\(\int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\) [616]

Optimal result

Integrand size = 29, antiderivative size = 172 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-3 a^3 x-\frac {15 a^3 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d} \]

[Out]

-3*a^3*x-15/16*a^3*arctanh(cos(d*x+c))/d+a^3*cos(d*x+c)/d-3*a^3*cot(d*x+c)/d+a^3*cot(d*x+c)^3/d-3/5*a^3*cot(d*
x+c)^5/d-1/7*a^3*cot(d*x+c)^7/d-15/16*a^3*cot(d*x+c)*csc(d*x+c)/d+11/8*a^3*cot(d*x+c)*csc(d*x+c)^3/d-1/2*a^3*c
ot(d*x+c)*csc(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2951, 3852, 8, 3853, 3855, 2718} \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {15 a^3 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^3(c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac {11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-3 a^3 x \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

-3*a^3*x - (15*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (a^3*Cos[c + d*x])/d - (3*a^3*Cot[c + d*x])/d + (a^3*Cot[c
+ d*x]^3)/d - (3*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/(7*d) - (15*a^3*Cot[c + d*x]*Csc[c + d*x])/(
16*d) + (11*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(2*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 (-3 a^9+8 a^9 \csc ^2(c+d x)+6 a^9 \csc ^3(c+d x)-6 a^9 \csc ^4(c+d x)-8 a^9 \csc ^5(c+d x)+3 a^9 \csc ^7(c+d x)+a^9 \csc ^8(c+d x)-a^9 \sin (c+d x)\right ) \, dx}{a^6} \\ & = -3 a^3 x+a^3 \int \csc ^8(c+d x) \, dx-a^3 \int \sin (c+d x) \, dx+\left (3 a^3\right ) \int \csc ^7(c+d x) \, dx+\left (6 a^3\right ) \int \csc ^3(c+d x) \, dx-\left (6 a^3\right ) \int \csc ^4(c+d x) \, dx+\left (8 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (8 a^3\right ) \int \csc ^5(c+d x) \, dx \\ & = -3 a^3 x+\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{d}+\frac {2 a^3 \cot (c+d x) \csc ^3(c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac {1}{2} \left (5 a^3\right ) \int \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \csc (c+d x) \, dx-\left (6 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac {\left (6 a^3\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (8 a^3\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d} \\ & = -3 a^3 x-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac {1}{8} \left (15 a^3\right ) \int \csc ^3(c+d x) \, dx-\left (3 a^3\right ) \int \csc (c+d x) \, dx \\ & = -3 a^3 x+\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac {1}{16} \left (15 a^3\right ) \int \csc (c+d x) \, dx \\ & = -3 a^3 x-\frac {15 a^3 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.67 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.70 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (-13440 c-13440 d x+4480 \cos (c+d x)-9984 \cot \left (\frac {1}{2} (c+d x)\right )-1050 \csc ^2\left (\frac {1}{2} (c+d x)\right )+350 \csc ^4\left (\frac {1}{2} (c+d x)\right )-35 \csc ^6\left (\frac {1}{2} (c+d x)\right )-4200 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+4200 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1050 \sec ^2\left (\frac {1}{2} (c+d x)\right )-350 \sec ^4\left (\frac {1}{2} (c+d x)\right )+35 \sec ^6\left (\frac {1}{2} (c+d x)\right )-7664 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+479 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-17 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-\frac {5}{2} \csc ^8\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+9984 \tan \left (\frac {1}{2} (c+d x)\right )+34 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )+5 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{4480 d} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(-13440*c - 13440*d*x + 4480*Cos[c + d*x] - 9984*Cot[(c + d*x)/2] - 1050*Csc[(c + d*x)/2]^2 + 350*Csc[(c
+ d*x)/2]^4 - 35*Csc[(c + d*x)/2]^6 - 4200*Log[Cos[(c + d*x)/2]] + 4200*Log[Sin[(c + d*x)/2]] + 1050*Sec[(c +
d*x)/2]^2 - 350*Sec[(c + d*x)/2]^4 + 35*Sec[(c + d*x)/2]^6 - 7664*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 479*Csc[
(c + d*x)/2]^4*Sin[c + d*x] - 17*Csc[(c + d*x)/2]^6*Sin[c + d*x] - (5*Csc[(c + d*x)/2]^8*Sin[c + d*x])/2 + 998
4*Tan[(c + d*x)/2] + 34*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] + 5*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2]))/(4480*d)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.42

method	result	size
risch	\(-3 a^{3} x +\frac {a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {a^{3} \left (-4480 i {\mathrm e}^{12 i \left (d x +c \right )}+525 \,{\mathrm e}^{13 i \left (d x +c \right )}+20160 i {\mathrm e}^{10 i \left (d x +c \right )}+980 \,{\mathrm e}^{11 i \left (d x +c \right )}-38080 i {\mathrm e}^{8 i \left (d x +c \right )}+945 \,{\mathrm e}^{9 i \left (d x +c \right )}+49280 i {\mathrm e}^{6 i \left (d x +c \right )}-32256 i {\mathrm e}^{4 i \left (d x +c \right )}-945 \,{\mathrm e}^{5 i \left (d x +c \right )}+12992 i {\mathrm e}^{2 i \left (d x +c \right )}-980 \,{\mathrm e}^{3 i \left (d x +c \right )}-2496 i-525 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{280 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}+\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}\)	\(244\)
parallelrisch	\(\frac {a^{3} \left (\csc ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-88200 \left (\sin \left (3 d x +3 c \right )-\frac {\sin \left (5 d x +5 c \right )}{3}+\frac {\sin \left (7 d x +7 c \right )}{21}-\frac {5 \sin \left (d x +c \right )}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+735 \left (384 d x +115\right ) \sin \left (3 d x +3 c \right )-94080 d x \sin \left (5 d x +5 c \right )+13440 d x \sin \left (7 d x +7 c \right )-470400 d x \sin \left (d x +c \right )+16240 \sin \left (2 d x +2 c \right )-47040 \sin \left (4 d x +4 c \right )-28175 \sin \left (5 d x +5 c \right )+5040 \sin \left (6 d x +6 c \right )+4025 \sin \left (7 d x +7 c \right )-2240 \sin \left (8 d x +8 c \right )-89600 \cos \left (d x +c \right )+96768 \cos \left (3 d x +3 c \right )-68096 \cos \left (5 d x +5 c \right )+19968 \cos \left (7 d x +7 c \right )-140875 \sin \left (d x +c \right )\right )}{36700160 d}\)	\(252\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}}{d}\)	\(261\)
default	\(\frac {a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}}{d}\)	\(261\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-3*a^3*x+1/2*a^3/d*exp(I*(d*x+c))+1/2*a^3/d*exp(-I*(d*x+c))+1/280*a^3*(-4480*I*exp(12*I*(d*x+c))+525*exp(13*I*
(d*x+c))+20160*I*exp(10*I*(d*x+c))+980*exp(11*I*(d*x+c))-38080*I*exp(8*I*(d*x+c))+945*exp(9*I*(d*x+c))+49280*I
*exp(6*I*(d*x+c))-32256*I*exp(4*I*(d*x+c))-945*exp(5*I*(d*x+c))+12992*I*exp(2*I*(d*x+c))-980*exp(3*I*(d*x+c))-
2496*I-525*exp(I*(d*x+c)))/d/(exp(2*I*(d*x+c))-1)^7+15/16*a^3/d*ln(exp(I*(d*x+c))-1)-15/16*a^3/d*ln(exp(I*(d*x
+c))+1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (160) = 320\).

Time = 0.28 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.95 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {4992 \, a^{3} \cos \left (d x + c\right )^{7} - 12992 \, a^{3} \cos \left (d x + c\right )^{5} + 11200 \, a^{3} \cos \left (d x + c\right )^{3} - 3360 \, a^{3} \cos \left (d x + c\right ) + 525 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 525 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 70 \, {\left (48 \, a^{3} d x \cos \left (d x + c\right )^{6} - 16 \, a^{3} \cos \left (d x + c\right )^{7} - 144 \, a^{3} d x \cos \left (d x + c\right )^{4} + 33 \, a^{3} \cos \left (d x + c\right )^{5} + 144 \, a^{3} d x \cos \left (d x + c\right )^{2} - 40 \, a^{3} \cos \left (d x + c\right )^{3} - 48 \, a^{3} d x + 15 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1120 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/1120*(4992*a^3*cos(d*x + c)^7 - 12992*a^3*cos(d*x + c)^5 + 11200*a^3*cos(d*x + c)^3 - 3360*a^3*cos(d*x + c)
+ 525*(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)*si
n(d*x + c) - 525*(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)*sin(d*x + c) + 70*(48*a^3*d*x*cos(d*x + c)^6 - 16*a^3*cos(d*x + c)^7 - 144*a^3*d*x*cos(d*x + c)^4 + 3
3*a^3*cos(d*x + c)^5 + 144*a^3*d*x*cos(d*x + c)^2 - 40*a^3*cos(d*x + c)^3 - 48*a^3*d*x + 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)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**8*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.35 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {224 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} - 35 \, a^{3} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 70 \, a^{3} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {160 \, a^{3}}{\tan \left (d x + c\right )^{7}}}{1120 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/1120*(224*(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 - 35*a^3*(2*(33*c
os(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)) + 70*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)) +
160*a^3/tan(d*x + c)^7)/d

Giac [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.69 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 49 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 245 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 875 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 455 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 13440 \, {\left (d x + c\right )} a^{3} + 4200 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 9065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {8960 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} - \frac {10890 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 455 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 875 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 245 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 49 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{4480 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/4480*(5*a^3*tan(1/2*d*x + 1/2*c)^7 + 35*a^3*tan(1/2*d*x + 1/2*c)^6 + 49*a^3*tan(1/2*d*x + 1/2*c)^5 - 245*a^3
*tan(1/2*d*x + 1/2*c)^4 - 875*a^3*tan(1/2*d*x + 1/2*c)^3 + 455*a^3*tan(1/2*d*x + 1/2*c)^2 - 13440*(d*x + c)*a^
3 + 4200*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 9065*a^3*tan(1/2*d*x + 1/2*c) + 8960*a^3/(tan(1/2*d*x + 1/2*c)^2
+ 1) - (10890*a^3*tan(1/2*d*x + 1/2*c)^7 + 9065*a^3*tan(1/2*d*x + 1/2*c)^6 + 455*a^3*tan(1/2*d*x + 1/2*c)^5 -
875*a^3*tan(1/2*d*x + 1/2*c)^4 - 245*a^3*tan(1/2*d*x + 1/2*c)^3 + 49*a^3*tan(1/2*d*x + 1/2*c)^2 + 35*a^3*tan(
1/2*d*x + 1/2*c) + 5*a^3)/tan(1/2*d*x + 1/2*c)^7)/d

Mupad [B] (verification not implemented)

Time = 10.72 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.26 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {25\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {15\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16\,d}+\frac {6\,a^3\,\mathrm {atan}\left (\frac {36\,a^6}{\frac {45\,a^6}{4}+36\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {45\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {45\,a^6}{4}+36\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{d}-\frac {259\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-243\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+234\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {118\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {54\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}+a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{7}}{d\,\left (128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}+\frac {259\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^3)/sin(c + d*x)^8,x)

[Out]

(13*a^3*tan(c/2 + (d*x)/2)^2)/(128*d) - (25*a^3*tan(c/2 + (d*x)/2)^3)/(128*d) - (7*a^3*tan(c/2 + (d*x)/2)^4)/(
128*d) + (7*a^3*tan(c/2 + (d*x)/2)^5)/(640*d) + (a^3*tan(c/2 + (d*x)/2)^6)/(128*d) + (a^3*tan(c/2 + (d*x)/2)^7
)/(896*d) + (15*a^3*log(tan(c/2 + (d*x)/2)))/(16*d) + (6*a^3*atan((36*a^6)/((45*a^6)/4 + 36*a^6*tan(c/2 + (d*x
)/2)) - (45*a^6*tan(c/2 + (d*x)/2))/(4*((45*a^6)/4 + 36*a^6*tan(c/2 + (d*x)/2)))))/d - ((54*a^3*tan(c/2 + (d*x
)/2)^2)/35 - 6*a^3*tan(c/2 + (d*x)/2)^3 - (118*a^3*tan(c/2 + (d*x)/2)^4)/5 + 6*a^3*tan(c/2 + (d*x)/2)^5 + 234*
a^3*tan(c/2 + (d*x)/2)^6 - 243*a^3*tan(c/2 + (d*x)/2)^7 + 259*a^3*tan(c/2 + (d*x)/2)^8 + a^3/7 + a^3*tan(c/2 +
(d*x)/2))/(d*(128*tan(c/2 + (d*x)/2)^7 + 128*tan(c/2 + (d*x)/2)^9)) + (259*a^3*tan(c/2 + (d*x)/2))/(128*d)