3.613 \(\int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=178 \[ -\frac{a^3 \cos ^3(c+d x)}{d}-\frac{5 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{d}+\frac{5 a^3 \cot (c+d x)}{d}+\frac{a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{45 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{45 a^3 x}{8} \]

[Out]

(45*a^3*x)/8 + (45*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (5*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/d + (5*a^3
*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/d - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]*Csc[c
+ d*x]^3)/(4*d) + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)

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Rubi [A]  time = 0.218743, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2872, 3770, 3767, 8, 3768, 2638, 2633, 2635} \[ -\frac{a^3 \cos ^3(c+d x)}{d}-\frac{5 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{d}+\frac{5 a^3 \cot (c+d x)}{d}+\frac{a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{45 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{45 a^3 x}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(45*a^3*x)/8 + (45*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (5*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/d + (5*a^3
*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/d - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]*Csc[c
+ d*x]^3)/(4*d) + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)

Rule 2872

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[1/a^p, Int[ExpandTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x]
)^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && ((GtQ[m,
0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 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 2638

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

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]

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]

Rubi steps

\begin{align*} \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\int \left (6 a^9-6 a^9 \csc (c+d x)-8 a^9 \csc ^2(c+d x)+3 a^9 \csc ^4(c+d x)+a^9 \csc ^5(c+d x)+8 a^9 \sin (c+d x)-3 a^9 \sin ^3(c+d x)-a^9 \sin ^4(c+d x)\right ) \, dx}{a^6}\\ &=6 a^3 x+a^3 \int \csc ^5(c+d x) \, dx-a^3 \int \sin ^4(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^4(c+d x) \, dx-\left (3 a^3\right ) \int \sin ^3(c+d x) \, dx-\left (6 a^3\right ) \int \csc (c+d x) \, dx-\left (8 a^3\right ) \int \csc ^2(c+d x) \, dx+\left (8 a^3\right ) \int \sin (c+d x) \, dx\\ &=6 a^3 x+\frac{6 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{8 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac{1}{4} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac{1}{4} \left (3 a^3\right ) \int \sin ^2(c+d x) \, dx+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac{\left (8 a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=6 a^3 x+\frac{6 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{5 a^3 \cos (c+d x)}{d}-\frac{a^3 \cos ^3(c+d x)}{d}+\frac{5 a^3 \cot (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{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)}{8 d}+\frac{a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}-\frac{1}{8} \left (3 a^3\right ) \int 1 \, dx+\frac{1}{8} \left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac{45 a^3 x}{8}+\frac{45 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{5 a^3 \cos (c+d x)}{d}-\frac{a^3 \cos ^3(c+d x)}{d}+\frac{5 a^3 \cot (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{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)}{8 d}+\frac{a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.85342, size = 235, normalized size = 1.32 \[ \frac{a^3 (\sin (c+d x)+1)^3 \left (360 (c+d x)+16 \sin (2 (c+d x))-2 \sin (4 (c+d x))-368 \cos (c+d x)-16 \cos (3 (c+d x))-192 \tan \left (\frac{1}{2} (c+d x)\right )+192 \cot \left (\frac{1}{2} (c+d x)\right )-\csc ^4\left (\frac{1}{2} (c+d x)\right )-6 \csc ^2\left (\frac{1}{2} (c+d x)\right )+\sec ^4\left (\frac{1}{2} (c+d x)\right )+6 \sec ^2\left (\frac{1}{2} (c+d x)\right )-360 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+360 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+64 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-4 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )\right )}{64 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(1 + Sin[c + d*x])^3*(360*(c + d*x) - 368*Cos[c + d*x] - 16*Cos[3*(c + d*x)] + 192*Cot[(c + d*x)/2] - 6*C
sc[(c + d*x)/2]^2 - Csc[(c + d*x)/2]^4 + 360*Log[Cos[(c + d*x)/2]] - 360*Log[Sin[(c + d*x)/2]] + 6*Sec[(c + d*
x)/2]^2 + Sec[(c + d*x)/2]^4 + 64*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 4*Csc[(c + d*x)/2]^4*Sin[c + d*x] + 16*S
in[2*(c + d*x)] - 2*Sin[4*(c + d*x)] - 192*Tan[(c + d*x)/2]))/(64*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

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Maple [A]  time = 0.096, size = 247, normalized size = 1.4 \begin{align*} 3\,{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{d\sin \left ( dx+c \right ) }}+3\,{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) }{d}}+{\frac{15\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4\,d}}+{\frac{45\,{a}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}+{\frac{45\,{a}^{3}x}{8}}+{\frac{45\,{a}^{3}c}{8\,d}}-{\frac{9\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{9\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d}}-{\frac{15\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d}}-{\frac{45\,{a}^{3}\cos \left ( dx+c \right ) }{8\,d}}-{\frac{45\,{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}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x)

[Out]

3/d*a^3/sin(d*x+c)*cos(d*x+c)^7+3*a^3*cos(d*x+c)^5*sin(d*x+c)/d+15/4*a^3*cos(d*x+c)^3*sin(d*x+c)/d+45/8*a^3*co
s(d*x+c)*sin(d*x+c)/d+45/8*a^3*x+45/8/d*a^3*c-9/8/d*a^3/sin(d*x+c)^2*cos(d*x+c)^7-9/8*a^3*cos(d*x+c)^5/d-15/8*
a^3*cos(d*x+c)^3/d-45/8*a^3*cos(d*x+c)/d-45/8/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)^
7-1/4/d*a^3/sin(d*x+c)^4*cos(d*x+c)^7

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Maxima [A]  time = 1.73312, size = 362, normalized size = 2.03 \begin{align*} -\frac{4 \,{\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} + 2 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3} - 8 \,{\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} + 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 )}}{16 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/16*(4*(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 + 2*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 25*tan(d*x + c)^2 + 8)/(tan(d*x + c)
^5 + 2*tan(d*x + c)^3 + tan(d*x + c)))*a^3 - 8*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 10*tan(d*x + c)^2 - 2)/(t
an(d*x + c)^5 + tan(d*x + c)^3))*a^3 + 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 = 1.32093, size = 664, normalized size = 3.73 \begin{align*} -\frac{16 \, a^{3} \cos \left (d x + c\right )^{7} - 90 \, a^{3} d x \cos \left (d x + c\right )^{4} + 48 \, a^{3} \cos \left (d x + c\right )^{5} + 180 \, a^{3} d x \cos \left (d x + c\right )^{2} - 150 \, a^{3} \cos \left (d x + c\right )^{3} - 90 \, a^{3} d x + 90 \, a^{3} \cos \left (d x + c\right ) - 45 \,{\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 ) + 45 \,{\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 ) + 2 \,{\left (2 \, a^{3} \cos \left (d x + c\right )^{7} - 9 \, a^{3} \cos \left (d x + c\right )^{5} + 60 \, a^{3} \cos \left (d x + c\right )^{3} - 45 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/16*(16*a^3*cos(d*x + c)^7 - 90*a^3*d*x*cos(d*x + c)^4 + 48*a^3*cos(d*x + c)^5 + 180*a^3*d*x*cos(d*x + c)^2
- 150*a^3*cos(d*x + c)^3 - 90*a^3*d*x + 90*a^3*cos(d*x + c) - 45*(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) + 45*(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) + 2*(2*a^3*cos(d*x + c)^7 - 9*a^3*cos(d*x + c)^5 + 60*a^3*cos(d*x + c)^3 - 45*a^3*cos(d*x + c))*sin(d*x
+ c))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)

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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)**5*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.35055, size = 423, normalized size = 2.38 \begin{align*} \frac{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 8 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 8 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 360 \,{\left (d x + c\right )} a^{3} - 360 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 184 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{250 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 136 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 32 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 552 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 837 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 1248 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 1100 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 736 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 556 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 152 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 8 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{3}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}^{4}}}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/64*(a^3*tan(1/2*d*x + 1/2*c)^4 + 8*a^3*tan(1/2*d*x + 1/2*c)^3 + 8*a^3*tan(1/2*d*x + 1/2*c)^2 + 360*(d*x + c)
*a^3 - 360*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 184*a^3*tan(1/2*d*x + 1/2*c) + (250*a^3*tan(1/2*d*x + 1/2*c)^1
2 + 136*a^3*tan(1/2*d*x + 1/2*c)^11 - 32*a^3*tan(1/2*d*x + 1/2*c)^10 + 552*a^3*tan(1/2*d*x + 1/2*c)^9 - 837*a^
3*tan(1/2*d*x + 1/2*c)^8 + 1248*a^3*tan(1/2*d*x + 1/2*c)^7 - 1100*a^3*tan(1/2*d*x + 1/2*c)^6 + 736*a^3*tan(1/2
*d*x + 1/2*c)^5 - 556*a^3*tan(1/2*d*x + 1/2*c)^4 + 152*a^3*tan(1/2*d*x + 1/2*c)^3 - 12*a^3*tan(1/2*d*x + 1/2*c
)^2 - 8*a^3*tan(1/2*d*x + 1/2*c) - a^3)/(tan(1/2*d*x + 1/2*c)^3 + tan(1/2*d*x + 1/2*c))^4)/d