3.622 \(\int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=286 \[ -\frac {a^3 \cot ^{13}(c+d x)}{13 d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {27 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}+\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d} \]

[Out]

27/1024*a^3*arctanh(cos(d*x+c))/d-4/7*a^3*cot(d*x+c)^7/d-a^3*cot(d*x+c)^9/d-6/11*a^3*cot(d*x+c)^11/d-1/13*a^3*
cot(d*x+c)^13/d+27/1024*a^3*cot(d*x+c)*csc(d*x+c)/d+9/512*a^3*cot(d*x+c)*csc(d*x+c)^3/d-3/128*a^3*cot(d*x+c)*c
sc(d*x+c)^5/d+1/16*a^3*cot(d*x+c)^3*csc(d*x+c)^5/d-1/10*a^3*cot(d*x+c)^5*csc(d*x+c)^5/d-3/64*a^3*cot(d*x+c)*cs
c(d*x+c)^7/d+1/8*a^3*cot(d*x+c)^3*csc(d*x+c)^7/d-1/4*a^3*cot(d*x+c)^5*csc(d*x+c)^7/d

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Rubi [A]  time = 0.46, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 21, 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} \[ -\frac {a^3 \cot ^{13}(c+d x)}{13 d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {27 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}+\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^8*(a + a*Sin[c + d*x])^3,x]

[Out]

(27*a^3*ArcTanh[Cos[c + d*x]])/(1024*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x]^9)/d - (6*a^3*Cot[c
+ d*x]^11)/(11*d) - (a^3*Cot[c + d*x]^13)/(13*d) + (27*a^3*Cot[c + d*x]*Csc[c + d*x])/(1024*d) + (9*a^3*Cot[c
+ d*x]*Csc[c + d*x]^3)/(512*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(128*d) + (a^3*Cot[c + d*x]^3*Csc[c + d*
x]^5)/(16*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^7)/(64*d) + (a^3*
Cot[c + d*x]^3*Csc[c + d*x]^7)/(8*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^7)/(4*d)

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 ^8(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^6(c+d x) \csc ^5(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^6(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^7(c+d x)+a^3 \cot ^6(c+d x) \csc ^8(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^8(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^7(c+d x) \, dx\\ &=-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {1}{2} a^3 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx-\frac {1}{4} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int x^6 \left (1+x^2\right )^3 \, 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 )^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}+\frac {1}{16} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {1}{8} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int \left (x^6+3 x^8+3 x^{10}+x^{12}\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {1}{32} a^3 \int \csc ^5(c+d x) \, dx-\frac {1}{64} \left (3 a^3\right ) \int \csc ^7(c+d x) \, dx\\ &=-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}+\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {1}{128} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{128} \left (5 a^3\right ) \int \csc ^5(c+d x) \, dx\\ &=-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{256 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {1}{256} \left (3 a^3\right ) \int \csc (c+d x) \, dx-\frac {1}{512} \left (15 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=\frac {3 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}+\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {\left (15 a^3\right ) \int \csc (c+d x) \, dx}{1024}\\ &=\frac {27 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}+\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]  time = 6.50, size = 283, normalized size = 0.99 \[ \frac {27 (a \sin (c+d x)+a)^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{1024 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}-\frac {27 (a \sin (c+d x)+a)^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{1024 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {\cot (c+d x) \csc ^{12}(c+d x) (a \sin (c+d x)+a)^3 (-194159966 \sin (c+d x)-182107926 \sin (3 (c+d x))-123736613 \sin (5 (c+d x))+4571567 \sin (7 (c+d x))+1846845 \sin (9 (c+d x))-135135 \sin (11 (c+d x))-243712000 \cos (2 (c+d x))-11079680 \cos (4 (c+d x))[phone] \cos (6 (c+d x))+9420800 \cos (8 (c+d x))-1433600 \cos (10 (c+d x))+102400 \cos (12 (c+d x))-200294400)}{5248122880 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^8*(a + a*Sin[c + d*x])^3,x]

[Out]

(27*Log[Cos[(c + d*x)/2]]*(a + a*Sin[c + d*x])^3)/(1024*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (27*Log[S
in[(c + d*x)/2]]*(a + a*Sin[c + d*x])^3)/(1024*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (Cot[c + d*x]*Csc[
c + d*x]^12*(a + a*Sin[c + d*x])^3*(-200294400 - 243712000*Cos[2*(c + d*x)] - 11079680*Cos[4*(c + d*x)] + 4329
4720*Cos[6*(c + d*x)] + 9420800*Cos[8*(c + d*x)] - 1433600*Cos[10*(c + d*x)] + 102400*Cos[12*(c + d*x)] - 1941
59966*Sin[c + d*x] - 182107926*Sin[3*(c + d*x)] - 123736613*Sin[5*(c + d*x)] + 4571567*Sin[7*(c + d*x)] + 1846
845*Sin[9*(c + d*x)] - 135135*Sin[11*(c + d*x)]))/(5248122880*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

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fricas [A]  time = 0.99, size = 417, normalized size = 1.46 \[ \frac {409600 \, a^{3} \cos \left (d x + c\right )^{13} - 2662400 \, a^{3} \cos \left (d x + c\right )^{11} + 7321600 \, a^{3} \cos \left (d x + c\right )^{9} - 5857280 \, a^{3} \cos \left (d x + c\right )^{7} + 135135 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 135135 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 2002 \, {\left (135 \, a^{3} \cos \left (d x + c\right )^{11} - 765 \, a^{3} \cos \left (d x + c\right )^{9} + 758 \, a^{3} \cos \left (d x + c\right )^{7} + 1782 \, a^{3} \cos \left (d x + c\right )^{5} - 765 \, a^{3} \cos \left (d x + c\right )^{3} + 135 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{10250240 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^14*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/10250240*(409600*a^3*cos(d*x + c)^13 - 2662400*a^3*cos(d*x + c)^11 + 7321600*a^3*cos(d*x + c)^9 - 5857280*a^
3*cos(d*x + c)^7 + 135135*(a^3*cos(d*x + c)^12 - 6*a^3*cos(d*x + c)^10 + 15*a^3*cos(d*x + c)^8 - 20*a^3*cos(d*
x + c)^6 + 15*a^3*cos(d*x + c)^4 - 6*a^3*cos(d*x + c)^2 + a^3)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 1351
35*(a^3*cos(d*x + c)^12 - 6*a^3*cos(d*x + c)^10 + 15*a^3*cos(d*x + c)^8 - 20*a^3*cos(d*x + c)^6 + 15*a^3*cos(d
*x + c)^4 - 6*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 2002*(135*a^3*cos(d*x + c)
^11 - 765*a^3*cos(d*x + c)^9 + 758*a^3*cos(d*x + c)^7 + 1782*a^3*cos(d*x + c)^5 - 765*a^3*cos(d*x + c)^3 + 135
*a^3*cos(d*x + c))*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)*sin(d*x + c))

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giac [A]  time = 0.50, size = 452, normalized size = 1.58 \[ \frac {770 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 5005 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 11830 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 8008 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 20020 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 65065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 94380 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 40040 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 150150 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385385 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 450450 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80080 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2162160 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 1401400 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {6875958 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1401400 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 80080 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 450450 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 385385 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 150150 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 40040 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 94380 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 65065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 20020 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8008 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 11830 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5005 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 770 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13}}}{82001920 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^14*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/82001920*(770*a^3*tan(1/2*d*x + 1/2*c)^13 + 5005*a^3*tan(1/2*d*x + 1/2*c)^12 + 11830*a^3*tan(1/2*d*x + 1/2*c
)^11 + 8008*a^3*tan(1/2*d*x + 1/2*c)^10 - 20020*a^3*tan(1/2*d*x + 1/2*c)^9 - 65065*a^3*tan(1/2*d*x + 1/2*c)^8
- 94380*a^3*tan(1/2*d*x + 1/2*c)^7 - 40040*a^3*tan(1/2*d*x + 1/2*c)^6 + 150150*a^3*tan(1/2*d*x + 1/2*c)^5 + 38
5385*a^3*tan(1/2*d*x + 1/2*c)^4 + 450450*a^3*tan(1/2*d*x + 1/2*c)^3 + 80080*a^3*tan(1/2*d*x + 1/2*c)^2 - 21621
60*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 1401400*a^3*tan(1/2*d*x + 1/2*c) + (6875958*a^3*tan(1/2*d*x + 1/2*c)^1
3 + 1401400*a^3*tan(1/2*d*x + 1/2*c)^12 - 80080*a^3*tan(1/2*d*x + 1/2*c)^11 - 450450*a^3*tan(1/2*d*x + 1/2*c)^
10 - 385385*a^3*tan(1/2*d*x + 1/2*c)^9 - 150150*a^3*tan(1/2*d*x + 1/2*c)^8 + 40040*a^3*tan(1/2*d*x + 1/2*c)^7
+ 94380*a^3*tan(1/2*d*x + 1/2*c)^6 + 65065*a^3*tan(1/2*d*x + 1/2*c)^5 + 20020*a^3*tan(1/2*d*x + 1/2*c)^4 - 800
8*a^3*tan(1/2*d*x + 1/2*c)^3 - 11830*a^3*tan(1/2*d*x + 1/2*c)^2 - 5005*a^3*tan(1/2*d*x + 1/2*c) - 770*a^3)/tan
(1/2*d*x + 1/2*c)^13)/d

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maple [A]  time = 0.41, size = 312, normalized size = 1.09 \[ -\frac {9 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{640 d \sin \left (d x +c \right )^{6}}+\frac {9 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{2560 d \sin \left (d x +c \right )^{4}}-\frac {27 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{5120 d \sin \left (d x +c \right )^{2}}-\frac {45 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{143 d \sin \left (d x +c \right )^{11}}-\frac {20 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{143 d \sin \left (d x +c \right )^{9}}-\frac {9 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{40 d \sin \left (d x +c \right )^{10}}-\frac {27 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{320 d \sin \left (d x +c \right )^{8}}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{12}}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{13 d \sin \left (d x +c \right )^{13}}-\frac {40 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{1001 d \sin \left (d x +c \right )^{7}}-\frac {27 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5120 d}-\frac {9 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{1024 d}-\frac {27 a^{3} \cos \left (d x +c \right )}{1024 d}-\frac {27 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^14*(a+a*sin(d*x+c))^3,x)

[Out]

-9/640/d*a^3/sin(d*x+c)^6*cos(d*x+c)^7+9/2560/d*a^3/sin(d*x+c)^4*cos(d*x+c)^7-27/5120/d*a^3/sin(d*x+c)^2*cos(d
*x+c)^7-45/143/d*a^3/sin(d*x+c)^11*cos(d*x+c)^7-20/143/d*a^3/sin(d*x+c)^9*cos(d*x+c)^7-9/40/d*a^3/sin(d*x+c)^1
0*cos(d*x+c)^7-27/320/d*a^3/sin(d*x+c)^8*cos(d*x+c)^7-1/4/d*a^3/sin(d*x+c)^12*cos(d*x+c)^7-1/13/d*a^3/sin(d*x+
c)^13*cos(d*x+c)^7-40/1001/d*a^3/sin(d*x+c)^7*cos(d*x+c)^7-27/5120*a^3*cos(d*x+c)^5/d-9/1024*a^3*cos(d*x+c)^3/
d-27/1024*a^3*cos(d*x+c)/d-27/1024/d*a^3*ln(csc(d*x+c)-cot(d*x+c))

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maxima [A]  time = 0.36, size = 368, normalized size = 1.29 \[ -\frac {15015 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{11} - 85 \, \cos \left (d x + c\right )^{9} + 198 \, \cos \left (d x + c\right )^{7} + 198 \, \cos \left (d x + c\right )^{5} - 85 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{12} - 6 \, \cos \left (d x + c\right )^{10} + 15 \, \cos \left (d x + c\right )^{8} - 20 \, \cos \left (d x + c\right )^{6} + 15 \, \cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12012 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {133120 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{3}}{\tan \left (d x + c\right )^{11}} + \frac {10240 \, {\left (429 \, \tan \left (d x + c\right )^{6} + 1001 \, \tan \left (d x + c\right )^{4} + 819 \, \tan \left (d x + c\right )^{2} + 231\right )} a^{3}}{\tan \left (d x + c\right )^{13}}}{30750720 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^14*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/30750720*(15015*a^3*(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)) + 12012
*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))/(co
s(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)) + 133120*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a^3/tan(d*x +
c)^11 + 10240*(429*tan(d*x + c)^6 + 1001*tan(d*x + c)^4 + 819*tan(d*x + c)^2 + 231)*a^3/tan(d*x + c)^13)/d

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mupad [B]  time = 10.90, size = 509, normalized size = 1.78 \[ \frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {45\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8192\,d}-\frac {77\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}-\frac {15\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8192\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {33\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{28672\,d}+\frac {13\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16384\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4096\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {13\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{90112\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{16384\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{106496\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {45\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8192\,d}+\frac {77\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}+\frac {15\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8192\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {33\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{28672\,d}-\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16384\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4096\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{90112\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{16384\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{106496\,d}-\frac {27\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{1024\,d}+\frac {35\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2048\,d}-\frac {35\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2048\,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)^14,x)

[Out]

(a^3*cot(c/2 + (d*x)/2)^6)/(2048*d) - (45*a^3*cot(c/2 + (d*x)/2)^3)/(8192*d) - (77*a^3*cot(c/2 + (d*x)/2)^4)/(
16384*d) - (15*a^3*cot(c/2 + (d*x)/2)^5)/(8192*d) - (a^3*cot(c/2 + (d*x)/2)^2)/(1024*d) + (33*a^3*cot(c/2 + (d
*x)/2)^7)/(28672*d) + (13*a^3*cot(c/2 + (d*x)/2)^8)/(16384*d) + (a^3*cot(c/2 + (d*x)/2)^9)/(4096*d) - (a^3*cot
(c/2 + (d*x)/2)^10)/(10240*d) - (13*a^3*cot(c/2 + (d*x)/2)^11)/(90112*d) - (a^3*cot(c/2 + (d*x)/2)^12)/(16384*
d) - (a^3*cot(c/2 + (d*x)/2)^13)/(106496*d) + (a^3*tan(c/2 + (d*x)/2)^2)/(1024*d) + (45*a^3*tan(c/2 + (d*x)/2)
^3)/(8192*d) + (77*a^3*tan(c/2 + (d*x)/2)^4)/(16384*d) + (15*a^3*tan(c/2 + (d*x)/2)^5)/(8192*d) - (a^3*tan(c/2
+ (d*x)/2)^6)/(2048*d) - (33*a^3*tan(c/2 + (d*x)/2)^7)/(28672*d) - (13*a^3*tan(c/2 + (d*x)/2)^8)/(16384*d) -
(a^3*tan(c/2 + (d*x)/2)^9)/(4096*d) + (a^3*tan(c/2 + (d*x)/2)^10)/(10240*d) + (13*a^3*tan(c/2 + (d*x)/2)^11)/(
90112*d) + (a^3*tan(c/2 + (d*x)/2)^12)/(16384*d) + (a^3*tan(c/2 + (d*x)/2)^13)/(106496*d) - (27*a^3*log(tan(c/
2 + (d*x)/2)))/(1024*d) + (35*a^3*cot(c/2 + (d*x)/2))/(2048*d) - (35*a^3*tan(c/2 + (d*x)/2))/(2048*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)**14*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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