3.604 \(\int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=270 \[ -\frac{2 a^2 \cot ^{11}(c+d x)}{11 d}-\frac{4 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{17 a^2 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac{11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac{17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac{17 a^2 \cot (c+d x) \csc (c+d x)}{1024 d} \]

[Out]

(17*a^2*ArcTanh[Cos[c + d*x]])/(1024*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (4*a^2*Cot[c + d*x]^9)/(9*d) - (2*a^2
*Cot[c + d*x]^11)/(11*d) + (17*a^2*Cot[c + d*x]*Csc[c + d*x])/(1024*d) + (17*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/
(1536*d) - (11*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(384*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^2*Co
t[c + d*x]^5*Csc[c + d*x]^5)/(10*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^7)/(64*d) + (a^2*Cot[c + d*x]^3*Csc[c + d
*x]^7)/(24*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^7)/(12*d)

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Rubi [A]  time = 0.426146, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 18, 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} \[ -\frac{2 a^2 \cot ^{11}(c+d x)}{11 d}-\frac{4 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{17 a^2 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac{11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac{17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac{17 a^2 \cot (c+d x) \csc (c+d x)}{1024 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]

[Out]

(17*a^2*ArcTanh[Cos[c + d*x]])/(1024*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (4*a^2*Cot[c + d*x]^9)/(9*d) - (2*a^2
*Cot[c + d*x]^11)/(11*d) + (17*a^2*Cot[c + d*x]*Csc[c + d*x])/(1024*d) + (17*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/
(1536*d) - (11*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(384*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^2*Co
t[c + d*x]^5*Csc[c + d*x]^5)/(10*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^7)/(64*d) + (a^2*Cot[c + d*x]^3*Csc[c + d
*x]^7)/(24*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^7)/(12*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 ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^6(c+d x) \csc ^5(c+d x)+2 a^2 \cot ^6(c+d x) \csc ^6(c+d x)+a^2 \cot ^6(c+d x) \csc ^7(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^7(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx\\ &=-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{1}{12} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx-\frac{1}{2} a^2 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac{a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}+\frac{1}{8} a^2 \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx+\frac{1}{16} \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 \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{4 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^{11}(c+d x)}{11 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac{a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{1}{64} a^2 \int \csc ^7(c+d x) \, dx-\frac{1}{32} a^2 \int \csc ^5(c+d x) \, dx\\ &=-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{4 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^{11}(c+d x)}{11 d}+\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac{a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{1}{384} \left (5 a^2\right ) \int \csc ^5(c+d x) \, dx-\frac{1}{128} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{4 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^{11}(c+d x)}{11 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{256 d}+\frac{17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac{11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac{a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{1}{512} \left (5 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac{1}{256} \left (3 a^2\right ) \int \csc (c+d x) \, dx\\ &=\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{4 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^{11}(c+d x)}{11 d}+\frac{17 a^2 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac{17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac{11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac{a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{\left (5 a^2\right ) \int \csc (c+d x) \, dx}{1024}\\ &=\frac{17 a^2 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{4 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^{11}(c+d x)}{11 d}+\frac{17 a^2 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac{17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac{11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac{a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}\\ \end{align*}

Mathematica [A]  time = 4.59543, size = 197, normalized size = 0.73 \[ \frac{a^2 (\sin (c+d x)+1)^2 \left (30159360 \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{11}(c+d x) (29655040 \sin (c+d x)[phone] \sin (3 (c+d x))[phone] \sin (5 (c+d x))+3235840 \sin (7 (c+d x))-532480 \sin (9 (c+d x))+40960 \sin (11 (c+d x))[phone] \cos (2 (c+d x))[phone] \cos (4 (c+d x))-6944091 \cos (6 (c+d x))-746130 \cos (8 (c+d x))+58905 \cos (10 (c+d x))[phone])\right )}{1816657920 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]

[Out]

(a^2*(1 + Sin[c + d*x])^2*(30159360*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x
]^11*(65553642 + 67499586*Cos[2*(c + d*x)] + 25966248*Cos[4*(c + d*x)] - 6944091*Cos[6*(c + d*x)] - 746130*Cos
[8*(c + d*x)] + 58905*Cos[10*(c + d*x)] + 29655040*Sin[c + d*x] + 51445760*Sin[3*(c + d*x)] + 25600000*Sin[5*(
c + d*x)] + 3235840*Sin[7*(c + d*x)] - 532480*Sin[9*(c + d*x)] + 40960*Sin[11*(c + d*x)])))/(1816657920*d*(Cos
[(c + d*x)/2] + Sin[(c + d*x)/2])^4)

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Maple [A]  time = 0.089, size = 288, normalized size = 1.1 \begin{align*} -{\frac{17\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{120\,d \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{17\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{320\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{17\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{1920\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{17\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7680\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{17\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{5120\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{17\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5120\,d}}-{\frac{17\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3072\,d}}-{\frac{17\,{a}^{2}\cos \left ( dx+c \right ) }{1024\,d}}-{\frac{17\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{1024\,d}}-{\frac{2\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{11\,d \left ( \sin \left ( dx+c \right ) \right ) ^{11}}}-{\frac{8\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{99\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{16\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{693\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{12\,d \left ( \sin \left ( dx+c \right ) \right ) ^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^13*(a+a*sin(d*x+c))^2,x)

[Out]

-17/120/d*a^2/sin(d*x+c)^10*cos(d*x+c)^7-17/320/d*a^2/sin(d*x+c)^8*cos(d*x+c)^7-17/1920/d*a^2/sin(d*x+c)^6*cos
(d*x+c)^7+17/7680/d*a^2/sin(d*x+c)^4*cos(d*x+c)^7-17/5120/d*a^2/sin(d*x+c)^2*cos(d*x+c)^7-17/5120*a^2*cos(d*x+
c)^5/d-17/3072*a^2*cos(d*x+c)^3/d-17/1024*a^2*cos(d*x+c)/d-17/1024/d*a^2*ln(csc(d*x+c)-cot(d*x+c))-2/11/d*a^2/
sin(d*x+c)^11*cos(d*x+c)^7-8/99/d*a^2/sin(d*x+c)^9*cos(d*x+c)^7-16/693/d*a^2/sin(d*x+c)^7*cos(d*x+c)^7-1/12/d*
a^2/sin(d*x+c)^12*cos(d*x+c)^7

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Maxima [A]  time = 1.0597, size = 436, normalized size = 1.61 \begin{align*} -\frac{1155 \, a^{2}{\left (\frac{2 \,{\left (15 \, \cos \left (d x + c\right )^{11} - 85 \, \cos \left (d x + c\right )^{9} + 198 \, \cos \left (d x + c\right )^{7} + 198 \, \cos \left (d x + c\right )^{5} - 85 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{12} - 6 \, \cos \left (d x + c\right )^{10} + 15 \, \cos \left (d x + c\right )^{8} - 20 \, \cos \left (d x + c\right )^{6} + 15 \, \cos \left (d x + c\right )^{4} - 6 \, \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 )} + 2772 \, 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 )} + \frac{20480 \,{\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{2}}{\tan \left (d x + c\right )^{11}}}{7096320 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^13*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/7096320*(1155*a^2*(2*(15*cos(d*x + c)^11 - 85*cos(d*x + c)^9 + 198*cos(d*x + c)^7 + 198*cos(d*x + c)^5 - 85
*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^12 - 6*cos(d*x + c)^10 + 15*cos(d*x + c)^8 - 20*cos(d*x + c)^
6 + 15*cos(d*x + c)^4 - 6*cos(d*x + c)^2 + 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 2772*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)) + 20480*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a^2/tan(d*x + c)^
11)/d

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Fricas [A]  time = 1.33592, size = 1044, normalized size = 3.87 \begin{align*} -\frac{117810 \, a^{2} \cos \left (d x + c\right )^{11} - 667590 \, a^{2} \cos \left (d x + c\right )^{9} + 135828 \, a^{2} \cos \left (d x + c\right )^{7} + 1555092 \, a^{2} \cos \left (d x + c\right )^{5} - 667590 \, a^{2} \cos \left (d x + c\right )^{3} + 117810 \, a^{2} \cos \left (d x + c\right ) - 58905 \,{\left (a^{2} \cos \left (d x + c\right )^{12} - 6 \, a^{2} \cos \left (d x + c\right )^{10} + 15 \, a^{2} \cos \left (d x + c\right )^{8} - 20 \, a^{2} \cos \left (d x + c\right )^{6} + 15 \, a^{2} \cos \left (d x + c\right )^{4} - 6 \, 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 ) + 58905 \,{\left (a^{2} \cos \left (d x + c\right )^{12} - 6 \, a^{2} \cos \left (d x + c\right )^{10} + 15 \, a^{2} \cos \left (d x + c\right )^{8} - 20 \, a^{2} \cos \left (d x + c\right )^{6} + 15 \, a^{2} \cos \left (d x + c\right )^{4} - 6 \, 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 ) + 20480 \,{\left (8 \, a^{2} \cos \left (d x + c\right )^{11} - 44 \, a^{2} \cos \left (d x + c\right )^{9} + 99 \, a^{2} \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{7096320 \,{\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, 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)^13*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/7096320*(117810*a^2*cos(d*x + c)^11 - 667590*a^2*cos(d*x + c)^9 + 135828*a^2*cos(d*x + c)^7 + 1555092*a^2*c
os(d*x + c)^5 - 667590*a^2*cos(d*x + c)^3 + 117810*a^2*cos(d*x + c) - 58905*(a^2*cos(d*x + c)^12 - 6*a^2*cos(d
*x + c)^10 + 15*a^2*cos(d*x + c)^8 - 20*a^2*cos(d*x + c)^6 + 15*a^2*cos(d*x + c)^4 - 6*a^2*cos(d*x + c)^2 + a^
2)*log(1/2*cos(d*x + c) + 1/2) + 58905*(a^2*cos(d*x + c)^12 - 6*a^2*cos(d*x + c)^10 + 15*a^2*cos(d*x + c)^8 -
20*a^2*cos(d*x + c)^6 + 15*a^2*cos(d*x + c)^4 - 6*a^2*cos(d*x + c)^2 + a^2)*log(-1/2*cos(d*x + c) + 1/2) + 204
80*(8*a^2*cos(d*x + c)^11 - 44*a^2*cos(d*x + c)^9 + 99*a^2*cos(d*x + c)^7)*sin(d*x + c))/(d*cos(d*x + c)^12 -
6*d*cos(d*x + c)^10 + 15*d*cos(d*x + c)^8 - 20*d*cos(d*x + c)^6 + 15*d*cos(d*x + c)^4 - 6*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)**13*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.36471, size = 567, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^13*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/56770560*(1155*a^2*tan(1/2*d*x + 1/2*c)^12 + 5040*a^2*tan(1/2*d*x + 1/2*c)^11 + 5544*a^2*tan(1/2*d*x + 1/2*c
)^10 - 6160*a^2*tan(1/2*d*x + 1/2*c)^9 - 24255*a^2*tan(1/2*d*x + 1/2*c)^8 - 39600*a^2*tan(1/2*d*x + 1/2*c)^7 -
27720*a^2*tan(1/2*d*x + 1/2*c)^6 + 55440*a^2*tan(1/2*d*x + 1/2*c)^5 + 162855*a^2*tan(1/2*d*x + 1/2*c)^4 + 184
800*a^2*tan(1/2*d*x + 1/2*c)^3 + 55440*a^2*tan(1/2*d*x + 1/2*c)^2 - 942480*a^2*log(abs(tan(1/2*d*x + 1/2*c)))
- 554400*a^2*tan(1/2*d*x + 1/2*c) + (2924714*a^2*tan(1/2*d*x + 1/2*c)^12 + 554400*a^2*tan(1/2*d*x + 1/2*c)^11
- 55440*a^2*tan(1/2*d*x + 1/2*c)^10 - 184800*a^2*tan(1/2*d*x + 1/2*c)^9 - 162855*a^2*tan(1/2*d*x + 1/2*c)^8 -
55440*a^2*tan(1/2*d*x + 1/2*c)^7 + 27720*a^2*tan(1/2*d*x + 1/2*c)^6 + 39600*a^2*tan(1/2*d*x + 1/2*c)^5 + 24255
*a^2*tan(1/2*d*x + 1/2*c)^4 + 6160*a^2*tan(1/2*d*x + 1/2*c)^3 - 5544*a^2*tan(1/2*d*x + 1/2*c)^2 - 5040*a^2*tan
(1/2*d*x + 1/2*c) - 1155*a^2)/tan(1/2*d*x + 1/2*c)^12)/d