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

Optimal. Leaf size=246 \[ -\frac {a^3 \cot ^{11}(c+d x)}{11 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {19 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]

[Out]

19/256*a^3*arctanh(cos(d*x+c))/d-4/7*a^3*cot(d*x+c)^7/d-5/9*a^3*cot(d*x+c)^9/d-1/11*a^3*cot(d*x+c)^11/d+19/256
*a^3*cot(d*x+c)*csc(d*x+c)/d-7/128*a^3*cot(d*x+c)*csc(d*x+c)^3/d+5/48*a^3*cot(d*x+c)^3*csc(d*x+c)^3/d-1/8*a^3*
cot(d*x+c)^5*csc(d*x+c)^3/d-3/32*a^3*cot(d*x+c)*csc(d*x+c)^5/d+3/16*a^3*cot(d*x+c)^3*csc(d*x+c)^5/d-3/10*a^3*c
ot(d*x+c)^5*csc(d*x+c)^5/d

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Rubi [A]  time = 0.44, antiderivative size = 246, normalized size of antiderivative = 1.00, 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, 2611, 3768, 3770, 2607, 14, 270} \[ -\frac {a^3 \cot ^{11}(c+d x)}{11 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {19 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(19*a^3*ArcTanh[Cos[c + d*x]])/(256*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (5*a^3*Cot[c + d*x]^9)/(9*d) - (a^3*Co
t[c + d*x]^11)/(11*d) + (19*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) + (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) - (3*a^3*Cot[c +
d*x]*Csc[c + d*x]^5)/(32*d) + (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (3*a^3*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 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]

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 ^6(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^6(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^5(c+d x)+a^3 \cot ^6(c+d x) \csc ^6(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{8} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx-\frac {1}{2} \left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{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 {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {1}{16} \left (5 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {1}{16} \left (9 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 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 {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{64} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{32} \left (3 a^3\right ) \int \csc ^5(c+d x) \, dx\\ &=-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 d}+\frac {5 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 {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 {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{128} \left (5 a^3\right ) \int \csc (c+d x) \, dx-\frac {1}{128} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=\frac {5 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 d}+\frac {19 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 {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 {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{256} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac {19 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 d}+\frac {19 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 {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 {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}\\ \end {align*}

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Mathematica [A]  time = 3.58, size = 187, normalized size = 0.76 \[ \frac {a^3 (\sin (c+d x)+1)^3 \left (16853760 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{10}(c+d x) (14477694 \sin (c+d x)+5875716 \sin (3 (c+d x))+7902972 \sin (5 (c+d x))-414645 \sin (7 (c+d x))-65835 \sin (9 (c+d x))[phone] \cos (2 (c+d x))+839680 \cos (4 (c+d x))-2149120 \cos (6 (c+d x))-568320 \cos (8 (c+d x))+47360 \cos (10 (c+d x))[phone])\right )}{227082240 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(1 + Sin[c + d*x])^3*(16853760*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x
]^10*(10050560 + 12423680*Cos[2*(c + d*x)] + 839680*Cos[4*(c + d*x)] - 2149120*Cos[6*(c + d*x)] - 568320*Cos[8
*(c + d*x)] + 47360*Cos[10*(c + d*x)] + 14477694*Sin[c + d*x] + 5875716*Sin[3*(c + d*x)] + 7902972*Sin[5*(c +
d*x)] - 414645*Sin[7*(c + d*x)] - 65835*Sin[9*(c + d*x)])))/(227082240*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])
^6)

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fricas [A]  time = 0.69, size = 360, normalized size = 1.46 \[ \frac {189440 \, a^{3} \cos \left (d x + c\right )^{11} - 1041920 \, a^{3} \cos \left (d x + c\right )^{9} + 1013760 \, a^{3} \cos \left (d x + c\right )^{7} + 65835 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 65835 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 462 \, {\left (285 \, a^{3} \cos \left (d x + c\right )^{9} - 50 \, a^{3} \cos \left (d x + c\right )^{7} - 2432 \, a^{3} \cos \left (d x + c\right )^{5} + 1330 \, a^{3} \cos \left (d x + c\right )^{3} - 285 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1774080 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/1774080*(189440*a^3*cos(d*x + c)^11 - 1041920*a^3*cos(d*x + c)^9 + 1013760*a^3*cos(d*x + c)^7 + 65835*(a^3*c
os(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)*sin(d*x + c) - 65835*(a^3*cos(d*x + c)^10 - 5*a^3*cos(d*x + c)^8 + 10*a^3*co
s(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)*sin(d*x + c) -
462*(285*a^3*cos(d*x + c)^9 - 50*a^3*cos(d*x + c)^7 - 2432*a^3*cos(d*x + c)^5 + 1330*a^3*cos(d*x + c)^3 - 285
*a^3*cos(d*x + c))*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)*sin(d*x + c))

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giac [A]  time = 0.46, size = 388, normalized size = 1.58 \[ \frac {630 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 4158 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 8470 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3465 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 40590 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 57750 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6930 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 138600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 244860 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 152460 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1053360 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 568260 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {3181018 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 568260 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 152460 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 244860 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 138600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 6930 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 57750 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40590 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3465 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8470 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4158 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 630 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}}}{14192640 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/14192640*(630*a^3*tan(1/2*d*x + 1/2*c)^11 + 4158*a^3*tan(1/2*d*x + 1/2*c)^10 + 8470*a^3*tan(1/2*d*x + 1/2*c)
^9 - 3465*a^3*tan(1/2*d*x + 1/2*c)^8 - 40590*a^3*tan(1/2*d*x + 1/2*c)^7 - 57750*a^3*tan(1/2*d*x + 1/2*c)^6 + 6
930*a^3*tan(1/2*d*x + 1/2*c)^5 + 138600*a^3*tan(1/2*d*x + 1/2*c)^4 + 244860*a^3*tan(1/2*d*x + 1/2*c)^3 + 15246
0*a^3*tan(1/2*d*x + 1/2*c)^2 - 1053360*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 568260*a^3*tan(1/2*d*x + 1/2*c) +
(3181018*a^3*tan(1/2*d*x + 1/2*c)^11 + 568260*a^3*tan(1/2*d*x + 1/2*c)^10 - 152460*a^3*tan(1/2*d*x + 1/2*c)^9
- 244860*a^3*tan(1/2*d*x + 1/2*c)^8 - 138600*a^3*tan(1/2*d*x + 1/2*c)^7 - 6930*a^3*tan(1/2*d*x + 1/2*c)^6 + 57
750*a^3*tan(1/2*d*x + 1/2*c)^5 + 40590*a^3*tan(1/2*d*x + 1/2*c)^4 + 3465*a^3*tan(1/2*d*x + 1/2*c)^3 - 8470*a^3
*tan(1/2*d*x + 1/2*c)^2 - 4158*a^3*tan(1/2*d*x + 1/2*c) - 630*a^3)/tan(1/2*d*x + 1/2*c)^11)/d

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maple [A]  time = 0.41, size = 264, normalized size = 1.07 \[ -\frac {19 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{80 d \sin \left (d x +c \right )^{8}}-\frac {19 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{480 d \sin \left (d x +c \right )^{6}}+\frac {19 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{1920 d \sin \left (d x +c \right )^{4}}-\frac {19 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{1280 d \sin \left (d x +c \right )^{2}}-\frac {19 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{1280 d}-\frac {19 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{768 d}-\frac {19 a^{3} \cos \left (d x +c \right )}{256 d}-\frac {19 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256 d}-\frac {37 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{99 d \sin \left (d x +c \right )^{9}}-\frac {74 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{693 d \sin \left (d x +c \right )^{7}}-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{10 d \sin \left (d x +c \right )^{10}}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{11 d \sin \left (d x +c \right )^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^3,x)

[Out]

-19/80/d*a^3/sin(d*x+c)^8*cos(d*x+c)^7-19/480/d*a^3/sin(d*x+c)^6*cos(d*x+c)^7+19/1920/d*a^3/sin(d*x+c)^4*cos(d
*x+c)^7-19/1280/d*a^3/sin(d*x+c)^2*cos(d*x+c)^7-19/1280*a^3*cos(d*x+c)^5/d-19/768*a^3*cos(d*x+c)^3/d-19/256*a^
3*cos(d*x+c)/d-19/256/d*a^3*ln(csc(d*x+c)-cot(d*x+c))-37/99/d*a^3/sin(d*x+c)^9*cos(d*x+c)^7-74/693/d*a^3/sin(d
*x+c)^7*cos(d*x+c)^7-3/10/d*a^3/sin(d*x+c)^10*cos(d*x+c)^7-1/11/d*a^3/sin(d*x+c)^11*cos(d*x+c)^7

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maxima [A]  time = 0.49, size = 308, normalized size = 1.25 \[ -\frac {2079 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2310 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {84480 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{3}}{\tan \left (d x + c\right )^{9}} + \frac {2560 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{3}}{\tan \left (d x + c\right )^{11}}}{1774080 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/1774080*(2079*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*c
os(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)) + 2310*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)) + 84480*(9*tan(d*x + c)^2 + 7)*a^3/tan(d*x + c)
^9 + 2560*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a^3/tan(d*x + c)^11)/d

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mupad [B]  time = 9.98, size = 433, normalized size = 1.76 \[ \frac {25\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{6144\,d}-\frac {53\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3072\,d}-\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {11\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {41\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {11\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{18432\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}+\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {53\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3072\,d}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {25\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{6144\,d}-\frac {41\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{18432\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}-\frac {19\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}+\frac {41\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1024\,d}-\frac {41\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1024\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^3)/sin(c + d*x)^12,x)

[Out]

(25*a^3*cot(c/2 + (d*x)/2)^6)/(6144*d) - (53*a^3*cot(c/2 + (d*x)/2)^3)/(3072*d) - (5*a^3*cot(c/2 + (d*x)/2)^4)
/(512*d) - (a^3*cot(c/2 + (d*x)/2)^5)/(2048*d) - (11*a^3*cot(c/2 + (d*x)/2)^2)/(1024*d) + (41*a^3*cot(c/2 + (d
*x)/2)^7)/(14336*d) + (a^3*cot(c/2 + (d*x)/2)^8)/(4096*d) - (11*a^3*cot(c/2 + (d*x)/2)^9)/(18432*d) - (3*a^3*c
ot(c/2 + (d*x)/2)^10)/(10240*d) - (a^3*cot(c/2 + (d*x)/2)^11)/(22528*d) + (11*a^3*tan(c/2 + (d*x)/2)^2)/(1024*
d) + (53*a^3*tan(c/2 + (d*x)/2)^3)/(3072*d) + (5*a^3*tan(c/2 + (d*x)/2)^4)/(512*d) + (a^3*tan(c/2 + (d*x)/2)^5
)/(2048*d) - (25*a^3*tan(c/2 + (d*x)/2)^6)/(6144*d) - (41*a^3*tan(c/2 + (d*x)/2)^7)/(14336*d) - (a^3*tan(c/2 +
(d*x)/2)^8)/(4096*d) + (11*a^3*tan(c/2 + (d*x)/2)^9)/(18432*d) + (3*a^3*tan(c/2 + (d*x)/2)^10)/(10240*d) + (a
^3*tan(c/2 + (d*x)/2)^11)/(22528*d) - (19*a^3*log(tan(c/2 + (d*x)/2)))/(256*d) + (41*a^3*cot(c/2 + (d*x)/2))/(
1024*d) - (41*a^3*tan(c/2 + (d*x)/2))/(1024*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)**12*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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