3.602 \(\int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=228 \[ -\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {13 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac {9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {13 a^2 \cot (c+d x) \csc (c+d x)}{256 d} \]

[Out]

13/256*a^2*arctanh(cos(d*x+c))/d-2/7*a^2*cot(d*x+c)^7/d-2/9*a^2*cot(d*x+c)^9/d+13/256*a^2*cot(d*x+c)*csc(d*x+c
)/d-9/128*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-1/32*a^2*cot(d*x+c)*csc(d*x+c)^5/d+1/16*a^2*cot(d*x+c)^3*csc(d*x+c)^5/d-1/10*a^2*cot(d*x+c)^5*csc(d*x+c)^5/
d

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Rubi [A]  time = 0.39, antiderivative size = 228, normalized size of antiderivative = 1.00, 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, 14} \[ -\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {13 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac {9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {13 a^2 \cot (c+d x) \csc (c+d x)}{256 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]

[Out]

(13*a^2*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) + (13*a^2
*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (9*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(128*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) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a
^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)

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 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 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 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 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]

Rubi steps

\begin {align*} \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^6(c+d x) \csc ^3(c+d x)+2 a^2 \cot ^6(c+d x) \csc ^4(c+d x)+a^2 \cot ^6(c+d x) \csc ^5(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{2} a^2 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx-\frac {1}{8} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{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 {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 {1}{16} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {1}{16} \left (5 a^2\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 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 {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 {1}{32} a^2 \int \csc ^5(c+d x) \, dx-\frac {1}{64} \left (5 a^2\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{128 d}-\frac {9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 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 {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 {1}{128} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{128} \left (5 a^2\right ) \int \csc (c+d x) \, dx\\ &=\frac {5 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}+\frac {13 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 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 {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 {1}{256} \left (3 a^2\right ) \int \csc (c+d x) \, dx\\ &=\frac {13 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}+\frac {13 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 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 {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}\\ \end {align*}

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Mathematica [A]  time = 1.29, size = 353, normalized size = 1.55 \[ -\frac {a^2 \csc ^{10}(c+d x) \left (1075200 \sin (2 (c+d x))+1044480 \sin (4 (c+d x))+414720 \sin (6 (c+d x))+51200 \sin (8 (c+d x))-5120 \sin (10 (c+d x))+2732940 \cos (c+d x)+1151640 \cos (3 (c+d x))+388248 \cos (5 (c+d x))-135870 \cos (7 (c+d x))-8190 \cos (9 (c+d x))+515970 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+859950 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-491400 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+184275 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-40950 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+4095 \cos (10 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-515970 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-859950 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+491400 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-184275 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+40950 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-4095 \cos (10 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )}{41287680 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]

[Out]

-1/41287680*(a^2*Csc[c + d*x]^10*(2732940*Cos[c + d*x] + 1151640*Cos[3*(c + d*x)] + 388248*Cos[5*(c + d*x)] -
135870*Cos[7*(c + d*x)] - 8190*Cos[9*(c + d*x)] - 515970*Log[Cos[(c + d*x)/2]] + 859950*Cos[2*(c + d*x)]*Log[C
os[(c + d*x)/2]] - 491400*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 184275*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2
]] - 40950*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 4095*Cos[10*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 515970*Log[
Sin[(c + d*x)/2]] - 859950*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 491400*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/
2]] - 184275*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 40950*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 4095*Cos[
10*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 1075200*Sin[2*(c + d*x)] + 1044480*Sin[4*(c + d*x)] + 414720*Sin[6*(c +
d*x)] + 51200*Sin[8*(c + d*x)] - 5120*Sin[10*(c + d*x)]))/d

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fricas [A]  time = 0.76, size = 327, normalized size = 1.43 \[ -\frac {8190 \, a^{2} \cos \left (d x + c\right )^{9} + 15540 \, a^{2} \cos \left (d x + c\right )^{7} - 69888 \, a^{2} \cos \left (d x + c\right )^{5} + 38220 \, a^{2} \cos \left (d x + c\right )^{3} - 8190 \, a^{2} \cos \left (d x + c\right ) - 4095 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 4095 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 5120 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{9} - 9 \, a^{2} \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{161280 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/161280*(8190*a^2*cos(d*x + c)^9 + 15540*a^2*cos(d*x + c)^7 - 69888*a^2*cos(d*x + c)^5 + 38220*a^2*cos(d*x +
c)^3 - 8190*a^2*cos(d*x + c) - 4095*(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) + 4095*(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) + 5120*(2*a^2*cos(d*x + c)^9 - 9*a^2*cos(d*x + c)^7)*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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giac [A]  time = 0.36, size = 324, normalized size = 1.42 \[ \frac {126 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 2160 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3990 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 7560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 13440 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 11340 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 65520 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 30240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {191906 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 30240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 11340 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 13440 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3990 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2160 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 126 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{1290240 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*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 + 315*a^2*tan(1/2*d*x + 1/2*c)^8 -
2160*a^2*tan(1/2*d*x + 1/2*c)^7 - 3990*a^2*tan(1/2*d*x + 1/2*c)^6 + 7560*a^2*tan(1/2*d*x + 1/2*c)^4 + 13440*a
^2*tan(1/2*d*x + 1/2*c)^3 + 11340*a^2*tan(1/2*d*x + 1/2*c)^2 - 65520*a^2*log(abs(tan(1/2*d*x + 1/2*c))) - 3024
0*a^2*tan(1/2*d*x + 1/2*c) + (191906*a^2*tan(1/2*d*x + 1/2*c)^10 + 30240*a^2*tan(1/2*d*x + 1/2*c)^9 - 11340*a^
2*tan(1/2*d*x + 1/2*c)^8 - 13440*a^2*tan(1/2*d*x + 1/2*c)^7 - 7560*a^2*tan(1/2*d*x + 1/2*c)^6 + 3990*a^2*tan(1
/2*d*x + 1/2*c)^4 + 2160*a^2*tan(1/2*d*x + 1/2*c)^3 - 315*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

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maple [A]  time = 0.40, size = 240, normalized size = 1.05 \[ -\frac {13 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{80 d \sin \left (d x +c \right )^{8}}-\frac {13 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{480 d \sin \left (d x +c \right )^{6}}+\frac {13 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{1920 d \sin \left (d x +c \right )^{4}}-\frac {13 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{1280 d \sin \left (d x +c \right )^{2}}-\frac {13 a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{1280 d}-\frac {13 a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{768 d}-\frac {13 a^{2} \cos \left (d x +c \right )}{256 d}-\frac {13 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256 d}-\frac {2 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{9 d \sin \left (d x +c \right )^{9}}-\frac {4 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{63 d \sin \left (d x +c \right )^{7}}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{10 d \sin \left (d x +c \right )^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x)

[Out]

-13/80/d*a^2/sin(d*x+c)^8*cos(d*x+c)^7-13/480/d*a^2/sin(d*x+c)^6*cos(d*x+c)^7+13/1920/d*a^2/sin(d*x+c)^4*cos(d
*x+c)^7-13/1280/d*a^2/sin(d*x+c)^2*cos(d*x+c)^7-13/1280*a^2*cos(d*x+c)^5/d-13/768*a^2*cos(d*x+c)^3/d-13/256*a^
2*cos(d*x+c)/d-13/256/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)^7-4/63/d*a^2/sin(d*x+c
)^7*cos(d*x+c)^7-1/10/d*a^2/sin(d*x+c)^10*cos(d*x+c)^7

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maxima [A]  time = 0.32, size = 273, normalized size = 1.20 \[ -\frac {63 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 210 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {5120 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*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)) + 210*a^2*(2*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 5
5*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)) + 5120*(9*tan(d*x + c)^2 + 7)*a^2/tan(d*x + c)^9)/d

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mupad [B]  time = 9.43, size = 357, normalized size = 1.57 \[ \frac {19\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{6144\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {9\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2304\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {9\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}+\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {19\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{6144\,d}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2304\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {13\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}+\frac {3\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}-\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^2)/sin(c + d*x)^11,x)

[Out]

(19*a^2*cot(c/2 + (d*x)/2)^6)/(6144*d) - (a^2*cot(c/2 + (d*x)/2)^3)/(96*d) - (3*a^2*cot(c/2 + (d*x)/2)^4)/(512
*d) - (9*a^2*cot(c/2 + (d*x)/2)^2)/(1024*d) + (3*a^2*cot(c/2 + (d*x)/2)^7)/(1792*d) - (a^2*cot(c/2 + (d*x)/2)^
8)/(4096*d) - (a^2*cot(c/2 + (d*x)/2)^9)/(2304*d) - (a^2*cot(c/2 + (d*x)/2)^10)/(10240*d) + (9*a^2*tan(c/2 + (
d*x)/2)^2)/(1024*d) + (a^2*tan(c/2 + (d*x)/2)^3)/(96*d) + (3*a^2*tan(c/2 + (d*x)/2)^4)/(512*d) - (19*a^2*tan(c
/2 + (d*x)/2)^6)/(6144*d) - (3*a^2*tan(c/2 + (d*x)/2)^7)/(1792*d) + (a^2*tan(c/2 + (d*x)/2)^8)/(4096*d) + (a^2
*tan(c/2 + (d*x)/2)^9)/(2304*d) + (a^2*tan(c/2 + (d*x)/2)^10)/(10240*d) - (13*a^2*log(tan(c/2 + (d*x)/2)))/(25
6*d) + (3*a^2*cot(c/2 + (d*x)/2))/(128*d) - (3*a^2*tan(c/2 + (d*x)/2))/(128*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**11*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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