3.391 \(\int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=218 \[ -\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{4 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{9 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac{9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{9 a^2 \cot (c+d x) \csc (c+d x)}{256 d} \]

[Out]

(-9*a^2*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (4*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*
Cot[c + d*x]^9)/(9*d) - (9*a^2*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (3*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d
) + (9*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(160*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) + (3*a^2*Cot[c + d
*x]*Csc[c + d*x]^7)/(80*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^7)/(10*d)

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Rubi [A]  time = 0.352659, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2873, 2611, 3768, 3770, 2607, 270} \[ -\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{4 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{9 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac{9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{9 a^2 \cot (c+d x) \csc (c+d x)}{256 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]

[Out]

(-9*a^2*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (4*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*
Cot[c + d*x]^9)/(9*d) - (9*a^2*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (3*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d
) + (9*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(160*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) + (3*a^2*Cot[c + d
*x]*Csc[c + d*x]^7)/(80*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^7)/(10*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 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 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]

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^4(c+d x) \csc ^5(c+d x)+2 a^2 \cot ^4(c+d x) \csc ^6(c+d x)+a^2 \cot ^4(c+d x) \csc ^7(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+a^2 \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^4(c+d x) \csc ^6(c+d x) \, dx\\ &=-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac{1}{10} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx-\frac{1}{8} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac{1}{80} \left (3 a^2\right ) \int \csc ^7(c+d x) \, dx+\frac{1}{16} a^2 \int \csc ^5(c+d x) \, dx+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{4 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac{1}{32} a^2 \int \csc ^5(c+d x) \, dx+\frac{1}{64} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{4 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{128 d}-\frac{3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac{1}{128} \left (3 a^2\right ) \int \csc (c+d x) \, dx+\frac{1}{128} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{4 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{9 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac{3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac{1}{256} \left (3 a^2\right ) \int \csc (c+d x) \, dx\\ &=-\frac{9 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{4 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{9 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac{3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}\\ \end{align*}

Mathematica [A]  time = 1.1789, size = 353, normalized size = 1.62 \[ -\frac{a^2 \csc ^{10}(c+d x) \left (1720320 \sin (2 (c+d x))+1228800 \sin (4 (c+d x))+184320 \sin (6 (c+d x))-40960 \sin (8 (c+d x))+4096 \sin (10 (c+d x))+3219300 \cos (c+d x)+1237320 \cos (3 (c+d x))-278712 \cos (5 (c+d x))-54810 \cos (7 (c+d x))+5670 \cos (9 (c+d x))-357210 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-595350 \cos (2 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+340200 \cos (4 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-127575 \cos (6 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+28350 \cos (8 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-2835 \cos (10 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+357210 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+595350 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-340200 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+127575 \cos (6 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-28350 \cos (8 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+2835 \cos (10 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{41287680 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]

[Out]

-(a^2*Csc[c + d*x]^10*(3219300*Cos[c + d*x] + 1237320*Cos[3*(c + d*x)] - 278712*Cos[5*(c + d*x)] - 54810*Cos[7
*(c + d*x)] + 5670*Cos[9*(c + d*x)] + 357210*Log[Cos[(c + d*x)/2]] - 595350*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)
/2]] + 340200*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 127575*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 28350*C
os[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 2835*Cos[10*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 357210*Log[Sin[(c + d*x
)/2]] + 595350*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 340200*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 127575
*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 28350*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 2835*Cos[10*(c + d*x)
]*Log[Sin[(c + d*x)/2]] + 1720320*Sin[2*(c + d*x)] + 1228800*Sin[4*(c + d*x)] + 184320*Sin[6*(c + d*x)] - 4096
0*Sin[8*(c + d*x)] + 4096*Sin[10*(c + d*x)]))/(41287680*d)

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Maple [A]  time = 0.085, size = 248, normalized size = 1.1 \begin{align*} -{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{32\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{128\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{256\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{256\,d}}+{\frac{9\,{a}^{2}\cos \left ( dx+c \right ) }{256\,d}}+{\frac{9\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{256\,d}}-{\frac{2\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{9\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{8\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{63\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{16\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{315\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{10\,d \left ( \sin \left ( dx+c \right ) \right ) ^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x)

[Out]

-3/16/d*a^2/sin(d*x+c)^8*cos(d*x+c)^5-3/32/d*a^2/sin(d*x+c)^6*cos(d*x+c)^5-3/128/d*a^2/sin(d*x+c)^4*cos(d*x+c)
^5+3/256/d*a^2/sin(d*x+c)^2*cos(d*x+c)^5+3/256*a^2*cos(d*x+c)^3/d+9/256*a^2*cos(d*x+c)/d+9/256/d*a^2*ln(csc(d*
x+c)-cot(d*x+c))-2/9/d*a^2/sin(d*x+c)^9*cos(d*x+c)^5-8/63/d*a^2/sin(d*x+c)^7*cos(d*x+c)^5-16/315/d*a^2/sin(d*x
+c)^5*cos(d*x+c)^5-1/10/d*a^2/sin(d*x+c)^10*cos(d*x+c)^5

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Maxima [A]  time = 1.15422, size = 382, normalized size = 1.75 \begin{align*} \frac{63 \, a^{2}{\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 )} + 630 \, a^{2}{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \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} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac{1024 \,{\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/161280*(63*a^2*(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)) + 630*a^2*(2*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*
cos(d*x + c)^3 + 3*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)
- 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 1024*(63*tan(d*x + c)^4 + 90*tan(d*x + c)^2 + 35)*a^2/
tan(d*x + c)^9)/d

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Fricas [A]  time = 1.54106, size = 887, normalized size = 4.07 \begin{align*} \frac{5670 \, a^{2} \cos \left (d x + c\right )^{9} - 26460 \, a^{2} \cos \left (d x + c\right )^{7} + 16128 \, a^{2} \cos \left (d x + c\right )^{5} + 26460 \, a^{2} \cos \left (d x + c\right )^{3} - 5670 \, a^{2} \cos \left (d x + c\right ) - 2835 \,{\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, 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 ) + 2835 \,{\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, 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 ) + 1024 \,{\left (8 \, a^{2} \cos \left (d x + c\right )^{9} - 36 \, a^{2} \cos \left (d x + c\right )^{7} + 63 \, a^{2} \cos \left (d x + c\right )^{5}\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/161280*(5670*a^2*cos(d*x + c)^9 - 26460*a^2*cos(d*x + c)^7 + 16128*a^2*cos(d*x + c)^5 + 26460*a^2*cos(d*x +
c)^3 - 5670*a^2*cos(d*x + c) - 2835*(a^2*cos(d*x + c)^10 - 5*a^2*cos(d*x + c)^8 + 10*a^2*cos(d*x + c)^6 - 10*a
^2*cos(d*x + c)^4 + 5*a^2*cos(d*x + c)^2 - a^2)*log(1/2*cos(d*x + c) + 1/2) + 2835*(a^2*cos(d*x + c)^10 - 5*a^
2*cos(d*x + c)^8 + 10*a^2*cos(d*x + c)^6 - 10*a^2*cos(d*x + c)^4 + 5*a^2*cos(d*x + c)^2 - a^2)*log(-1/2*cos(d*
x + c) + 1/2) + 1024*(8*a^2*cos(d*x + c)^9 - 36*a^2*cos(d*x + c)^7 + 63*a^2*cos(d*x + c)^5)*sin(d*x + c))/(d*c
os(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*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)**4*csc(d*x+c)**11*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.48593, size = 482, normalized size = 2.21 \begin{align*} \frac{126 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 560 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 945 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 720 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 630 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 4032 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 7560 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 6720 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1260 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 45360 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 30240 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{132858 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 30240 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 1260 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 6720 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 7560 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 4032 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 630 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 720 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 945 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 560 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 126 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10}}}{1290240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1290240*(126*a^2*tan(1/2*d*x + 1/2*c)^10 + 560*a^2*tan(1/2*d*x + 1/2*c)^9 + 945*a^2*tan(1/2*d*x + 1/2*c)^8 +
720*a^2*tan(1/2*d*x + 1/2*c)^7 - 630*a^2*tan(1/2*d*x + 1/2*c)^6 - 4032*a^2*tan(1/2*d*x + 1/2*c)^5 - 7560*a^2*
tan(1/2*d*x + 1/2*c)^4 - 6720*a^2*tan(1/2*d*x + 1/2*c)^3 + 1260*a^2*tan(1/2*d*x + 1/2*c)^2 + 45360*a^2*log(abs
(tan(1/2*d*x + 1/2*c))) + 30240*a^2*tan(1/2*d*x + 1/2*c) - (132858*a^2*tan(1/2*d*x + 1/2*c)^10 + 30240*a^2*tan
(1/2*d*x + 1/2*c)^9 + 1260*a^2*tan(1/2*d*x + 1/2*c)^8 - 6720*a^2*tan(1/2*d*x + 1/2*c)^7 - 7560*a^2*tan(1/2*d*x
+ 1/2*c)^6 - 4032*a^2*tan(1/2*d*x + 1/2*c)^5 - 630*a^2*tan(1/2*d*x + 1/2*c)^4 + 720*a^2*tan(1/2*d*x + 1/2*c)^
3 + 945*a^2*tan(1/2*d*x + 1/2*c)^2 + 560*a^2*tan(1/2*d*x + 1/2*c) + 126*a^2)/tan(1/2*d*x + 1/2*c)^10)/d