3.84 \(\int \frac{\csc ^3(e+f x)}{(a+b \tan ^2(e+f x))^3} \, dx\)

Optimal. Leaf size=205 \[ -\frac{\sqrt{b} \left (15 a^2-40 a b+24 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right )}{8 a^4 f (a-b)^{3/2}}-\frac{b (11 a-12 b) \sec (e+f x)}{8 a^3 f (a-b) \left (a+b \sec ^2(e+f x)-b\right )}-\frac{3 b \sec (e+f x)}{4 a^2 f \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac{(a-6 b) \tanh ^{-1}(\cos (e+f x))}{2 a^4 f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left (a+b \sec ^2(e+f x)-b\right )^2} \]

[Out]

-(Sqrt[b]*(15*a^2 - 40*a*b + 24*b^2)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*a^4*(a - b)^(3/2)*f) - ((a
- 6*b)*ArcTanh[Cos[e + f*x]])/(2*a^4*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f*(a - b + b*Sec[e + f*x]^2)^2) -
(3*b*Sec[e + f*x])/(4*a^2*f*(a - b + b*Sec[e + f*x]^2)^2) - ((11*a - 12*b)*b*Sec[e + f*x])/(8*a^3*(a - b)*f*(a
- b + b*Sec[e + f*x]^2))

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Rubi [A]  time = 0.292662, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3664, 471, 527, 522, 207, 205} \[ -\frac{\sqrt{b} \left (15 a^2-40 a b+24 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right )}{8 a^4 f (a-b)^{3/2}}-\frac{b (11 a-12 b) \sec (e+f x)}{8 a^3 f (a-b) \left (a+b \sec ^2(e+f x)-b\right )}-\frac{3 b \sec (e+f x)}{4 a^2 f \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac{(a-6 b) \tanh ^{-1}(\cos (e+f x))}{2 a^4 f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left (a+b \sec ^2(e+f x)-b\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3,x]

[Out]

-(Sqrt[b]*(15*a^2 - 40*a*b + 24*b^2)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*a^4*(a - b)^(3/2)*f) - ((a
- 6*b)*ArcTanh[Cos[e + f*x]])/(2*a^4*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f*(a - b + b*Sec[e + f*x]^2)^2) -
(3*b*Sec[e + f*x])/(4*a^2*f*(a - b + b*Sec[e + f*x]^2)^2) - ((11*a - 12*b)*b*Sec[e + f*x])/(8*a^3*(a - b)*f*(a
- b + b*Sec[e + f*x]^2))

Rule 3664

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sec[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a - b + b*ff^2*x^2)^p)/x^(
m + 1), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 471

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (-1+x^2\right )^2 \left (a-b+b x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left (a-b+b \sec ^2(e+f x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{a-b-5 b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{2 a f}\\ &=-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac{3 b \sec (e+f x)}{4 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{2 (2 a-3 b) (a-b)-18 (a-b) b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{8 a^2 (a-b) f}\\ &=-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac{3 b \sec (e+f x)}{4 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac{(11 a-12 b) b \sec (e+f x)}{8 a^3 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{2 (a-b) \left (4 a^2-17 a b+12 b^2\right )-2 (11 a-12 b) (a-b) b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\sec (e+f x)\right )}{16 a^3 (a-b)^2 f}\\ &=-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac{3 b \sec (e+f x)}{4 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac{(11 a-12 b) b \sec (e+f x)}{8 a^3 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )}+\frac{(a-6 b) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{2 a^4 f}-\frac{\left (b \left (15 a^2-40 a b+24 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a-b+b x^2} \, dx,x,\sec (e+f x)\right )}{8 a^4 (a-b) f}\\ &=-\frac{\sqrt{b} \left (15 a^2-40 a b+24 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right )}{8 a^4 (a-b)^{3/2} f}-\frac{(a-6 b) \tanh ^{-1}(\cos (e+f x))}{2 a^4 f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac{3 b \sec (e+f x)}{4 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac{(11 a-12 b) b \sec (e+f x)}{8 a^3 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )}\\ \end{align*}

Mathematica [A]  time = 6.53343, size = 286, normalized size = 1.4 \[ \frac{\frac{8 a^2 b^2 \cos (e+f x)}{(a-b) ((a-b) \cos (2 (e+f x))+a+b)^2}+\frac{\sqrt{b} \left (15 a^2-40 a b+24 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b}-\sqrt{a} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{b}}\right )}{(a-b)^{3/2}}+\frac{\sqrt{b} \left (15 a^2-40 a b+24 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b}+\sqrt{a} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{b}}\right )}{(a-b)^{3/2}}-\frac{2 a b (9 a-8 b) \cos (e+f x)}{(a-b) ((a-b) \cos (2 (e+f x))+a+b)}+4 (a-6 b) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )\right )-4 (a-6 b) \log \left (\cos \left (\frac{1}{2} (e+f x)\right )\right )-a \csc ^2\left (\frac{1}{2} (e+f x)\right )+a \sec ^2\left (\frac{1}{2} (e+f x)\right )}{8 a^4 f} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3,x]

[Out]

((Sqrt[b]*(15*a^2 - 40*a*b + 24*b^2)*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(3/2) +
(Sqrt[b]*(15*a^2 - 40*a*b + 24*b^2)*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(3/2) +
(8*a^2*b^2*Cos[e + f*x])/((a - b)*(a + b + (a - b)*Cos[2*(e + f*x)])^2) - (2*a*(9*a - 8*b)*b*Cos[e + f*x])/((
a - b)*(a + b + (a - b)*Cos[2*(e + f*x)])) - a*Csc[(e + f*x)/2]^2 - 4*(a - 6*b)*Log[Cos[(e + f*x)/2]] + 4*(a -
6*b)*Log[Sin[(e + f*x)/2]] + a*Sec[(e + f*x)/2]^2)/(8*a^4*f)

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Maple [B]  time = 0.109, size = 435, normalized size = 2.1 \begin{align*}{\frac{1}{4\,f{a}^{3} \left ( \cos \left ( fx+e \right ) +1 \right ) }}-{\frac{\ln \left ( \cos \left ( fx+e \right ) +1 \right ) }{4\,f{a}^{3}}}+{\frac{3\,\ln \left ( \cos \left ( fx+e \right ) +1 \right ) b}{2\,f{a}^{4}}}-{\frac{9\,b \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{8\,f{a}^{2} \left ( a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b \right ) ^{2}}}+{\frac{{b}^{2} \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{f{a}^{3} \left ( a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b \right ) ^{2}}}-{\frac{7\,{b}^{2}\cos \left ( fx+e \right ) }{8\,f{a}^{2} \left ( a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b \right ) ^{2} \left ( a-b \right ) }}+{\frac{{b}^{3}\cos \left ( fx+e \right ) }{f{a}^{3} \left ( a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b \right ) ^{2} \left ( a-b \right ) }}+{\frac{15\,b}{8\,f{a}^{2} \left ( a-b \right ) }\arctan \left ({ \left ( a-b \right ) \cos \left ( fx+e \right ){\frac{1}{\sqrt{b \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( a-b \right ) }}}}-5\,{\frac{{b}^{2}}{f{a}^{3} \left ( a-b \right ) \sqrt{b \left ( a-b \right ) }}\arctan \left ({\frac{ \left ( a-b \right ) \cos \left ( fx+e \right ) }{\sqrt{b \left ( a-b \right ) }}} \right ) }+3\,{\frac{{b}^{3}}{f{a}^{4} \left ( a-b \right ) \sqrt{b \left ( a-b \right ) }}\arctan \left ({\frac{ \left ( a-b \right ) \cos \left ( fx+e \right ) }{\sqrt{b \left ( a-b \right ) }}} \right ) }+{\frac{1}{4\,f{a}^{3} \left ( \cos \left ( fx+e \right ) -1 \right ) }}+{\frac{\ln \left ( \cos \left ( fx+e \right ) -1 \right ) }{4\,f{a}^{3}}}-{\frac{3\,\ln \left ( \cos \left ( fx+e \right ) -1 \right ) b}{2\,f{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^3/(a+b*tan(f*x+e)^2)^3,x)

[Out]

1/4/f/a^3/(cos(f*x+e)+1)-1/4/f/a^3*ln(cos(f*x+e)+1)+3/2/f/a^4*ln(cos(f*x+e)+1)*b-9/8/f*b/a^2/(a*cos(f*x+e)^2-c
os(f*x+e)^2*b+b)^2*cos(f*x+e)^3+1/f*b^2/a^3/(a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)^2*cos(f*x+e)^3-7/8/f*b^2/a^2/(a*
cos(f*x+e)^2-cos(f*x+e)^2*b+b)^2/(a-b)*cos(f*x+e)+1/f*b^3/a^3/(a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)^2/(a-b)*cos(f*
x+e)+15/8/f*b/a^2/(a-b)/(b*(a-b))^(1/2)*arctan((a-b)*cos(f*x+e)/(b*(a-b))^(1/2))-5/f*b^2/a^3/(a-b)/(b*(a-b))^(
1/2)*arctan((a-b)*cos(f*x+e)/(b*(a-b))^(1/2))+3/f*b^3/a^4/(a-b)/(b*(a-b))^(1/2)*arctan((a-b)*cos(f*x+e)/(b*(a-
b))^(1/2))+1/4/f/a^3/(cos(f*x+e)-1)+1/4/f/a^3*ln(cos(f*x+e)-1)-3/2/f/a^4*ln(cos(f*x+e)-1)*b

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^3/(a+b*tan(f*x+e)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.79051, size = 3227, normalized size = 15.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^3/(a+b*tan(f*x+e)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(2*(4*a^4 - 21*a^3*b + 29*a^2*b^2 - 12*a*b^3)*cos(f*x + e)^5 + 2*(17*a^3*b - 40*a^2*b^2 + 24*a*b^3)*cos(
f*x + e)^3 - ((15*a^4 - 70*a^3*b + 119*a^2*b^2 - 88*a*b^3 + 24*b^4)*cos(f*x + e)^6 - (15*a^4 - 100*a^3*b + 229
*a^2*b^2 - 216*a*b^3 + 72*b^4)*cos(f*x + e)^4 - 15*a^2*b^2 + 40*a*b^3 - 24*b^4 - (30*a^3*b - 125*a^2*b^2 + 168
*a*b^3 - 72*b^4)*cos(f*x + e)^2)*sqrt(-b/(a - b))*log(-((a - b)*cos(f*x + e)^2 - 2*(a - b)*sqrt(-b/(a - b))*co
s(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) + 2*(11*a^2*b^2 - 12*a*b^3)*cos(f*x + e) - 4*((a^4 - 9*a^3*b + 2
1*a^2*b^2 - 19*a*b^3 + 6*b^4)*cos(f*x + e)^6 - (a^4 - 11*a^3*b + 37*a^2*b^2 - 45*a*b^3 + 18*b^4)*cos(f*x + e)^
4 - a^2*b^2 + 7*a*b^3 - 6*b^4 - (2*a^3*b - 17*a^2*b^2 + 33*a*b^3 - 18*b^4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e
) + 1/2) + 4*((a^4 - 9*a^3*b + 21*a^2*b^2 - 19*a*b^3 + 6*b^4)*cos(f*x + e)^6 - (a^4 - 11*a^3*b + 37*a^2*b^2 -
45*a*b^3 + 18*b^4)*cos(f*x + e)^4 - a^2*b^2 + 7*a*b^3 - 6*b^4 - (2*a^3*b - 17*a^2*b^2 + 33*a*b^3 - 18*b^4)*cos
(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f*cos(f*x + e)^6 - (a^7 - 5*
a^6*b + 7*a^5*b^2 - 3*a^4*b^3)*f*cos(f*x + e)^4 - (2*a^6*b - 5*a^5*b^2 + 3*a^4*b^3)*f*cos(f*x + e)^2 - (a^5*b^
2 - a^4*b^3)*f), 1/8*((4*a^4 - 21*a^3*b + 29*a^2*b^2 - 12*a*b^3)*cos(f*x + e)^5 + (17*a^3*b - 40*a^2*b^2 + 24*
a*b^3)*cos(f*x + e)^3 - ((15*a^4 - 70*a^3*b + 119*a^2*b^2 - 88*a*b^3 + 24*b^4)*cos(f*x + e)^6 - (15*a^4 - 100*
a^3*b + 229*a^2*b^2 - 216*a*b^3 + 72*b^4)*cos(f*x + e)^4 - 15*a^2*b^2 + 40*a*b^3 - 24*b^4 - (30*a^3*b - 125*a^
2*b^2 + 168*a*b^3 - 72*b^4)*cos(f*x + e)^2)*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) +
(11*a^2*b^2 - 12*a*b^3)*cos(f*x + e) - 2*((a^4 - 9*a^3*b + 21*a^2*b^2 - 19*a*b^3 + 6*b^4)*cos(f*x + e)^6 - (a^
4 - 11*a^3*b + 37*a^2*b^2 - 45*a*b^3 + 18*b^4)*cos(f*x + e)^4 - a^2*b^2 + 7*a*b^3 - 6*b^4 - (2*a^3*b - 17*a^2*
b^2 + 33*a*b^3 - 18*b^4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 2*((a^4 - 9*a^3*b + 21*a^2*b^2 - 19*a*b
^3 + 6*b^4)*cos(f*x + e)^6 - (a^4 - 11*a^3*b + 37*a^2*b^2 - 45*a*b^3 + 18*b^4)*cos(f*x + e)^4 - a^2*b^2 + 7*a*
b^3 - 6*b^4 - (2*a^3*b - 17*a^2*b^2 + 33*a*b^3 - 18*b^4)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 -
3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f*cos(f*x + e)^6 - (a^7 - 5*a^6*b + 7*a^5*b^2 - 3*a^4*b^3)*f*cos(f*x + e)^4 -
(2*a^6*b - 5*a^5*b^2 + 3*a^4*b^3)*f*cos(f*x + e)^2 - (a^5*b^2 - a^4*b^3)*f)]

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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(csc(f*x+e)**3/(a+b*tan(f*x+e)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.6436, size = 836, normalized size = 4.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^3/(a+b*tan(f*x+e)^2)^3,x, algorithm="giac")

[Out]

-1/8*((15*a^2*b - 40*a*b^2 + 24*b^3)*arctan(-(a*cos(f*x + e) - b*cos(f*x + e) - b)/(sqrt(a*b - b^2)*cos(f*x +
e) + sqrt(a*b - b^2)))/((a^5 - a^4*b)*sqrt(a*b - b^2)) + 2*(9*a^3*b - 10*a^2*b^2 + 27*a^3*b*(cos(f*x + e) - 1)
/(cos(f*x + e) + 1) - 80*a^2*b^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 56*a*b^3*(cos(f*x + e) - 1)/(cos(f*x
+ e) + 1) + 27*a^3*b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 102*a^2*b^2*(cos(f*x + e) - 1)^2/(cos(f*x + e
) + 1)^2 + 152*a*b^3*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 80*b^4*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1
)^2 + 9*a^3*b*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 - 32*a^2*b^2*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3
+ 24*a*b^3*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3)/((a^5 - a^4*b)*(a + 2*a*(cos(f*x + e) - 1)/(cos(f*x + e
) + 1) - 4*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + a*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2)^2) - 2*(a -
6*b)*log(-(cos(f*x + e) - 1)/(cos(f*x + e) + 1))/a^4 - (a - 2*a*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 12*b*(
cos(f*x + e) - 1)/(cos(f*x + e) + 1))*(cos(f*x + e) + 1)/(a^4*(cos(f*x + e) - 1)) + (cos(f*x + e) - 1)/(a^3*(c
os(f*x + e) + 1)))/f