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

Optimal. Leaf size=175 \[ -\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}+\frac{5 a^3 \cot (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{25 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{13 a^3 x}{2} \]

[Out]

(13*a^3*x)/2 - (25*a^3*ArcTanh[Cos[c + d*x]])/(8*d) + (a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/(3*d) + (5*a
^3*Cot[c + d*x])/d - (2*a^3*Cot[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x]^5)/(5*d) + (23*a^3*Cot[c + d*x]*Csc[c +
d*x])/(8*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)

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Rubi [A]  time = 0.298944, antiderivative size = 175, normalized size of antiderivative = 1., 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 = {2709, 3770, 3767, 8, 3768, 2635, 2633} \[ -\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}+\frac{5 a^3 \cot (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{25 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{13 a^3 x}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(13*a^3*x)/2 - (25*a^3*ArcTanh[Cos[c + d*x]])/(8*d) + (a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/(3*d) + (5*a
^3*Cot[c + d*x])/d - (2*a^3*Cot[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x]^5)/(5*d) + (23*a^3*Cot[c + d*x]*Csc[c +
d*x])/(8*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)

Rule 2709

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_), x_Symbol] :> Dist[a^p, Int[Expan
dIntegrand[(Sin[e + f*x]^p*(a + b*Sin[e + f*x])^(m - p/2))/(a - b*Sin[e + f*x])^(p/2), x], x], x] /; FreeQ[{a,
b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\int \left (8 a^9+6 a^9 \csc (c+d x)-6 a^9 \csc ^2(c+d x)-8 a^9 \csc ^3(c+d x)+3 a^9 \csc ^5(c+d x)+a^9 \csc ^6(c+d x)-3 a^9 \sin ^2(c+d x)-a^9 \sin ^3(c+d x)\right ) \, dx}{a^6}\\ &=8 a^3 x+a^3 \int \csc ^6(c+d x) \, dx-a^3 \int \sin ^3(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^5(c+d x) \, dx-\left (3 a^3\right ) \int \sin ^2(c+d x) \, dx+\left (6 a^3\right ) \int \csc (c+d x) \, dx-\left (6 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (8 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=8 a^3 x-\frac{6 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{4 a^3 \cot (c+d x) \csc (c+d x)}{d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac{1}{2} \left (3 a^3\right ) \int 1 \, dx+\frac{1}{4} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx-\left (4 a^3\right ) \int \csc (c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{a^3 \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac{\left (6 a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=\frac{13 a^3 x}{2}-\frac{2 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{5 a^3 \cot (c+d x)}{d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{8} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac{13 a^3 x}{2}-\frac{25 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{5 a^3 \cot (c+d x)}{d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 1.63668, size = 271, normalized size = 1.55 \[ \frac{a^3 (\sin (c+d x)+1)^3 \left (6240 (c+d x)+720 \sin (2 (c+d x))+720 \cos (c+d x)-80 \cos (3 (c+d x))-2624 \tan \left (\frac{1}{2} (c+d x)\right )+2624 \cot \left (\frac{1}{2} (c+d x)\right )-45 \csc ^4\left (\frac{1}{2} (c+d x)\right )+690 \csc ^2\left (\frac{1}{2} (c+d x)\right )+45 \sec ^4\left (\frac{1}{2} (c+d x)\right )-690 \sec ^2\left (\frac{1}{2} (c+d x)\right )+3000 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-3000 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+304 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-3 \sin (c+d x) \csc ^6\left (\frac{1}{2} (c+d x)\right )-19 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )+6 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )\right )}{960 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(1 + Sin[c + d*x])^3*(6240*(c + d*x) + 720*Cos[c + d*x] - 80*Cos[3*(c + d*x)] + 2624*Cot[(c + d*x)/2] + 6
90*Csc[(c + d*x)/2]^2 - 45*Csc[(c + d*x)/2]^4 - 3000*Log[Cos[(c + d*x)/2]] + 3000*Log[Sin[(c + d*x)/2]] - 690*
Sec[(c + d*x)/2]^2 + 45*Sec[(c + d*x)/2]^4 + 304*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 19*Csc[(c + d*x)/2]^4*Sin
[c + d*x] - 3*Csc[(c + d*x)/2]^6*Sin[c + d*x] + 720*Sin[2*(c + d*x)] - 2624*Tan[(c + d*x)/2] + 6*Sec[(c + d*x)
/2]^4*Tan[(c + d*x)/2]))/(960*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

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Maple [A]  time = 0.095, size = 293, normalized size = 1.7 \begin{align*}{\frac{5\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{5\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d}}+{\frac{25\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{25\,{a}^{3}\cos \left ( dx+c \right ) }{8\,d}}+{\frac{25\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+4\,{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{d\sin \left ( dx+c \right ) }}+4\,{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) }{d}}+5\,{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{d}}+{\frac{15\,{a}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{13\,{a}^{3}x}{2}}+{\frac{13\,{a}^{3}c}{2\,d}}-{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/8/d*a^3/sin(d*x+c)^2*cos(d*x+c)^7+5/8*a^3*cos(d*x+c)^5/d+25/24*a^3*cos(d*x+c)^3/d+25/8*a^3*cos(d*x+c)/d+25/8
/d*a^3*ln(csc(d*x+c)-cot(d*x+c))-1/d*a^3/sin(d*x+c)^3*cos(d*x+c)^7+4/d*a^3/sin(d*x+c)*cos(d*x+c)^7+4*a^3*cos(d
*x+c)^5*sin(d*x+c)/d+5*a^3*cos(d*x+c)^3*sin(d*x+c)/d+15/2*a^3*cos(d*x+c)*sin(d*x+c)/d+13/2*a^3*x+13/2/d*a^3*c-
3/4/d*a^3/sin(d*x+c)^4*cos(d*x+c)^7-1/5*a^3*cot(d*x+c)^5/d+1/3*a^3*cot(d*x+c)^3/d-a^3*cot(d*x+c)/d

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Maxima [A]  time = 1.7904, size = 338, normalized size = 1.93 \begin{align*} -\frac{20 \,{\left (4 \, \cos \left (d x + c\right )^{3} - \frac{6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 120 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{3} + 16 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} + 45 \, a^{3}{\left (\frac{2 \,{\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/240*(20*(4*cos(d*x + c)^3 - 6*cos(d*x + c)/(cos(d*x + c)^2 - 1) + 24*cos(d*x + c) - 15*log(cos(d*x + c) + 1
) + 15*log(cos(d*x + c) - 1))*a^3 - 120*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 10*tan(d*x + c)^2 - 2)/(tan(d*x
+ c)^5 + tan(d*x + c)^3))*a^3 + 16*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)
*a^3 + 45*a^3*(2*(9*cos(d*x + c)^3 - 7*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - 16*cos(d*x + c)
+ 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d

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Fricas [A]  time = 1.28919, size = 738, normalized size = 4.22 \begin{align*} -\frac{360 \, a^{3} \cos \left (d x + c\right )^{7} - 2392 \, a^{3} \cos \left (d x + c\right )^{5} + 3640 \, a^{3} \cos \left (d x + c\right )^{3} - 1560 \, a^{3} \cos \left (d x + c\right ) + 375 \,{\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, 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 ) - 375 \,{\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, 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 ) + 10 \,{\left (8 \, a^{3} \cos \left (d x + c\right )^{7} - 156 \, a^{3} d x \cos \left (d x + c\right )^{4} - 40 \, a^{3} \cos \left (d x + c\right )^{5} + 312 \, a^{3} d x \cos \left (d x + c\right )^{2} + 125 \, a^{3} \cos \left (d x + c\right )^{3} - 156 \, a^{3} d x - 75 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/240*(360*a^3*cos(d*x + c)^7 - 2392*a^3*cos(d*x + c)^5 + 3640*a^3*cos(d*x + c)^3 - 1560*a^3*cos(d*x + c) + 3
75*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^2 + a^3)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 375*(a^3*cos(d
*x + c)^4 - 2*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 10*(8*a^3*cos(d*x + c)^7 -
156*a^3*d*x*cos(d*x + c)^4 - 40*a^3*cos(d*x + c)^5 + 312*a^3*d*x*cos(d*x + c)^2 + 125*a^3*cos(d*x + c)^3 - 15
6*a^3*d*x - 75*a^3*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**6*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.24863, size = 373, normalized size = 2.13 \begin{align*} \frac{6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 45 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 50 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 600 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 6240 \,{\left (d x + c\right )} a^{3} + 3000 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 2580 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{320 \,{\left (9 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 12 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 9 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 4 \, a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3}} - \frac{6850 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 2580 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 600 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 50 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 45 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{960 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/960*(6*a^3*tan(1/2*d*x + 1/2*c)^5 + 45*a^3*tan(1/2*d*x + 1/2*c)^4 + 50*a^3*tan(1/2*d*x + 1/2*c)^3 - 600*a^3*
tan(1/2*d*x + 1/2*c)^2 + 6240*(d*x + c)*a^3 + 3000*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 2580*a^3*tan(1/2*d*x +
1/2*c) - 320*(9*a^3*tan(1/2*d*x + 1/2*c)^5 - 12*a^3*tan(1/2*d*x + 1/2*c)^2 - 9*a^3*tan(1/2*d*x + 1/2*c) - 4*a
^3)/(tan(1/2*d*x + 1/2*c)^2 + 1)^3 - (6850*a^3*tan(1/2*d*x + 1/2*c)^5 - 2580*a^3*tan(1/2*d*x + 1/2*c)^4 - 600*
a^3*tan(1/2*d*x + 1/2*c)^3 + 50*a^3*tan(1/2*d*x + 1/2*c)^2 + 45*a^3*tan(1/2*d*x + 1/2*c) + 6*a^3)/tan(1/2*d*x
+ 1/2*c)^5)/d