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

Optimal. Leaf size=238 \[ -\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{125 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-a^3 x \]

[Out]

-(a^3*x) + (125*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (a^3*Cot[c + d*x])/d + (a^3*Cot[c + d*x]^3)/(3*d) - (a^3*
Cot[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x]^7)/(7*d) - (115*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^3*Co
t[c + d*x]^3*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x])/(2*d) - (5*a^3*Cot[c + d*x]*Csc[c + d*x]^
3)/(64*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d)

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Rubi [A]  time = 0.355159, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2873, 3473, 8, 2611, 3770, 2607, 30, 3768} \[ -\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{125 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-a^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3*x) + (125*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (a^3*Cot[c + d*x])/d + (a^3*Cot[c + d*x]^3)/(3*d) - (a^3*
Cot[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x]^7)/(7*d) - (115*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^3*Co
t[c + d*x]^3*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x])/(2*d) - (5*a^3*Cot[c + d*x]*Csc[c + d*x]^
3)/(64*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d)

Rule 2873

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 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

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

Rule 2611

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 3770

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

Rule 2607

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 30

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

Rule 3768

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]

Rubi steps

\begin{align*} \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^6(c+d x)+3 a^3 \cot ^6(c+d x) \csc (c+d x)+3 a^3 \cot ^6(c+d x) \csc ^2(c+d x)+a^3 \cot ^6(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^6(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{1}{8} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx-a^3 \int \cot ^4(c+d x) \, dx-\frac{1}{2} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a^3 \cot ^7(c+d x)}{7 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{1}{16} \left (5 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+a^3 \int \cot ^2(c+d x) \, dx+\frac{1}{8} \left (15 a^3\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx\\ &=-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{1}{64} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac{1}{16} \left (15 a^3\right ) \int \csc (c+d x) \, dx-a^3 \int 1 \, dx\\ &=-a^3 x+\frac{15 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{1}{128} \left (5 a^3\right ) \int \csc (c+d x) \, dx\\ &=-a^3 x+\frac{125 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 1.17163, size = 279, normalized size = 1.17 \[ \frac{a^3 \left (118784 \tan \left (\frac{1}{2} (c+d x)\right )-118784 \cot \left (\frac{1}{2} (c+d x)\right )-108780 \csc ^2\left (\frac{1}{2} (c+d x)\right )+105 \sec ^8\left (\frac{1}{2} (c+d x)\right )+700 \sec ^6\left (\frac{1}{2} (c+d x)\right )-17010 \sec ^4\left (\frac{1}{2} (c+d x)\right )+108780 \sec ^2\left (\frac{1}{2} (c+d x)\right )-210000 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+210000 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+71936 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-15 (24 \sin (c+d x)+7) \csc ^8\left (\frac{1}{2} (c+d x)\right )+4 (732 \sin (c+d x)-175) \csc ^6\left (\frac{1}{2} (c+d x)\right )+(17010-4496 \sin (c+d x)) \csc ^4\left (\frac{1}{2} (c+d x)\right )+720 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^6\left (\frac{1}{2} (c+d x)\right )-5856 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )-215040 c-215040 d x\right )}{215040 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(-215040*c - 215040*d*x - 118784*Cot[(c + d*x)/2] - 108780*Csc[(c + d*x)/2]^2 + 210000*Log[Cos[(c + d*x)/
2]] - 210000*Log[Sin[(c + d*x)/2]] + 108780*Sec[(c + d*x)/2]^2 - 17010*Sec[(c + d*x)/2]^4 + 700*Sec[(c + d*x)/
2]^6 + 105*Sec[(c + d*x)/2]^8 + 71936*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + Csc[(c + d*x)/2]^4*(17010 - 4496*Sin
[c + d*x]) - 15*Csc[(c + d*x)/2]^8*(7 + 24*Sin[c + d*x]) + 4*Csc[(c + d*x)/2]^6*(-175 + 732*Sin[c + d*x]) + 11
8784*Tan[(c + d*x)/2] - 5856*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] + 720*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2]))/(
215040*d)

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Maple [A]  time = 0.096, size = 253, normalized size = 1.1 \begin{align*} -{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}}-{a}^{3}x-{\frac{{a}^{3}c}{d}}-{\frac{25\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{48\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{25\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{192\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{25\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{128\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{25\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{128\,d}}-{\frac{125\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{384\,d}}-{\frac{125\,{a}^{3}\cos \left ( dx+c \right ) }{128\,d}}-{\frac{125\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}}-{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c))^3,x)

[Out]

-1/5*a^3*cot(d*x+c)^5/d+1/3*a^3*cot(d*x+c)^3/d-a^3*cot(d*x+c)/d-a^3*x-1/d*a^3*c-25/48/d*a^3/sin(d*x+c)^6*cos(d
*x+c)^7+25/192/d*a^3/sin(d*x+c)^4*cos(d*x+c)^7-25/128/d*a^3/sin(d*x+c)^2*cos(d*x+c)^7-25/128*a^3*cos(d*x+c)^5/
d-125/384*a^3*cos(d*x+c)^3/d-125/128*a^3*cos(d*x+c)/d-125/128/d*a^3*ln(csc(d*x+c)-cot(d*x+c))-3/7/d*a^3/sin(d*
x+c)^7*cos(d*x+c)^7-1/8/d*a^3/sin(d*x+c)^8*cos(d*x+c)^7

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Maxima [A]  time = 1.67002, size = 358, normalized size = 1.5 \begin{align*} -\frac{1792 \,{\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 (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 )} - 840 \, 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 )} + \frac{11520 \, a^{3}}{\tan \left (d x + c\right )^{7}}}{26880 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/26880*(1792*(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*(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)) - 840*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*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) + 11520*a^3/tan(d*x + c)^7)/d

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Fricas [A]  time = 1.23547, size = 983, normalized size = 4.13 \begin{align*} -\frac{26880 \, a^{3} d x \cos \left (d x + c\right )^{8} - 107520 \, a^{3} d x \cos \left (d x + c\right )^{6} - 54390 \, a^{3} \cos \left (d x + c\right )^{7} + 161280 \, a^{3} d x \cos \left (d x + c\right )^{4} + 127750 \, a^{3} \cos \left (d x + c\right )^{5} - 107520 \, a^{3} d x \cos \left (d x + c\right )^{2} - 96250 \, a^{3} \cos \left (d x + c\right )^{3} + 26880 \, a^{3} d x + 26250 \, a^{3} \cos \left (d x + c\right ) - 13125 \,{\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, 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 ) + 13125 \,{\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, 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 ) - 256 \,{\left (116 \, a^{3} \cos \left (d x + c\right )^{7} - 406 \, a^{3} \cos \left (d x + c\right )^{5} + 350 \, a^{3} \cos \left (d x + c\right )^{3} - 105 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{26880 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/26880*(26880*a^3*d*x*cos(d*x + c)^8 - 107520*a^3*d*x*cos(d*x + c)^6 - 54390*a^3*cos(d*x + c)^7 + 161280*a^3
*d*x*cos(d*x + c)^4 + 127750*a^3*cos(d*x + c)^5 - 107520*a^3*d*x*cos(d*x + c)^2 - 96250*a^3*cos(d*x + c)^3 + 2
6880*a^3*d*x + 26250*a^3*cos(d*x + c) - 13125*(a^3*cos(d*x + c)^8 - 4*a^3*cos(d*x + c)^6 + 6*a^3*cos(d*x + c)^
4 - 4*a^3*cos(d*x + c)^2 + a^3)*log(1/2*cos(d*x + c) + 1/2) + 13125*(a^3*cos(d*x + c)^8 - 4*a^3*cos(d*x + c)^6
+ 6*a^3*cos(d*x + c)^4 - 4*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d*x + c) + 1/2) - 256*(116*a^3*cos(d*x + c)
^7 - 406*a^3*cos(d*x + c)^5 + 350*a^3*cos(d*x + c)^3 - 105*a^3*cos(d*x + c))*sin(d*x + c))/(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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**9*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.26472, size = 408, normalized size = 1.71 \begin{align*} \frac{105 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 720 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1120 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 3696 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 14280 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 560 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 77280 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 215040 \,{\left (d x + c\right )} a^{3} - 210000 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 122640 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{570750 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 122640 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 77280 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 560 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 14280 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 3696 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1120 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 720 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 105 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8}}}{215040 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/215040*(105*a^3*tan(1/2*d*x + 1/2*c)^8 + 720*a^3*tan(1/2*d*x + 1/2*c)^7 + 1120*a^3*tan(1/2*d*x + 1/2*c)^6 -
3696*a^3*tan(1/2*d*x + 1/2*c)^5 - 14280*a^3*tan(1/2*d*x + 1/2*c)^4 - 560*a^3*tan(1/2*d*x + 1/2*c)^3 + 77280*a^
3*tan(1/2*d*x + 1/2*c)^2 - 215040*(d*x + c)*a^3 - 210000*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 122640*a^3*tan(1
/2*d*x + 1/2*c) + (570750*a^3*tan(1/2*d*x + 1/2*c)^8 - 122640*a^3*tan(1/2*d*x + 1/2*c)^7 - 77280*a^3*tan(1/2*d
*x + 1/2*c)^6 + 560*a^3*tan(1/2*d*x + 1/2*c)^5 + 14280*a^3*tan(1/2*d*x + 1/2*c)^4 + 3696*a^3*tan(1/2*d*x + 1/2
*c)^3 - 1120*a^3*tan(1/2*d*x + 1/2*c)^2 - 720*a^3*tan(1/2*d*x + 1/2*c) - 105*a^3)/tan(1/2*d*x + 1/2*c)^8)/d