3.2.6 \(\int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx\) [106]

Optimal. Leaf size=127 \[ \frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {10 \cot ^7(c+d x)}{7 a^3 d}+\frac {11 \cot ^9(c+d x)}{9 a^3 d}+\frac {4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac {3 \csc ^7(c+d x)}{7 a^3 d}+\frac {7 \csc ^9(c+d x)}{9 a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d} \]

[Out]

3/5*cot(d*x+c)^5/a^3/d+10/7*cot(d*x+c)^7/a^3/d+11/9*cot(d*x+c)^9/a^3/d+4/11*cot(d*x+c)^11/a^3/d-3/7*csc(d*x+c)
^7/a^3/d+7/9*csc(d*x+c)^9/a^3/d-4/11*csc(d*x+c)^11/a^3/d

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Rubi [A]
time = 0.29, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3957, 2954, 2952, 2687, 276, 2686, 14} \begin {gather*} \frac {4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac {11 \cot ^9(c+d x)}{9 a^3 d}+\frac {10 \cot ^7(c+d x)}{7 a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac {7 \csc ^9(c+d x)}{9 a^3 d}-\frac {3 \csc ^7(c+d x)}{7 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Cot[c + d*x]^5)/(5*a^3*d) + (10*Cot[c + d*x]^7)/(7*a^3*d) + (11*Cot[c + d*x]^9)/(9*a^3*d) + (4*Cot[c + d*x]
^11)/(11*a^3*d) - (3*Csc[c + d*x]^7)/(7*a^3*d) + (7*Csc[c + d*x]^9)/(9*a^3*d) - (4*Csc[c + d*x]^11)/(11*a^3*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 276

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 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2687

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 2952

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 2954

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(a/g)^(2*m), Int[(g*Cos[e + f*x])^(2*m + 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] && ILtQ[m, 0]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac {\cot ^3(c+d x) \csc ^3(c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=-\frac {\int (-a+a \cos (c+d x))^3 \cot ^3(c+d x) \csc ^9(c+d x) \, dx}{a^6}\\ &=\frac {\int \left (-a^3 \cot ^6(c+d x) \csc ^6(c+d x)+3 a^3 \cot ^5(c+d x) \csc ^7(c+d x)-3 a^3 \cot ^4(c+d x) \csc ^8(c+d x)+a^3 \cot ^3(c+d x) \csc ^9(c+d x)\right ) \, dx}{a^6}\\ &=-\frac {\int \cot ^6(c+d x) \csc ^6(c+d x) \, dx}{a^3}+\frac {\int \cot ^3(c+d x) \csc ^9(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^5(c+d x) \csc ^7(c+d x) \, dx}{a^3}-\frac {3 \int \cot ^4(c+d x) \csc ^8(c+d x) \, dx}{a^3}\\ &=-\frac {\text {Subst}\left (\int x^8 \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {\text {Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^6 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^4 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=-\frac {\text {Subst}\left (\int \left (-x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {\text {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (x^4+3 x^6+3 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {10 \cot ^7(c+d x)}{7 a^3 d}+\frac {11 \cot ^9(c+d x)}{9 a^3 d}+\frac {4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac {3 \csc ^7(c+d x)}{7 a^3 d}+\frac {7 \csc ^9(c+d x)}{9 a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d}\\ \end {align*}

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Mathematica [A]
time = 0.81, size = 223, normalized size = 1.76 \begin {gather*} \frac {\csc (c) \csc ^5(c+d x) \sec ^3(c+d x) (-3886080 \sin (c)+563200 \sin (d x)+524150 \sin (c+d x)+314490 \sin (2 (c+d x))-162010 \sin (3 (c+d x))-238250 \sin (4 (c+d x))-47650 \sin (5 (c+d x))+47650 \sin (6 (c+d x))+28590 \sin (7 (c+d x))+4765 \sin (8 (c+d x))-2027520 \sin (2 c+d x)+1486848 \sin (c+2 d x)-2365440 \sin (3 c+2 d x)+452608 \sin (2 c+3 d x)+665600 \sin (3 c+4 d x)+133120 \sin (4 c+5 d x)-133120 \sin (5 c+6 d x)-79872 \sin (6 c+7 d x)-13312 \sin (7 c+8 d x))}{56770560 a^3 d (1+\sec (c+d x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[c]*Csc[c + d*x]^5*Sec[c + d*x]^3*(-3886080*Sin[c] + 563200*Sin[d*x] + 524150*Sin[c + d*x] + 314490*Sin[2*
(c + d*x)] - 162010*Sin[3*(c + d*x)] - 238250*Sin[4*(c + d*x)] - 47650*Sin[5*(c + d*x)] + 47650*Sin[6*(c + d*x
)] + 28590*Sin[7*(c + d*x)] + 4765*Sin[8*(c + d*x)] - 2027520*Sin[2*c + d*x] + 1486848*Sin[c + 2*d*x] - 236544
0*Sin[3*c + 2*d*x] + 452608*Sin[2*c + 3*d*x] + 665600*Sin[3*c + 4*d*x] + 133120*Sin[4*c + 5*d*x] - 133120*Sin[
5*c + 6*d*x] - 79872*Sin[6*c + 7*d*x] - 13312*Sin[7*c + 8*d*x]))/(56770560*a^3*d*(1 + Sec[c + d*x])^3)

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Maple [A]
time = 0.13, size = 112, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {-\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}-\frac {2 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}}{256 d \,a^{3}}\)	\(112\)
default	\(\frac {-\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}-\frac {2 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}}{256 d \,a^{3}}\)	\(112\)
risch	\(-\frac {16 i \left (2310 \,{\mathrm e}^{10 i \left (d x +c \right )}+1980 \,{\mathrm e}^{9 i \left (d x +c \right )}+3795 \,{\mathrm e}^{8 i \left (d x +c \right )}+550 \,{\mathrm e}^{7 i \left (d x +c \right )}+1452 \,{\mathrm e}^{6 i \left (d x +c \right )}+442 \,{\mathrm e}^{5 i \left (d x +c \right )}+650 \,{\mathrm e}^{4 i \left (d x +c \right )}+130 \,{\mathrm e}^{3 i \left (d x +c \right )}-130 \,{\mathrm e}^{2 i \left (d x +c \right )}-78 \,{\mathrm e}^{i \left (d x +c \right )}-13\right )}{3465 a^{3} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{11} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{5}}\)	\(148\)
norman	\(\frac {-\frac {1}{1280 a d}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 a d}+\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a d}-\frac {3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d}+\frac {3 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{640 a d}+\frac {\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )}{896 a d}-\frac {\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )}{1152 a d}-\frac {\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2816 a d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{2}}\)	\(158\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^6/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/256/d/a^3*(-1/11*tan(1/2*d*x+1/2*c)^11-2/9*tan(1/2*d*x+1/2*c)^9+2/7*tan(1/2*d*x+1/2*c)^7+6/5*tan(1/2*d*x+1/2
*c)^5-6*tan(1/2*d*x+1/2*c)+2/tan(1/2*d*x+1/2*c)-2/3/tan(1/2*d*x+1/2*c)^3-1/5/tan(1/2*d*x+1/2*c)^5)

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Maxima [A]
time = 0.27, size = 174, normalized size = 1.37 \begin {gather*} -\frac {\frac {\frac {20790 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {4158 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {990 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {770 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {315 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a^{3}} + \frac {231 \, {\left (\frac {10 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {30 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{3} \sin \left (d x + c\right )^{5}}}{887040 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/887040*((20790*sin(d*x + c)/(cos(d*x + c) + 1) - 4158*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 990*sin(d*x + c
)^7/(cos(d*x + c) + 1)^7 + 770*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 315*sin(d*x + c)^11/(cos(d*x + c) + 1)^11
)/a^3 + 231*(10*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 30*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 3)*(cos(d*x + c
) + 1)^5/(a^3*sin(d*x + c)^5))/d

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Fricas [A]
time = 3.45, size = 191, normalized size = 1.50 \begin {gather*} \frac {104 \, \cos \left (d x + c\right )^{8} + 312 \, \cos \left (d x + c\right )^{7} + 52 \, \cos \left (d x + c\right )^{6} - 676 \, \cos \left (d x + c\right )^{5} - 585 \, \cos \left (d x + c\right )^{4} + 325 \, \cos \left (d x + c\right )^{3} - 25 \, \cos \left (d x + c\right )^{2} - 150 \, \cos \left (d x + c\right ) - 50}{3465 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/3465*(104*cos(d*x + c)^8 + 312*cos(d*x + c)^7 + 52*cos(d*x + c)^6 - 676*cos(d*x + c)^5 - 585*cos(d*x + c)^4
+ 325*cos(d*x + c)^3 - 25*cos(d*x + c)^2 - 150*cos(d*x + c) - 50)/((a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c
)^6 + a^3*d*cos(d*x + c)^5 - 5*a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 + 3*a^3*d*
cos(d*x + c) + a^3*d)*sin(d*x + c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\csc ^{6}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**6/(a+a*sec(d*x+c))**3,x)

[Out]

Integral(csc(c + d*x)**6/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3

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Giac [A]
time = 0.58, size = 134, normalized size = 1.06 \begin {gather*} \frac {\frac {231 \, {\left (30 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - \frac {315 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 770 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 990 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 4158 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 20790 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{33}}}{887040 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/887040*(231*(30*tan(1/2*d*x + 1/2*c)^4 - 10*tan(1/2*d*x + 1/2*c)^2 - 3)/(a^3*tan(1/2*d*x + 1/2*c)^5) - (315*
a^30*tan(1/2*d*x + 1/2*c)^11 + 770*a^30*tan(1/2*d*x + 1/2*c)^9 - 990*a^30*tan(1/2*d*x + 1/2*c)^7 - 4158*a^30*t
an(1/2*d*x + 1/2*c)^5 + 20790*a^30*tan(1/2*d*x + 1/2*c))/a^33)/d

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Mupad [B]
time = 1.43, size = 201, normalized size = 1.58 \begin {gather*} -\frac {693\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+2310\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-6930\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+20790\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-4158\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-990\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+770\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+315\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{887040\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(c + d*x)^6*(a + a/cos(c + d*x))^3),x)

[Out]

-(693*cos(c/2 + (d*x)/2)^16 + 315*sin(c/2 + (d*x)/2)^16 + 770*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^14 - 990
*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^12 - 4158*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^10 + 20790*cos(c/2
+ (d*x)/2)^10*sin(c/2 + (d*x)/2)^6 - 6930*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^4 + 2310*cos(c/2 + (d*x)/2)
^14*sin(c/2 + (d*x)/2)^2)/(887040*a^3*d*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^5)

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