3.406 \(\int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=216 \[ -\frac{a^3 \cot ^9(c+d x)}{3 d}-\frac{a^3 \cot ^7(c+d x)}{d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{21 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac{29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{21 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]

[Out]

(-21*a^3*ArcTanh[Cos[c + d*x]])/(256*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/d - (a^3*Cot[c +
d*x]^9)/(3*d) - (21*a^3*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (7*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (2
9*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(160*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) + (3*a^3*Cot[c + d*x]
*Csc[c + d*x]^7)/(80*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^7)/(10*d)

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Rubi [A]  time = 0.389845, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2873, 2607, 14, 2611, 3768, 3770, 270} \[ -\frac{a^3 \cot ^9(c+d x)}{3 d}-\frac{a^3 \cot ^7(c+d x)}{d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{21 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac{29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{21 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]

[Out]

(-21*a^3*ArcTanh[Cos[c + d*x]])/(256*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/d - (a^3*Cot[c +
d*x]^9)/(3*d) - (21*a^3*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (7*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (2
9*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(160*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) + (3*a^3*Cot[c + d*x]
*Csc[c + d*x]^7)/(80*d) - (a^3*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 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 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 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 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))^3 \, dx &=\int \left (a^3 \cot ^4(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^5(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^6(c+d x)+a^3 \cot ^4(c+d x) \csc ^7(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx+a^3 \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^6(c+d x) \, dx\\ &=-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac{1}{10} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx-\frac{1}{8} \left (9 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac{1}{80} \left (3 a^3\right ) \int \csc ^7(c+d x) \, dx+\frac{1}{16} \left (3 a^3\right ) \int \csc ^5(c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{d}-\frac{a^3 \cot ^9(c+d x)}{3 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac{1}{32} a^3 \int \csc ^5(c+d x) \, dx+\frac{1}{64} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{d}-\frac{a^3 \cot ^9(c+d x)}{3 d}-\frac{9 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac{1}{128} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\frac{1}{128} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac{9 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{d}-\frac{a^3 \cot ^9(c+d x)}{3 d}-\frac{21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac{1}{256} \left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac{21 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{d}-\frac{a^3 \cot ^9(c+d x)}{3 d}-\frac{21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}\\ \end{align*}

Mathematica [A]  time = 2.14824, size = 366, normalized size = 1.69 \[ \frac{a^3 (\sin (c+d x)+1)^3 \left (4096 \tan \left (\frac{1}{2} (c+d x)\right )-4096 \cot \left (\frac{1}{2} (c+d x)\right )-1260 \csc ^2\left (\frac{1}{2} (c+d x)\right )+6 \sec ^{10}\left (\frac{1}{2} (c+d x)\right )+75 \sec ^8\left (\frac{1}{2} (c+d x)\right )-390 \sec ^6\left (\frac{1}{2} (c+d x)\right )-180 \sec ^4\left (\frac{1}{2} (c+d x)\right )+1260 \sec ^2\left (\frac{1}{2} (c+d x)\right )+5040 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-5040 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+64 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-2 (10 \sin (c+d x)+3) \csc ^{10}\left (\frac{1}{2} (c+d x)\right )+5 (4 \sin (c+d x)-15) \csc ^8\left (\frac{1}{2} (c+d x)\right )+6 (42 \sin (c+d x)+65) \csc ^6\left (\frac{1}{2} (c+d x)\right )-4 (\sin (c+d x)-45) \csc ^4\left (\frac{1}{2} (c+d x)\right )+40 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^8\left (\frac{1}{2} (c+d x)\right )-40 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^6\left (\frac{1}{2} (c+d x)\right )-504 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )\right )}{61440 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(1 + Sin[c + d*x])^3*(-4096*Cot[(c + d*x)/2] - 1260*Csc[(c + d*x)/2]^2 - 5040*Log[Cos[(c + d*x)/2]] + 504
0*Log[Sin[(c + d*x)/2]] + 1260*Sec[(c + d*x)/2]^2 - 180*Sec[(c + d*x)/2]^4 - 390*Sec[(c + d*x)/2]^6 + 75*Sec[(
c + d*x)/2]^8 + 6*Sec[(c + d*x)/2]^10 + 64*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 4*Csc[(c + d*x)/2]^4*(-45 + Sin
[c + d*x]) + 5*Csc[(c + d*x)/2]^8*(-15 + 4*Sin[c + d*x]) - 2*Csc[(c + d*x)/2]^10*(3 + 10*Sin[c + d*x]) + 6*Csc
[(c + d*x)/2]^6*(65 + 42*Sin[c + d*x]) + 4096*Tan[(c + d*x)/2] - 504*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] - 40*
Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2] + 40*Sec[(c + d*x)/2]^8*Tan[(c + d*x)/2]))/(61440*d*(Cos[(c + d*x)/2] + Si
n[(c + d*x)/2])^6)

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Maple [A]  time = 0.096, size = 248, normalized size = 1.2 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{2\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{15\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{7\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{7\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{32\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{7\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{128\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{7\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{256\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{7\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{256\,d}}+{\frac{21\,{a}^{3}\cos \left ( dx+c \right ) }{256\,d}}+{\frac{21\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{256\,d}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{{a}^{3} \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))^3,x)

[Out]

-1/3/d*a^3/sin(d*x+c)^7*cos(d*x+c)^5-2/15/d*a^3/sin(d*x+c)^5*cos(d*x+c)^5-7/16/d*a^3/sin(d*x+c)^8*cos(d*x+c)^5
-7/32/d*a^3/sin(d*x+c)^6*cos(d*x+c)^5-7/128/d*a^3/sin(d*x+c)^4*cos(d*x+c)^5+7/256/d*a^3/sin(d*x+c)^2*cos(d*x+c
)^5+7/256*a^3*cos(d*x+c)^3/d+21/256*a^3*cos(d*x+c)/d+21/256/d*a^3*ln(csc(d*x+c)-cot(d*x+c))-1/3/d*a^3/sin(d*x+
c)^9*cos(d*x+c)^5-1/10/d*a^3/sin(d*x+c)^10*cos(d*x+c)^5

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Maxima [A]  time = 1.09919, size = 416, normalized size = 1.93 \begin{align*} \frac{21 \, a^{3}{\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^{3}{\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{1536 \,{\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}} - \frac{512 \,{\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{53760 \, 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))^3,x, algorithm="maxima")

[Out]

1/53760*(21*a^3*(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^3*(2*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*c
os(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)) - 1536*(7*tan(d*x + c)^2 + 5)*a^3/tan(d*x + c)^7 - 512*(6
3*tan(d*x + c)^4 + 90*tan(d*x + c)^2 + 35)*a^3/tan(d*x + c)^9)/d

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Fricas [A]  time = 1.27438, size = 871, normalized size = 4.03 \begin{align*} \frac{630 \, a^{3} \cos \left (d x + c\right )^{9} - 2940 \, a^{3} \cos \left (d x + c\right )^{7} + 768 \, a^{3} \cos \left (d x + c\right )^{5} + 2940 \, a^{3} \cos \left (d x + c\right )^{3} - 630 \, a^{3} \cos \left (d x + c\right ) - 315 \,{\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, 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 ) + 315 \,{\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, 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 ) + 512 \,{\left (2 \, a^{3} \cos \left (d x + c\right )^{9} - 9 \, a^{3} \cos \left (d x + c\right )^{7} + 12 \, a^{3} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{7680 \,{\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))^3,x, algorithm="fricas")

[Out]

1/7680*(630*a^3*cos(d*x + c)^9 - 2940*a^3*cos(d*x + c)^7 + 768*a^3*cos(d*x + c)^5 + 2940*a^3*cos(d*x + c)^3 -
630*a^3*cos(d*x + c) - 315*(a^3*cos(d*x + c)^10 - 5*a^3*cos(d*x + c)^8 + 10*a^3*cos(d*x + c)^6 - 10*a^3*cos(d*
x + c)^4 + 5*a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2) + 315*(a^3*cos(d*x + c)^10 - 5*a^3*cos(d*x
+ c)^8 + 10*a^3*cos(d*x + c)^6 - 10*a^3*cos(d*x + c)^4 + 5*a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos(d*x + c) + 1
/2) + 512*(2*a^3*cos(d*x + c)^9 - 9*a^3*cos(d*x + c)^7 + 12*a^3*cos(d*x + c)^5)*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*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))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.53611, size = 482, normalized size = 2.23 \begin{align*} \frac{6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 40 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 105 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 120 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 30 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 384 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 840 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 960 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 60 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 5040 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 3600 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{14762 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 3600 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 60 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 960 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 840 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 384 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 30 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 120 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 105 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 40 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10}}}{61440 \, 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))^3,x, algorithm="giac")

[Out]

1/61440*(6*a^3*tan(1/2*d*x + 1/2*c)^10 + 40*a^3*tan(1/2*d*x + 1/2*c)^9 + 105*a^3*tan(1/2*d*x + 1/2*c)^8 + 120*
a^3*tan(1/2*d*x + 1/2*c)^7 - 30*a^3*tan(1/2*d*x + 1/2*c)^6 - 384*a^3*tan(1/2*d*x + 1/2*c)^5 - 840*a^3*tan(1/2*
d*x + 1/2*c)^4 - 960*a^3*tan(1/2*d*x + 1/2*c)^3 + 60*a^3*tan(1/2*d*x + 1/2*c)^2 + 5040*a^3*log(abs(tan(1/2*d*x
+ 1/2*c))) + 3600*a^3*tan(1/2*d*x + 1/2*c) - (14762*a^3*tan(1/2*d*x + 1/2*c)^10 + 3600*a^3*tan(1/2*d*x + 1/2*
c)^9 + 60*a^3*tan(1/2*d*x + 1/2*c)^8 - 960*a^3*tan(1/2*d*x + 1/2*c)^7 - 840*a^3*tan(1/2*d*x + 1/2*c)^6 - 384*a
^3*tan(1/2*d*x + 1/2*c)^5 - 30*a^3*tan(1/2*d*x + 1/2*c)^4 + 120*a^3*tan(1/2*d*x + 1/2*c)^3 + 105*a^3*tan(1/2*d
*x + 1/2*c)^2 + 40*a^3*tan(1/2*d*x + 1/2*c) + 6*a^3)/tan(1/2*d*x + 1/2*c)^10)/d