3.621 \(\int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=270 \[ -\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {41 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac {41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac {41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d} \]

[Out]

41/1024*a^3*arctanh(cos(d*x+c))/d-4/7*a^3*cot(d*x+c)^7/d-7/9*a^3*cot(d*x+c)^9/d-3/11*a^3*cot(d*x+c)^11/d+41/10
24*a^3*cot(d*x+c)*csc(d*x+c)/d+41/1536*a^3*cot(d*x+c)*csc(d*x+c)^3/d-35/384*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*cot(d*x+c)^5*csc(d*x+c)^5/d-1/64*a^3*cot(d*x+c)*csc(d*x+c)^7/d+1/24*
a^3*cot(d*x+c)^3*csc(d*x+c)^7/d-1/12*a^3*cot(d*x+c)^5*csc(d*x+c)^7/d

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Rubi [A]  time = 0.46, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 2607, 14, 2611, 3768, 3770, 270} \[ -\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {41 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac {41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac {41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]

[Out]

(41*a^3*ArcTanh[Cos[c + d*x]])/(1024*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (7*a^3*Cot[c + d*x]^9)/(9*d) - (3*a^3
*Cot[c + d*x]^11)/(11*d) + (41*a^3*Cot[c + d*x]*Csc[c + d*x])/(1024*d) + (41*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/
(1536*d) - (35*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(384*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) - (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)/(24*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^7)/(12*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 ^7(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (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)+3 a^3 \cot ^6(c+d x) \csc ^6(c+d x)+a^3 \cot ^6(c+d x) \csc ^7(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^7(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx\\ &=-\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 ^7(c+d x)}{12 d}-\frac {1}{12} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^7(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 ) \, 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 {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 {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}+\frac {1}{8} a^3 \int \cot ^2(c+d x) \csc ^7(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+x^8\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 {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 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 {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)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {1}{64} a^3 \int \csc ^7(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 {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 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 {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)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {1}{384} \left (5 a^3\right ) \int \csc ^5(c+d x) \, dx-\frac {1}{128} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {9 a^3 \cot (c+d x) \csc (c+d x)}{256 d}+\frac {41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 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 {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)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {1}{512} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{256} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac {9 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 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 {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)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {\left (5 a^3\right ) \int \csc (c+d x) \, dx}{1024}\\ &=\frac {41 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 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 {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)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}\\ \end {align*}

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Mathematica [A]  time = 4.72, size = 197, normalized size = 0.73 \[ \frac {a^3 (\sin (c+d x)+1)^3 \left (72737280 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{11}(c+d x) (49776640 \sin (c+d x)[phone] \sin (3 (c+d x))[phone] \sin (5 (c+d x))+2206720 \sin (7 (c+d x))-1530880 \sin (9 (c+d x))+117760 \sin (11 (c+d x))[phone] \cos (2 (c+d x))[phone] \cos (4 (c+d x))-23426403 \cos (6 (c+d x))-1799490 \cos (8 (c+d x))+142065 \cos (10 (c+d x))[phone])\right )}{1816657920 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(1 + Sin[c + d*x])^3*(72737280*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x
]^11*(91311066 + 62609778*Cos[2*(c + d*x)] + 22551144*Cos[4*(c + d*x)] - 23426403*Cos[6*(c + d*x)] - 1799490*C
os[8*(c + d*x)] + 142065*Cos[10*(c + d*x)] + 49776640*Sin[c + d*x] + 84039680*Sin[3*(c + d*x)] + 38118400*Sin[
5*(c + d*x)] + 2206720*Sin[7*(c + d*x)] - 1530880*Sin[9*(c + d*x)] + 117760*Sin[11*(c + d*x)])))/(1816657920*d
*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

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fricas [A]  time = 1.01, size = 384, normalized size = 1.42 \[ -\frac {284130 \, a^{3} \cos \left (d x + c\right )^{11} - 1610070 \, a^{3} \cos \left (d x + c\right )^{9} - 507276 \, a^{3} \cos \left (d x + c\right )^{7} + 3750516 \, a^{3} \cos \left (d x + c\right )^{5} - 1610070 \, a^{3} \cos \left (d x + c\right )^{3} + 284130 \, a^{3} \cos \left (d x + c\right ) - 142065 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 142065 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 10240 \, {\left (46 \, a^{3} \cos \left (d x + c\right )^{11} - 253 \, a^{3} \cos \left (d x + c\right )^{9} + 396 \, a^{3} \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{7096320 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

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))^3,x, algorithm="fricas")

[Out]

-1/7096320*(284130*a^3*cos(d*x + c)^11 - 1610070*a^3*cos(d*x + c)^9 - 507276*a^3*cos(d*x + c)^7 + 3750516*a^3*
cos(d*x + c)^5 - 1610070*a^3*cos(d*x + c)^3 + 284130*a^3*cos(d*x + c) - 142065*(a^3*cos(d*x + c)^12 - 6*a^3*co
s(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) + 142065*(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) +
10240*(46*a^3*cos(d*x + c)^11 - 253*a^3*cos(d*x + c)^9 + 396*a^3*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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giac [A]  time = 0.54, size = 420, normalized size = 1.56 \[ \frac {1155 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 7560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 16632 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 3080 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 51975 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 106920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 83160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 83160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 384615 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 572880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 166320 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2273040 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 1496880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {7053722 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 1496880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 166320 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 572880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 384615 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 83160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 83160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 106920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 51975 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3080 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16632 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1155 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12}}}{56770560 \, d} \]

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))^3,x, algorithm="giac")

[Out]

1/56770560*(1155*a^3*tan(1/2*d*x + 1/2*c)^12 + 7560*a^3*tan(1/2*d*x + 1/2*c)^11 + 16632*a^3*tan(1/2*d*x + 1/2*
c)^10 + 3080*a^3*tan(1/2*d*x + 1/2*c)^9 - 51975*a^3*tan(1/2*d*x + 1/2*c)^8 - 106920*a^3*tan(1/2*d*x + 1/2*c)^7
- 83160*a^3*tan(1/2*d*x + 1/2*c)^6 + 83160*a^3*tan(1/2*d*x + 1/2*c)^5 + 384615*a^3*tan(1/2*d*x + 1/2*c)^4 + 5
72880*a^3*tan(1/2*d*x + 1/2*c)^3 + 166320*a^3*tan(1/2*d*x + 1/2*c)^2 - 2273040*a^3*log(abs(tan(1/2*d*x + 1/2*c
))) - 1496880*a^3*tan(1/2*d*x + 1/2*c) + (7053722*a^3*tan(1/2*d*x + 1/2*c)^12 + 1496880*a^3*tan(1/2*d*x + 1/2*
c)^11 - 166320*a^3*tan(1/2*d*x + 1/2*c)^10 - 572880*a^3*tan(1/2*d*x + 1/2*c)^9 - 384615*a^3*tan(1/2*d*x + 1/2*
c)^8 - 83160*a^3*tan(1/2*d*x + 1/2*c)^7 + 83160*a^3*tan(1/2*d*x + 1/2*c)^6 + 106920*a^3*tan(1/2*d*x + 1/2*c)^5
+ 51975*a^3*tan(1/2*d*x + 1/2*c)^4 - 3080*a^3*tan(1/2*d*x + 1/2*c)^3 - 16632*a^3*tan(1/2*d*x + 1/2*c)^2 - 756
0*a^3*tan(1/2*d*x + 1/2*c) - 1155*a^3)/tan(1/2*d*x + 1/2*c)^12)/d

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maple [A]  time = 0.41, size = 288, normalized size = 1.07 \[ -\frac {23 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{99 d \sin \left (d x +c \right )^{9}}-\frac {46 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{693 d \sin \left (d x +c \right )^{7}}-\frac {41 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{120 d \sin \left (d x +c \right )^{10}}-\frac {41 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{320 d \sin \left (d x +c \right )^{8}}-\frac {41 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{1920 d \sin \left (d x +c \right )^{6}}+\frac {41 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7680 d \sin \left (d x +c \right )^{4}}-\frac {41 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{5120 d \sin \left (d x +c \right )^{2}}-\frac {41 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5120 d}-\frac {41 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{3072 d}-\frac {41 a^{3} \cos \left (d x +c \right )}{1024 d}-\frac {41 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024 d}-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{11 d \sin \left (d x +c \right )^{11}}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{12 d \sin \left (d x +c \right )^{12}} \]

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))^3,x)

[Out]

-23/99/d*a^3/sin(d*x+c)^9*cos(d*x+c)^7-46/693/d*a^3/sin(d*x+c)^7*cos(d*x+c)^7-41/120/d*a^3/sin(d*x+c)^10*cos(d
*x+c)^7-41/320/d*a^3/sin(d*x+c)^8*cos(d*x+c)^7-41/1920/d*a^3/sin(d*x+c)^6*cos(d*x+c)^7+41/7680/d*a^3/sin(d*x+c
)^4*cos(d*x+c)^7-41/5120/d*a^3/sin(d*x+c)^2*cos(d*x+c)^7-41/5120*a^3*cos(d*x+c)^5/d-41/3072*a^3*cos(d*x+c)^3/d
-41/1024*a^3*cos(d*x+c)/d-41/1024/d*a^3*ln(csc(d*x+c)-cot(d*x+c))-3/11/d*a^3/sin(d*x+c)^11*cos(d*x+c)^7-1/12/d
*a^3/sin(d*x+c)^12*cos(d*x+c)^7

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maxima [A]  time = 0.40, size = 348, normalized size = 1.29 \[ -\frac {1155 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{11} - 85 \, \cos \left (d x + c\right )^{9} + 198 \, \cos \left (d x + c\right )^{7} + 198 \, \cos \left (d x + c\right )^{5} - 85 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{12} - 6 \, \cos \left (d x + c\right )^{10} + 15 \, \cos \left (d x + c\right )^{8} - 20 \, \cos \left (d x + c\right )^{6} + 15 \, \cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 8316 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {112640 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{3}}{\tan \left (d x + c\right )^{9}} + \frac {30720 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{3}}{\tan \left (d x + c\right )^{11}}}{7096320 \, d} \]

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))^3,x, algorithm="maxima")

[Out]

-1/7096320*(1155*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)) + 8316*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))/(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)) + 112640*(9*tan(d*x + c)^2 + 7)*a^3/tan(d*x + c)^9 + 30720*(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 = 10.40, size = 471, normalized size = 1.74 \[ \frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {31\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3072\,d}-\frac {111\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {27\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}+\frac {15\,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}{18432\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{49152\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {31\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3072\,d}+\frac {111\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {27\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}-\frac {15\,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}{18432\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{49152\,d}-\frac {41\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{1024\,d}+\frac {27\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1024\,d}-\frac {27\,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)^13,x)

[Out]

(3*a^3*cot(c/2 + (d*x)/2)^6)/(2048*d) - (31*a^3*cot(c/2 + (d*x)/2)^3)/(3072*d) - (111*a^3*cot(c/2 + (d*x)/2)^4
)/(16384*d) - (3*a^3*cot(c/2 + (d*x)/2)^5)/(2048*d) - (3*a^3*cot(c/2 + (d*x)/2)^2)/(1024*d) + (27*a^3*cot(c/2
+ (d*x)/2)^7)/(14336*d) + (15*a^3*cot(c/2 + (d*x)/2)^8)/(16384*d) - (a^3*cot(c/2 + (d*x)/2)^9)/(18432*d) - (3*
a^3*cot(c/2 + (d*x)/2)^10)/(10240*d) - (3*a^3*cot(c/2 + (d*x)/2)^11)/(22528*d) - (a^3*cot(c/2 + (d*x)/2)^12)/(
49152*d) + (3*a^3*tan(c/2 + (d*x)/2)^2)/(1024*d) + (31*a^3*tan(c/2 + (d*x)/2)^3)/(3072*d) + (111*a^3*tan(c/2 +
(d*x)/2)^4)/(16384*d) + (3*a^3*tan(c/2 + (d*x)/2)^5)/(2048*d) - (3*a^3*tan(c/2 + (d*x)/2)^6)/(2048*d) - (27*a
^3*tan(c/2 + (d*x)/2)^7)/(14336*d) - (15*a^3*tan(c/2 + (d*x)/2)^8)/(16384*d) + (a^3*tan(c/2 + (d*x)/2)^9)/(184
32*d) + (3*a^3*tan(c/2 + (d*x)/2)^10)/(10240*d) + (3*a^3*tan(c/2 + (d*x)/2)^11)/(22528*d) + (a^3*tan(c/2 + (d*
x)/2)^12)/(49152*d) - (41*a^3*log(tan(c/2 + (d*x)/2)))/(1024*d) + (27*a^3*cot(c/2 + (d*x)/2))/(1024*d) - (27*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)**13*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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