3*a*cot(c/2 + (d*x)/2))/(256*d) - (3*a*tan(c/2 + (d*x)/2))/(256*d) - (a*cot(c/2 + (d*x)/2)^2)/(1024*d) - (a*c
ot(c/2 + (d*x)/2)^3)/(192*d) - (a*cot(c/2 + (d*x)/2)^4)/(512*d) + (a*cot(c/2 + (d*x)/2)^6)/(2048*d) + (3*a*cot
(c/2 + (d*x)/2)^7)/(3584*d) + (a*cot(c/2 + (d*x)/2)^8)/(4096*d) - (a*cot(c/2 + (d*x)/2)^9)/(4608*d) - (a*cot(c
/2 + (d*x)/2)^10)/(10240*d) + (a*tan(c/2 + (d*x)/2)^2)/(1024*d) + (a*tan(c/2 + (d*x)/2)^3)/(192*d) + (a*tan(c/
2 + (d*x)/2)^4)/(512*d) - (a*tan(c/2 + (d*x)/2)^6)/(2048*d) - (3*a*tan(c/2 + (d*x)/2)^7)/(3584*d) - (a*tan(c/2
+ (d*x)/2)^8)/(4096*d) + (a*tan(c/2 + (d*x)/2)^9)/(4608*d) + (a*tan(c/2 + (d*x)/2)^10)/(10240*d) - (3*a*log(t
an(c/2 + (d*x)/2)))/(256*d)