3.404 \(\int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=176 \[ -\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {27 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{128 d} \]

[Out]

-27/128*a^3*arctanh(cos(d*x+c))/d-4/5*a^3*cot(d*x+c)^5/d-3/7*a^3*cot(d*x+c)^7/d-27/128*a^3*cot(d*x+c)*csc(d*x+
c)/d+23/64*a^3*cot(d*x+c)*csc(d*x+c)^3/d-1/2*a^3*cot(d*x+c)^3*csc(d*x+c)^3/d+1/16*a^3*cot(d*x+c)*csc(d*x+c)^5/
d-1/8*a^3*cot(d*x+c)^3*csc(d*x+c)^5/d

________________________________________________________________________________________

Rubi [A]  time = 0.34, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 2607, 30, 2611, 3768, 3770, 14} \[ -\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {27 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{128 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

(-27*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x]^7)/(7*d) - (27*a^
3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (23*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^3*Cot[c + d*x]^3*Csc[c
+ d*x]^3)/(2*d) + (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^4(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^4(c+d x)+a^3 \cot ^4(c+d x) \csc ^5(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx+a^3 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}-\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {1}{8} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx-\frac {1}{2} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int x^4 \, 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 ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {1}{16} a^3 \int \csc ^5(c+d x) \, dx+\frac {1}{8} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {1}{64} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\frac {1}{16} \left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac {3 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {1}{128} \left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac {27 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 5.09, size = 313, normalized size = 1.78 \[ -\frac {a^3 \sin (c+d x) (\sin (c+d x)+1)^3 \left (10 (7 \csc (c+d x)+24) \csc ^8\left (\frac {1}{2} (c+d x)\right )+8 (105 \csc (c+d x)-76) \csc ^6\left (\frac {1}{2} (c+d x)\right )-4 (1715 \csc (c+d x)+856) \csc ^4\left (\frac {1}{2} (c+d x)\right )+8 (945 \csc (c+d x)+1664) \csc ^2\left (\frac {1}{2} (c+d x)\right )-4 \left ((1056 \cos (c+d x)+517 \cos (2 (c+d x))+104 \cos (3 (c+d x))+703) \sec ^8\left (\frac {1}{2} (c+d x)\right )+4480 \sin ^8\left (\frac {1}{2} (c+d x)\right ) \csc ^9(c+d x)+13440 \sin ^6\left (\frac {1}{2} (c+d x)\right ) \csc ^7(c+d x)-27440 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^5(c+d x)+7560 \sin ^2\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)-7560 \csc (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )\right )}{143360 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

-1/143360*(a^3*(10*Csc[(c + d*x)/2]^8*(24 + 7*Csc[c + d*x]) + 8*Csc[(c + d*x)/2]^6*(-76 + 105*Csc[c + d*x]) +
8*Csc[(c + d*x)/2]^2*(1664 + 945*Csc[c + d*x]) - 4*Csc[(c + d*x)/2]^4*(856 + 1715*Csc[c + d*x]) - 4*(-7560*Csc
[c + d*x]*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + (703 + 1056*Cos[c + d*x] + 517*Cos[2*(c + d*x)] +
104*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^8 + 7560*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2 - 27440*Csc[c + d*x]^5*Sin[(
c + d*x)/2]^4 + 13440*Csc[c + d*x]^7*Sin[(c + d*x)/2]^6 + 4480*Csc[c + d*x]^9*Sin[(c + d*x)/2]^8))*Sin[c + d*x
]*(1 + Sin[c + d*x])^3)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 271, normalized size = 1.54 \[ \frac {1890 \, a^{3} \cos \left (d x + c\right )^{7} + 2030 \, a^{3} \cos \left (d x + c\right )^{5} - 6930 \, a^{3} \cos \left (d x + c\right )^{3} + 1890 \, a^{3} \cos \left (d x + c\right ) - 945 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 945 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 256 \, {\left (13 \, a^{3} \cos \left (d x + c\right )^{7} - 28 \, a^{3} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{8960 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/8960*(1890*a^3*cos(d*x + c)^7 + 2030*a^3*cos(d*x + c)^5 - 6930*a^3*cos(d*x + c)^3 + 1890*a^3*cos(d*x + c) -
945*(a^3*cos(d*x + c)^8 - 4*a^3*cos(d*x + c)^6 + 6*a^3*cos(d*x + c)^4 - 4*a^3*cos(d*x + c)^2 + a^3)*log(1/2*co
s(d*x + c) + 1/2) + 945*(a^3*cos(d*x + c)^8 - 4*a^3*cos(d*x + c)^6 + 6*a^3*cos(d*x + c)^4 - 4*a^3*cos(d*x + c)
^2 + a^3)*log(-1/2*cos(d*x + c) + 1/2) + 256*(13*a^3*cos(d*x + c)^7 - 28*a^3*cos(d*x + c)^5)*sin(d*x + c))/(d*
cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)

________________________________________________________________________________________

giac [A]  time = 0.38, size = 293, normalized size = 1.66 \[ \frac {35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 240 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 112 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1960 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1680 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15120 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 9520 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {41094 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 9520 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1680 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1960 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 112 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 35 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{71680 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/71680*(35*a^3*tan(1/2*d*x + 1/2*c)^8 + 240*a^3*tan(1/2*d*x + 1/2*c)^7 + 560*a^3*tan(1/2*d*x + 1/2*c)^6 + 112
*a^3*tan(1/2*d*x + 1/2*c)^5 - 1960*a^3*tan(1/2*d*x + 1/2*c)^4 - 3920*a^3*tan(1/2*d*x + 1/2*c)^3 - 1680*a^3*tan
(1/2*d*x + 1/2*c)^2 + 15120*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 9520*a^3*tan(1/2*d*x + 1/2*c) - (41094*a^3*ta
n(1/2*d*x + 1/2*c)^8 + 9520*a^3*tan(1/2*d*x + 1/2*c)^7 - 1680*a^3*tan(1/2*d*x + 1/2*c)^6 - 3920*a^3*tan(1/2*d*
x + 1/2*c)^5 - 1960*a^3*tan(1/2*d*x + 1/2*c)^4 + 112*a^3*tan(1/2*d*x + 1/2*c)^3 + 560*a^3*tan(1/2*d*x + 1/2*c)
^2 + 240*a^3*tan(1/2*d*x + 1/2*c) + 35*a^3)/tan(1/2*d*x + 1/2*c)^8)/d

________________________________________________________________________________________

maple [A]  time = 0.39, size = 200, normalized size = 1.14 \[ -\frac {13 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{35 d \sin \left (d x +c \right )^{5}}-\frac {9 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{6}}-\frac {9 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{64 d \sin \left (d x +c \right )^{4}}+\frac {9 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{128 d \sin \left (d x +c \right )^{2}}+\frac {9 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{128 d}+\frac {27 a^{3} \cos \left (d x +c \right )}{128 d}+\frac {27 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128 d}-\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^3,x)

[Out]

-13/35/d*a^3/sin(d*x+c)^5*cos(d*x+c)^5-9/16/d*a^3/sin(d*x+c)^6*cos(d*x+c)^5-9/64/d*a^3/sin(d*x+c)^4*cos(d*x+c)
^5+9/128/d*a^3/sin(d*x+c)^2*cos(d*x+c)^5+9/128*a^3*cos(d*x+c)^3/d+27/128*a^3*cos(d*x+c)/d+27/128/d*a^3*ln(csc(
d*x+c)-cot(d*x+c))-3/7/d*a^3/sin(d*x+c)^7*cos(d*x+c)^5-1/8/d*a^3/sin(d*x+c)^8*cos(d*x+c)^5

________________________________________________________________________________________

maxima [A]  time = 0.33, size = 246, normalized size = 1.40 \[ \frac {35 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 280 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1792 \, a^{3}}{\tan \left (d x + c\right )^{5}} - \frac {768 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}}}{8960 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/8960*(35*a^3*(2*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*cos(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)) + 280*a^3*(2*(3*cos(d*x + c)^5 + 8*cos(d*x + c)^3 - 3*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 +
3*cos(d*x + c)^2 - 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 1792*a^3/tan(d*x + c)^5 - 768*(7*
tan(d*x + c)^2 + 5)*a^3/tan(d*x + c)^7)/d

________________________________________________________________________________________

mupad [B]  time = 9.22, size = 319, normalized size = 1.81 \[ \frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}+\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {27\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d}-\frac {17\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {17\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^3)/sin(c + d*x)^9,x)

[Out]

(3*a^3*cot(c/2 + (d*x)/2)^2)/(128*d) + (7*a^3*cot(c/2 + (d*x)/2)^3)/(128*d) + (7*a^3*cot(c/2 + (d*x)/2)^4)/(25
6*d) - (a^3*cot(c/2 + (d*x)/2)^5)/(640*d) - (a^3*cot(c/2 + (d*x)/2)^6)/(128*d) - (3*a^3*cot(c/2 + (d*x)/2)^7)/
(896*d) - (a^3*cot(c/2 + (d*x)/2)^8)/(2048*d) - (3*a^3*tan(c/2 + (d*x)/2)^2)/(128*d) - (7*a^3*tan(c/2 + (d*x)/
2)^3)/(128*d) - (7*a^3*tan(c/2 + (d*x)/2)^4)/(256*d) + (a^3*tan(c/2 + (d*x)/2)^5)/(640*d) + (a^3*tan(c/2 + (d*
x)/2)^6)/(128*d) + (3*a^3*tan(c/2 + (d*x)/2)^7)/(896*d) + (a^3*tan(c/2 + (d*x)/2)^8)/(2048*d) + (27*a^3*log(ta
n(c/2 + (d*x)/2)))/(128*d) - (17*a^3*cot(c/2 + (d*x)/2))/(128*d) + (17*a^3*tan(c/2 + (d*x)/2))/(128*d)

________________________________________________________________________________________

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)**4*csc(d*x+c)**9*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

________________________________________________________________________________________