\(\int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx\) [600]

Optimal result

Integrand size = 29, antiderivative size = 182 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {45 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {35 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d} \]

[Out]

45/128*a^2*arctanh(cos(d*x+c))/d-2/7*a^2*cot(d*x+c)^7/d-35/128*a^2*cot(d*x+c)*csc(d*x+c)/d+5/24*a^2*cot(d*x+c)
^3*csc(d*x+c)/d-1/6*a^2*cot(d*x+c)^5*csc(d*x+c)/d-5/64*a^2*cot(d*x+c)*csc(d*x+c)^3/d+5/48*a^2*cot(d*x+c)^3*csc
(d*x+c)^3/d-1/8*a^2*cot(d*x+c)^5*csc(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2952, 2691, 3855, 2687, 30, 3853} \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {45 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {35 a^2 \cot (c+d x) \csc (c+d x)}{128 d} \]

[In]

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

[Out]

(45*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (35*a^2*Cot[c + d*x]*Csc[c + d*x])/(12
8*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x])/(24*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x])/(6*d) - (5*a^2*Cot[c + d
*x]*Csc[c + d*x]^3)/(64*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^3
)/(8*d)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 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]

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}& = \int \left (a^2 \cot ^6(c+d x) \csc (c+d x)+2 a^2 \cot ^6(c+d x) \csc ^2(c+d x)+a^2 \cot ^6(c+d x) \csc ^3(c+d x)\right ) \, dx \\ & = a^2 \int \cot ^6(c+d x) \csc (c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx \\ & = -\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{8} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx-\frac {1}{6} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\frac {\left (2 a^2\right ) \text {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {1}{16} \left (5 a^2\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {1}{8} \left (5 a^2\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx \\ & = -\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{64} \left (5 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{16} \left (5 a^2\right ) \int \csc (c+d x) \, dx \\ & = \frac {5 a^2 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {35 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{128} \left (5 a^2\right ) \int \csc (c+d x) \, dx \\ & = \frac {45 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {35 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(401\) vs. \(2(182)=364\).

Time = 0.29 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.20 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=a^2 \left (\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{7 d}-\frac {83 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {19 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{224 d}+\frac {17 \csc ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {5 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{224 d}+\frac {\csc ^6\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^6\left (\frac {1}{2} (c+d x)\right )}{448 d}-\frac {\csc ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}+\frac {45 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}-\frac {45 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}+\frac {83 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {17 \sec ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {\sec ^6\left (\frac {1}{2} (c+d x)\right )}{512 d}+\frac {\sec ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}-\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{7 d}+\frac {19 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{224 d}-\frac {5 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{224 d}+\frac {\sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{448 d}\right ) \]

[In]

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

[Out]

a^2*(Cot[(c + d*x)/2]/(7*d) - (83*Csc[(c + d*x)/2]^2)/(512*d) - (19*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(224*
d) + (17*Csc[(c + d*x)/2]^4)/(1024*d) + (5*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(224*d) + Csc[(c + d*x)/2]^6/(
512*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^6)/(448*d) - Csc[(c + d*x)/2]^8/(2048*d) + (45*Log[Cos[(c + d*x)/2
]])/(128*d) - (45*Log[Sin[(c + d*x)/2]])/(128*d) + (83*Sec[(c + d*x)/2]^2)/(512*d) - (17*Sec[(c + d*x)/2]^4)/(
1024*d) - Sec[(c + d*x)/2]^6/(512*d) + Sec[(c + d*x)/2]^8/(2048*d) - Tan[(c + d*x)/2]/(7*d) + (19*Sec[(c + d*x
)/2]^2*Tan[(c + d*x)/2])/(224*d) - (5*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(224*d) + (Sec[(c + d*x)/2]^6*Tan[(
c + d*x)/2])/(448*d))

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.15

method	result	size
parallelrisch	\(\frac {\left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {32 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-32 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+96 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+256 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-160 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-720 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+160 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-256 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-96 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+40 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{7}-1\right ) a^{2}}{2048 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}\)	\(210\)
risch	\(\frac {a^{2} \left (581 \,{\mathrm e}^{15 i \left (d x +c \right )}-2065 \,{\mathrm e}^{13 i \left (d x +c \right )}+8960 i {\mathrm e}^{10 i \left (d x +c \right )}+21 \,{\mathrm e}^{11 i \left (d x +c \right )}+1792 i {\mathrm e}^{14 i \left (d x +c \right )}-5705 \,{\mathrm e}^{9 i \left (d x +c \right )}-5376 i {\mathrm e}^{4 i \left (d x +c \right )}-5705 \,{\mathrm e}^{7 i \left (d x +c \right )}-8960 i {\mathrm e}^{8 i \left (d x +c \right )}+21 \,{\mathrm e}^{5 i \left (d x +c \right )}+256 i {\mathrm e}^{2 i \left (d x +c \right )}-2065 \,{\mathrm e}^{3 i \left (d x +c \right )}+5376 i {\mathrm e}^{6 i \left (d x +c \right )}+581 \,{\mathrm e}^{i \left (d x +c \right )}-1792 i {\mathrm e}^{12 i \left (d x +c \right )}-256 i\right )}{448 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}+\frac {45 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}-\frac {45 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}\)	\(238\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {2 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\)	\(255\)
default	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {2 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\)	\(255\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/2048*(tan(1/2*d*x+1/2*c)^16+32/7*tan(1/2*d*x+1/2*c)^15-32*tan(1/2*d*x+1/2*c)^13-40*tan(1/2*d*x+1/2*c)^12+96*
tan(1/2*d*x+1/2*c)^11+256*tan(1/2*d*x+1/2*c)^10-160*tan(1/2*d*x+1/2*c)^9-720*ln(tan(1/2*d*x+1/2*c))*tan(1/2*d*
x+1/2*c)^8+160*tan(1/2*d*x+1/2*c)^7-256*tan(1/2*d*x+1/2*c)^6-96*tan(1/2*d*x+1/2*c)^5+40*tan(1/2*d*x+1/2*c)^4+3
2*tan(1/2*d*x+1/2*c)^3-32/7*tan(1/2*d*x+1/2*c)-1)*a^2/d/tan(1/2*d*x+1/2*c)^8

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.40 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {512 \, a^{2} \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - 1162 \, a^{2} \cos \left (d x + c\right )^{7} + 3066 \, a^{2} \cos \left (d x + c\right )^{5} - 2310 \, a^{2} \cos \left (d x + c\right )^{3} + 630 \, a^{2} \cos \left (d x + c\right ) - 315 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 315 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{1792 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/1792*(512*a^2*cos(d*x + c)^7*sin(d*x + c) - 1162*a^2*cos(d*x + c)^7 + 3066*a^2*cos(d*x + c)^5 - 2310*a^2*co
s(d*x + c)^3 + 630*a^2*cos(d*x + c) - 315*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a^2*cos(d*x + c)^4 -
4*a^2*cos(d*x + c)^2 + a^2)*log(1/2*cos(d*x + c) + 1/2) + 315*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a
^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + a^2)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^8 - 4*d*cos(d*x
+ c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**9*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.21 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {7 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 56 \, a^{2} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {1536 \, a^{2}}{\tan \left (d x + c\right )^{7}}}{5376 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/5376*(7*a^2*(2*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 55*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^
8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x +
c) - 1)) - 56*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)) + 1536*a^2/tan(d*x + c)^7)/
d

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.43 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {7 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 32 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 224 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 280 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 672 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1792 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5040 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 1120 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {13698 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1120 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1792 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 672 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 280 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 224 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 32 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{14336 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/14336*(7*a^2*tan(1/2*d*x + 1/2*c)^8 + 32*a^2*tan(1/2*d*x + 1/2*c)^7 - 224*a^2*tan(1/2*d*x + 1/2*c)^5 - 280*a
^2*tan(1/2*d*x + 1/2*c)^4 + 672*a^2*tan(1/2*d*x + 1/2*c)^3 + 1792*a^2*tan(1/2*d*x + 1/2*c)^2 - 5040*a^2*log(ab
s(tan(1/2*d*x + 1/2*c))) - 1120*a^2*tan(1/2*d*x + 1/2*c) + (13698*a^2*tan(1/2*d*x + 1/2*c)^8 + 1120*a^2*tan(1/
2*d*x + 1/2*c)^7 - 1792*a^2*tan(1/2*d*x + 1/2*c)^6 - 672*a^2*tan(1/2*d*x + 1/2*c)^5 + 280*a^2*tan(1/2*d*x + 1/
2*c)^4 + 224*a^2*tan(1/2*d*x + 1/2*c)^3 - 32*a^2*tan(1/2*d*x + 1/2*c) - 7*a^2)/tan(1/2*d*x + 1/2*c)^8)/d

Mupad [B] (verification not implemented)

Time = 13.00 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.13 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2\,\left (7\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-32\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+224\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+280\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-1792\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+1120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-1120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+1792\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-280\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-224\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+5040\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\right )}{14336\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^2)/sin(c + d*x)^9,x)

[Out]

-(a^2*(7*cos(c/2 + (d*x)/2)^16 - 7*sin(c/2 + (d*x)/2)^16 - 32*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^15 + 32*co
s(c/2 + (d*x)/2)^15*sin(c/2 + (d*x)/2) + 224*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^13 + 280*cos(c/2 + (d*x)/
2)^4*sin(c/2 + (d*x)/2)^12 - 672*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^11 - 1792*cos(c/2 + (d*x)/2)^6*sin(c/
2 + (d*x)/2)^10 + 1120*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^9 - 1120*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2
)^7 + 1792*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^6 + 672*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^5 - 280*c
os(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^4 - 224*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)^3 + 5040*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)^8))/(14336*d*cos(c/2 + (d*x)/2)^8*sin(c
/2 + (d*x)/2)^8)