\(\int \frac {\cot (e+f x)}{a+b \sec ^3(e+f x)} \, dx\) [460]

Optimal result

Integrand size = 21, antiderivative size = 295 \[ \int \frac {\cot (e+f x)}{a+b \sec ^3(e+f x)} \, dx=-\frac {b^{2/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} \cos (e+f x)}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{a} \left (a^{4/3}+a^{2/3} b^{2/3}+b^{4/3}\right ) f}+\frac {\log (1-\cos (e+f x))}{2 (a+b) f}+\frac {\log (1+\cos (e+f x))}{2 (a-b) f}-\frac {\left (a^{2/3}+b^{2/3}\right ) b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \cos (e+f x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right ) f}+\frac {\left (a^{2/3}+b^{2/3}\right ) b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \cos (e+f x)+a^{2/3} \cos ^2(e+f x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right ) f}-\frac {b^2 \log \left (b+a \cos ^3(e+f x)\right )}{3 a \left (a^2-b^2\right ) f} \]

[Out]

1/2*ln(1-cos(f*x+e))/(a+b)/f+1/2*ln(1+cos(f*x+e))/(a-b)/f-1/3*(a^(2/3)+b^(2/3))*b^(2/3)*ln(b^(1/3)+a^(1/3)*cos
(f*x+e))/a^(1/3)/(a^2-b^2)/f+1/6*(a^(2/3)+b^(2/3))*b^(2/3)*ln(b^(2/3)-a^(1/3)*b^(1/3)*cos(f*x+e)+a^(2/3)*cos(f
*x+e)^2)/a^(1/3)/(a^2-b^2)/f-1/3*b^2*ln(b+a*cos(f*x+e)^3)/a/(a^2-b^2)/f-1/3*b^(2/3)*arctan(1/3*(b^(1/3)-2*a^(1
/3)*cos(f*x+e))/b^(1/3)*3^(1/2))/a^(1/3)/(a^(4/3)+a^(2/3)*b^(2/3)+b^(4/3))/f*3^(1/2)

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {4223, 6857, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {\cot (e+f x)}{a+b \sec ^3(e+f x)} \, dx=-\frac {b^{2/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} \cos (e+f x)}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{a} f \left (a^{2/3} b^{2/3}+a^{4/3}+b^{4/3}\right )}-\frac {b^2 \log \left (a \cos ^3(e+f x)+b\right )}{3 a f \left (a^2-b^2\right )}+\frac {b^{2/3} \left (a^{2/3}+b^{2/3}\right ) \log \left (a^{2/3} \cos ^2(e+f x)-\sqrt [3]{a} \sqrt [3]{b} \cos (e+f x)+b^{2/3}\right )}{6 \sqrt [3]{a} f \left (a^2-b^2\right )}-\frac {b^{2/3} \left (a^{2/3}+b^{2/3}\right ) \log \left (\sqrt [3]{a} \cos (e+f x)+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} f \left (a^2-b^2\right )}+\frac {\log (1-\cos (e+f x))}{2 f (a+b)}+\frac {\log (\cos (e+f x)+1)}{2 f (a-b)} \]

[In]

Int[Cot[e + f*x]/(a + b*Sec[e + f*x]^3),x]

[Out]

-((b^(2/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*Cos[e + f*x])/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(1/3)*(a^(4/3) + a^(2/3)*b
^(2/3) + b^(4/3))*f)) + Log[1 - Cos[e + f*x]]/(2*(a + b)*f) + Log[1 + Cos[e + f*x]]/(2*(a - b)*f) - ((a^(2/3)
+ b^(2/3))*b^(2/3)*Log[b^(1/3) + a^(1/3)*Cos[e + f*x]])/(3*a^(1/3)*(a^2 - b^2)*f) + ((a^(2/3) + b^(2/3))*b^(2/
3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*Cos[e + f*x] + a^(2/3)*Cos[e + f*x]^2])/(6*a^(1/3)*(a^2 - b^2)*f) - (b^2*Log[
b + a*Cos[e + f*x]^3])/(3*a*(a^2 - b^2)*f)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1874

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, Dist[(-r)*((B*r - A*s)/(3*a*s)), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) +
s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[
a/b]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 4223

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Module[{ff =
FreeFactors[Cos[e + f*x], x]}, Dist[-(f*ff^(m + n*p - 1))^(-1), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*
(ff*x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n] && IntegerQ[p]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right ) \left (b+a x^3\right )} \, dx,x,\cos (e+f x)\right )}{f} \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {1}{2 (a+b) (-1+x)}-\frac {1}{2 (a-b) (1+x)}-\frac {b \left (b-a x+b x^2\right )}{\left (-a^2+b^2\right ) \left (b+a x^3\right )}\right ) \, dx,x,\cos (e+f x)\right )}{f} \\ & = \frac {\log (1-\cos (e+f x))}{2 (a+b) f}+\frac {\log (1+\cos (e+f x))}{2 (a-b) f}-\frac {b \text {Subst}\left (\int \frac {b-a x+b x^2}{b+a x^3} \, dx,x,\cos (e+f x)\right )}{\left (a^2-b^2\right ) f} \\ & = \frac {\log (1-\cos (e+f x))}{2 (a+b) f}+\frac {\log (1+\cos (e+f x))}{2 (a-b) f}-\frac {b \text {Subst}\left (\int \frac {b-a x}{b+a x^3} \, dx,x,\cos (e+f x)\right )}{\left (a^2-b^2\right ) f}-\frac {b^2 \text {Subst}\left (\int \frac {x^2}{b+a x^3} \, dx,x,\cos (e+f x)\right )}{\left (a^2-b^2\right ) f} \\ & = \frac {\log (1-\cos (e+f x))}{2 (a+b) f}+\frac {\log (1+\cos (e+f x))}{2 (a-b) f}-\frac {b^2 \log \left (b+a \cos ^3(e+f x)\right )}{3 a \left (a^2-b^2\right ) f}-\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {\sqrt [3]{b} \left (-a \sqrt [3]{b}+2 \sqrt [3]{a} b\right )+\sqrt [3]{a} \left (-a \sqrt [3]{b}-\sqrt [3]{a} b\right ) x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\cos (e+f x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right ) f}-\frac {\left (\left (a^{2/3}+b^{2/3}\right ) b^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,\cos (e+f x)\right )}{3 \left (a^2-b^2\right ) f} \\ & = \frac {\log (1-\cos (e+f x))}{2 (a+b) f}+\frac {\log (1+\cos (e+f x))}{2 (a-b) f}-\frac {\left (a^{2/3}+b^{2/3}\right ) b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \cos (e+f x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right ) f}-\frac {b^2 \log \left (b+a \cos ^3(e+f x)\right )}{3 a \left (a^2-b^2\right ) f}+\frac {b \text {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\cos (e+f x)\right )}{2 \left (a^{4/3}+a^{2/3} b^{2/3}+b^{4/3}\right ) f}+\frac {\left (\left (a^{2/3}+b^{2/3}\right ) b^{2/3}\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\cos (e+f x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right ) f} \\ & = \frac {\log (1-\cos (e+f x))}{2 (a+b) f}+\frac {\log (1+\cos (e+f x))}{2 (a-b) f}-\frac {\left (a^{2/3}+b^{2/3}\right ) b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \cos (e+f x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right ) f}+\frac {\left (a^{2/3}+b^{2/3}\right ) b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \cos (e+f x)+a^{2/3} \cos ^2(e+f x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right ) f}-\frac {b^2 \log \left (b+a \cos ^3(e+f x)\right )}{3 a \left (a^2-b^2\right ) f}+\frac {b^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} \cos (e+f x)}{\sqrt [3]{b}}\right )}{\sqrt [3]{a} \left (a^{4/3}+a^{2/3} b^{2/3}+b^{4/3}\right ) f} \\ & = -\frac {b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} \cos (e+f x)}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \left (a^{4/3}+a^{2/3} b^{2/3}+b^{4/3}\right ) f}+\frac {\log (1-\cos (e+f x))}{2 (a+b) f}+\frac {\log (1+\cos (e+f x))}{2 (a-b) f}-\frac {\left (a^{2/3}+b^{2/3}\right ) b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \cos (e+f x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right ) f}+\frac {\left (a^{2/3}+b^{2/3}\right ) b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \cos (e+f x)+a^{2/3} \cos ^2(e+f x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right ) f}-\frac {b^2 \log \left (b+a \cos ^3(e+f x)\right )}{3 a \left (a^2-b^2\right ) f} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.69 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.73 \[ \int \frac {\cot (e+f x)}{a+b \sec ^3(e+f x)} \, dx=\frac {\log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{(a-b) f}+\frac {\log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{(a+b) f}+\frac {b \left (3 b \log \left (\sec ^2\left (\frac {1}{2} (e+f x)\right )\right )+(-a+b) \text {RootSum}\left [-8 a+12 a \text {$\#$1}-6 a \text {$\#$1}^2+a \text {$\#$1}^3-b \text {$\#$1}^3\&,\frac {-4 a \log \left (1-\text {$\#$1}+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+2 a \log \left (1-\text {$\#$1}+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \text {$\#$1}+b \log \left (1-\text {$\#$1}+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \text {$\#$1}^2}{4 a-4 a \text {$\#$1}+a \text {$\#$1}^2-b \text {$\#$1}^2}\&\right ]\right )}{3 \left (a^3-a b^2\right ) f} \]

[In]

Integrate[Cot[e + f*x]/(a + b*Sec[e + f*x]^3),x]

[Out]

Log[Cos[(e + f*x)/2]]/((a - b)*f) + Log[Sin[(e + f*x)/2]]/((a + b)*f) + (b*(3*b*Log[Sec[(e + f*x)/2]^2] + (-a
+ b)*RootSum[-8*a + 12*a*#1 - 6*a*#1^2 + a*#1^3 - b*#1^3 & , (-4*a*Log[1 - #1 + Tan[(e + f*x)/2]^2] + 2*a*Log[
1 - #1 + Tan[(e + f*x)/2]^2]*#1 + b*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1^2)/(4*a - 4*a*#1 + a*#1^2 - b*#1^2) &
]))/(3*(a^3 - a*b^2)*f)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.43 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.98

method	result	size
risch	\(\frac {i x}{a}-\frac {i x}{a +b}-\frac {i e}{f \left (a +b \right )}-\frac {i x}{a -b}-\frac {i e}{f \left (a -b \right )}-\frac {2 i a^{2} b^{2} f^{3} x}{-a^{5} f^{3}+a^{3} b^{2} f^{3}}-\frac {2 i a^{2} b^{2} f^{2} e}{-a^{5} f^{3}+a^{3} b^{2} f^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f \left (a +b \right )}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{f \left (a -b \right )}+i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (27 a^{5} f^{3}-27 a^{3} b^{2} f^{3}\right ) \textit {\_Z}^{3}-27 i b^{2} a^{2} f^{2} \textit {\_Z}^{2}+9 \textit {\_Z} a \,b^{2} f +i b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\left (\left (-\frac {18 a^{3} f^{2}}{b}+18 a b \,f^{2}\right ) \textit {\_R}^{2}+18 i b f \textit {\_R} -\frac {4 b}{a}\right ) {\mathrm e}^{i \left (f x +e \right )}+1\right )\right )\)	\(288\)
derivativedivides	\(\frac {\frac {\ln \left (1+\cos \left (f x +e \right )\right )}{2 a -2 b}+\frac {\ln \left (-1+\cos \left (f x +e \right )\right )}{2 a +2 b}+\frac {\left (-b \left (\frac {\ln \left (\cos \left (f x +e \right )+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (\cos \left (f x +e \right )^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} \cos \left (f x +e \right )+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \cos \left (f x +e \right )}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\right )+a \left (-\frac {\ln \left (\cos \left (f x +e \right )+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\ln \left (\cos \left (f x +e \right )^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} \cos \left (f x +e \right )+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \cos \left (f x +e \right )}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )-\frac {b \ln \left (b +a \cos \left (f x +e \right )^{3}\right )}{3 a}\right ) b}{\left (a +b \right ) \left (a -b \right )}}{f}\)	\(303\)
default	\(\frac {\frac {\ln \left (1+\cos \left (f x +e \right )\right )}{2 a -2 b}+\frac {\ln \left (-1+\cos \left (f x +e \right )\right )}{2 a +2 b}+\frac {\left (-b \left (\frac {\ln \left (\cos \left (f x +e \right )+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (\cos \left (f x +e \right )^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} \cos \left (f x +e \right )+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \cos \left (f x +e \right )}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\right )+a \left (-\frac {\ln \left (\cos \left (f x +e \right )+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\ln \left (\cos \left (f x +e \right )^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} \cos \left (f x +e \right )+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \cos \left (f x +e \right )}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )-\frac {b \ln \left (b +a \cos \left (f x +e \right )^{3}\right )}{3 a}\right ) b}{\left (a +b \right ) \left (a -b \right )}}{f}\)	\(303\)

[In]

int(cot(f*x+e)/(a+b*sec(f*x+e)^3),x,method=_RETURNVERBOSE)

[Out]

I*x/a-I/(a+b)*x-I/f/(a+b)*e-I/(a-b)*x-I/f/(a-b)*e-2*I*a^2*b^2*f^3/(-a^5*f^3+a^3*b^2*f^3)*x-2*I*a^2*b^2*f^2/(-a
^5*f^3+a^3*b^2*f^3)*e+1/f/(a+b)*ln(exp(I*(f*x+e))-1)+1/f/(a-b)*ln(exp(I*(f*x+e))+1)+I*sum(_R*ln(exp(2*I*(f*x+e
))+((-18*a^3/b*f^2+18*a*b*f^2)*_R^2+18*I*b*f*_R-4*b/a)*exp(I*(f*x+e))+1),_R=RootOf((27*a^5*f^3-27*a^3*b^2*f^3)
*_Z^3-27*I*b^2*a^2*f^2*_Z^2+9*_Z*a*b^2*f+I*b^2))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.06 (sec) , antiderivative size = 6482, normalized size of antiderivative = 21.97 \[ \int \frac {\cot (e+f x)}{a+b \sec ^3(e+f x)} \, dx=\text {Too large to display} \]

[In]

integrate(cot(f*x+e)/(a+b*sec(f*x+e)^3),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\cot (e+f x)}{a+b \sec ^3(e+f x)} \, dx=\int \frac {\cot {\left (e + f x \right )}}{a + b \sec ^{3}{\left (e + f x \right )}}\, dx \]

[In]

integrate(cot(f*x+e)/(a+b*sec(f*x+e)**3),x)

[Out]

Integral(cot(e + f*x)/(a + b*sec(e + f*x)**3), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.04 \[ \int \frac {\cot (e+f x)}{a+b \sec ^3(e+f x)} \, dx=-\frac {\frac {2 \, \sqrt {3} {\left (a b^{2} {\left (3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}} + \frac {2 \, b}{a}\right )} - 3 \, a^{2} b \left (\frac {b}{a}\right )^{\frac {2}{3}} - 2 \, b^{3}\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {b}{a}\right )^{\frac {1}{3}} - 2 \, \cos \left (f x + e\right )\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{{\left (a^{4} \left (\frac {b}{a}\right )^{\frac {2}{3}} - a^{2} b^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {3 \, {\left (b^{2} {\left (2 \, \left (\frac {b}{a}\right )^{\frac {2}{3}} - 1\right )} - a b \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )} \log \left (\cos \left (f x + e\right )^{2} - \left (\frac {b}{a}\right )^{\frac {1}{3}} \cos \left (f x + e\right ) + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}} - a b^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {6 \, {\left (b^{2} {\left (\left (\frac {b}{a}\right )^{\frac {2}{3}} + 1\right )} + a b \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )} \log \left (\left (\frac {b}{a}\right )^{\frac {1}{3}} + \cos \left (f x + e\right )\right )}{a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}} - a b^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {9 \, \log \left (\cos \left (f x + e\right ) + 1\right )}{a - b} - \frac {9 \, \log \left (\cos \left (f x + e\right ) - 1\right )}{a + b}}{18 \, f} \]

[In]

integrate(cot(f*x+e)/(a+b*sec(f*x+e)^3),x, algorithm="maxima")

[Out]

-1/18*(2*sqrt(3)*(a*b^2*(3*(b/a)^(1/3) + 2*b/a) - 3*a^2*b*(b/a)^(2/3) - 2*b^3)*arctan(-1/3*sqrt(3)*((b/a)^(1/3
) - 2*cos(f*x + e))/(b/a)^(1/3))/((a^4*(b/a)^(2/3) - a^2*b^2*(b/a)^(2/3))*(b/a)^(1/3)) + 3*(b^2*(2*(b/a)^(2/3)
- 1) - a*b*(b/a)^(1/3))*log(cos(f*x + e)^2 - (b/a)^(1/3)*cos(f*x + e) + (b/a)^(2/3))/(a^3*(b/a)^(2/3) - a*b^2
*(b/a)^(2/3)) + 6*(b^2*((b/a)^(2/3) + 1) + a*b*(b/a)^(1/3))*log((b/a)^(1/3) + cos(f*x + e))/(a^3*(b/a)^(2/3) -
a*b^2*(b/a)^(2/3)) - 9*log(cos(f*x + e) + 1)/(a - b) - 9*log(cos(f*x + e) - 1)/(a + b))/f

Giac [F]

\[ \int \frac {\cot (e+f x)}{a+b \sec ^3(e+f x)} \, dx=\int { \frac {\cot \left (f x + e\right )}{b \sec \left (f x + e\right )^{3} + a} \,d x } \]

[In]

integrate(cot(f*x+e)/(a+b*sec(f*x+e)^3),x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 23.78 (sec) , antiderivative size = 11182, normalized size of antiderivative = 37.91 \[ \int \frac {\cot (e+f x)}{a+b \sec ^3(e+f x)} \, dx=\text {Too large to display} \]

[In]

int(cot(e + f*x)/(a + b/cos(e + f*x)^3),x)

[Out]

log(sin(e/2 + (f*x)/2)/cos(e/2 + (f*x)/2))/(f*(a + b)) - log(1/cos(e/2 + (f*x)/2)^2)/(f*(a + b)) + (a*symsum(l
og((262144*(832*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*b^7 - 22*a*b^5 - 84
0*b^6*cos(e + f*x) + 440*b^6 - 264*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^
2*b^8 + 16*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*b^9 + 1823*root(27*a^3
*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*a^2*b^5 - 21*root(27*a^3*b^2*z^3 - 27*a^5*z^3
- 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*a^3*b^4 - 8864*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9
*a*b^2*z - b^2, z, k)^2*a*b^7 + 3092*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k
)^3*a*b^8 - 192*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a*b^9 + 88*root(2
7*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*b^8*cos(e + f*x) - a^2*b^4*cos(e + f*x)
+ 65221*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*a^2*b^6 - 32708*root(27*
a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*a^3*b^5 + 2859*root(27*a^3*b^2*z^3 - 27*a
^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*a^4*b^4 - 9*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z
^2 + 9*a*b^2*z - b^2, z, k)^2*a^5*b^3 + 26274*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z -
b^2, z, k)^3*a^2*b^7 - 212230*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a^3
*b^6 + 216667*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a^4*b^5 - 44745*roo
t(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a^5*b^4 + 1584*root(27*a^3*b^2*z^3 -
27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a^6*b^3 - 12720*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*
a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^2*b^8 + 14028*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a
*b^2*z - b^2, z, k)^4*a^3*b^7 + 156387*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z,
k)^4*a^4*b^6 - 457125*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^5*b^5 +
228117*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^6*b^4 - 24723*root(27*a^
3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^7*b^3 + 486*root(27*a^3*b^2*z^3 - 27*a^5*
z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^8*b^2 + 864*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^
2 + 9*a*b^2*z - b^2, z, k)^5*a^2*b^9 + 18792*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b
^2, z, k)^5*a^3*b^8 - 151488*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^4*
b^7 + 577008*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^5*b^6 - 414504*roo
t(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^6*b^5 - 144432*root(27*a^3*b^2*z^3
- 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^7*b^4 + 63702*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 2
7*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^8*b^3 + 486*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a
*b^2*z - b^2, z, k)^5*a^9*b^2 - 1728*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k
)^6*a^3*b^9 + 3672*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^4*b^8 + 6944
4*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^5*b^7 - 637794*root(27*a^3*b^
2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^6*b^6 + 1468908*root(27*a^3*b^2*z^3 - 27*a^5*
z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^7*b^5 - 1112400*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^
2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^8*b^4 + 210384*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*
z - b^2, z, k)^6*a^9*b^3 - 486*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^
10*b^2 + 1296*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^4*b^9 - 23004*roo
t(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^5*b^8 + 195534*root(27*a^3*b^2*z^3
- 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^6*b^7 - 778734*root(27*a^3*b^2*z^3 - 27*a^5*z^3 -
27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^7*b^6 + 1175796*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2
+ 9*a*b^2*z - b^2, z, k)^7*a^8*b^5 - 690768*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^
2, z, k)^7*a^9*b^4 + 120366*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^10*
b^3 - 486*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^11*b^2 - 8702*root(27
*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*a*b^6 - 272*root(27*a^3*b^2*z^3 - 27*a^5*z
^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*b^7*cos(e + f*x) + 62*a*b^5*cos(e + f*x) + 13774*root(27*a^3*b^2*
z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*a*b^6*cos(e + f*x) - 4098*root(27*a^3*b^2*z^3 - 27*
a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*a^2*b^5*cos(e + f*x) + 122*root(27*a^3*b^2*z^3 - 27*a^5*z^3
- 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*a^3*b^4*cos(e + f*x) + 2088*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^
2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*a*b^7*cos(e + f*x) - 980*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^
2 + 9*a*b^2*z - b^2, z, k)^3*a*b^8*cos(e + f*x) - 85013*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*
a*b^2*z - b^2, z, k)^2*a^2*b^6*cos(e + f*x) + 55956*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^
2*z - b^2, z, k)^2*a^3*b^5*cos(e + f*x) - 8075*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z -
b^2, z, k)^2*a^4*b^4*cos(e + f*x) + 117*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2,
z, k)^2*a^5*b^3*cos(e + f*x) + 818*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^
3*a^2*b^7*cos(e + f*x) + 217434*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a
^3*b^6*cos(e + f*x) - 285091*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a^4*
b^5*cos(e + f*x) + 82633*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a^5*b^4*
cos(e + f*x) - 6984*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a^6*b^3*cos(e
+ f*x) + 3792*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^2*b^8*cos(e + f*
x) - 42132*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^3*b^7*cos(e + f*x) -
54423*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^4*b^6*cos(e + f*x) + 435
417*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^5*b^5*cos(e + f*x) - 280113
*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^6*b^4*cos(e + f*x) + 49239*roo
t(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^7*b^3*cos(e + f*x) - 3402*root(27*
a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^8*b^2*cos(e + f*x) - 4968*root(27*a^3*b
^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^3*b^8*cos(e + f*x) + 99864*root(27*a^3*b^2*z
^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^4*b^7*cos(e + f*x) - 643536*root(27*a^3*b^2*z^3
- 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^5*b^6*cos(e + f*x) + 636552*root(27*a^3*b^2*z^3 - 2
7*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^6*b^5*cos(e + f*x) - 936*root(27*a^3*b^2*z^3 - 27*a^5*
z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^7*b^4*cos(e + f*x) - 28170*root(27*a^3*b^2*z^3 - 27*a^5*z^3
- 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^8*b^3*cos(e + f*x) - 3402*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*
a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^9*b^2*cos(e + f*x) - 2376*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b
^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^4*b^8*cos(e + f*x) + 972*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2
+ 9*a*b^2*z - b^2, z, k)^6*a^5*b^7*cos(e + f*x) + 457758*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 +
9*a*b^2*z - b^2, z, k)^6*a^6*b^6*cos(e + f*x) - 1352916*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*
a*b^2*z - b^2, z, k)^6*a^7*b^5*cos(e + f*x) + 1122336*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*
b^2*z - b^2, z, k)^6*a^8*b^4*cos(e + f*x) - 229176*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2
*z - b^2, z, k)^6*a^9*b^3*cos(e + f*x) + 3402*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z -
b^2, z, k)^6*a^10*b^2*cos(e + f*x) + 7452*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2,
z, k)^7*a^5*b^8*cos(e + f*x) - 139482*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z,
k)^7*a^6*b^7*cos(e + f*x) + 729810*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)
^7*a^7*b^6*cos(e + f*x) - 1208844*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7
*a^8*b^5*cos(e + f*x) + 752328*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^
9*b^4*cos(e + f*x) - 144666*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^10*
b^3*cos(e + f*x) + 3402*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^11*b^2*
cos(e + f*x)))/cos(e/2 + (f*x)/2)^2)*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k
), k, 1, 3))/(f*(a + b)) + (b*symsum(log((262144*(832*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*
b^2*z - b^2, z, k)*b^7 - 22*a*b^5 - 840*b^6*cos(e + f*x) + 440*b^6 - 264*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27
*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*b^8 + 16*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z
- b^2, z, k)^3*b^9 + 1823*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*a^2*b^5
- 21*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*a^3*b^4 - 8864*root(27*a^3*b^2
*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*a*b^7 + 3092*root(27*a^3*b^2*z^3 - 27*a^5*z^3 -
27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a*b^8 - 192*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*
b^2*z - b^2, z, k)^4*a*b^9 + 88*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*b
^8*cos(e + f*x) - a^2*b^4*cos(e + f*x) + 65221*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z -
b^2, z, k)^2*a^2*b^6 - 32708*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*a^3
*b^5 + 2859*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*a^4*b^4 - 9*root(27*a
^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*a^5*b^3 + 26274*root(27*a^3*b^2*z^3 - 27*a
^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a^2*b^7 - 212230*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*
b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a^3*b^6 + 216667*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^
2*z - b^2, z, k)^3*a^4*b^5 - 44745*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^
3*a^5*b^4 + 1584*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a^6*b^3 - 12720*
root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^2*b^8 + 14028*root(27*a^3*b^2*z
^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^3*b^7 + 156387*root(27*a^3*b^2*z^3 - 27*a^5*z^3
- 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^4*b^6 - 457125*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2
+ 9*a*b^2*z - b^2, z, k)^4*a^5*b^5 + 228117*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b
^2, z, k)^4*a^6*b^4 - 24723*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^7*b
^3 + 486*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^8*b^2 + 864*root(27*a^
3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^2*b^9 + 18792*root(27*a^3*b^2*z^3 - 27*a^
5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^3*b^8 - 151488*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b
^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^4*b^7 + 577008*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2
*z - b^2, z, k)^5*a^5*b^6 - 414504*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^
5*a^6*b^5 - 144432*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^7*b^4 + 6370
2*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^8*b^3 + 486*root(27*a^3*b^2*z
^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^9*b^2 - 1728*root(27*a^3*b^2*z^3 - 27*a^5*z^3 -
27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^3*b^9 + 3672*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9
*a*b^2*z - b^2, z, k)^6*a^4*b^8 + 69444*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z
, k)^6*a^5*b^7 - 637794*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^6*b^6 +
1468908*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^7*b^5 - 1112400*root(2
7*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^8*b^4 + 210384*root(27*a^3*b^2*z^3 -
27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^9*b^3 - 486*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2
*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^10*b^2 + 1296*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^
2*z - b^2, z, k)^7*a^4*b^9 - 23004*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^
7*a^5*b^8 + 195534*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^6*b^7 - 7787
34*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^7*b^6 + 1175796*root(27*a^3*
b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^8*b^5 - 690768*root(27*a^3*b^2*z^3 - 27*a^5
*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^9*b^4 + 120366*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^
2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^10*b^3 - 486*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z
- b^2, z, k)^7*a^11*b^2 - 8702*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*a*b^
6 - 272*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*b^7*cos(e + f*x) + 62*a*b^5
*cos(e + f*x) + 13774*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*a*b^6*cos(e +
f*x) - 4098*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*a^2*b^5*cos(e + f*x) +
122*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)*a^3*b^4*cos(e + f*x) + 2088*ro
ot(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*a*b^7*cos(e + f*x) - 980*root(27*a^
3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a*b^8*cos(e + f*x) - 85013*root(27*a^3*b^2*
z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*a^2*b^6*cos(e + f*x) + 55956*root(27*a^3*b^2*z^3
- 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*a^3*b^5*cos(e + f*x) - 8075*root(27*a^3*b^2*z^3 - 27*
a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*a^4*b^4*cos(e + f*x) + 117*root(27*a^3*b^2*z^3 - 27*a^5*z^
3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^2*a^5*b^3*cos(e + f*x) + 818*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27
*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a^2*b^7*cos(e + f*x) + 217434*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^
2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a^3*b^6*cos(e + f*x) - 285091*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b
^2*z^2 + 9*a*b^2*z - b^2, z, k)^3*a^4*b^5*cos(e + f*x) + 82633*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z
^2 + 9*a*b^2*z - b^2, z, k)^3*a^5*b^4*cos(e + f*x) - 6984*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 +
9*a*b^2*z - b^2, z, k)^3*a^6*b^3*cos(e + f*x) + 3792*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b
^2*z - b^2, z, k)^4*a^2*b^8*cos(e + f*x) - 42132*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z
- b^2, z, k)^4*a^3*b^7*cos(e + f*x) - 54423*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b
^2, z, k)^4*a^4*b^6*cos(e + f*x) + 435417*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2,
z, k)^4*a^5*b^5*cos(e + f*x) - 280113*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z,
k)^4*a^6*b^4*cos(e + f*x) + 49239*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^
4*a^7*b^3*cos(e + f*x) - 3402*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^4*a^8
*b^2*cos(e + f*x) - 4968*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^3*b^8*
cos(e + f*x) + 99864*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^4*b^7*cos(
e + f*x) - 643536*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^5*b^6*cos(e +
f*x) + 636552*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^6*b^5*cos(e + f*
x) - 936*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^7*b^4*cos(e + f*x) - 2
8170*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^8*b^3*cos(e + f*x) - 3402*
root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^5*a^9*b^2*cos(e + f*x) - 2376*root(
27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^4*b^8*cos(e + f*x) + 972*root(27*a^3
*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^5*b^7*cos(e + f*x) + 457758*root(27*a^3*b^
2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^6*b^6*cos(e + f*x) - 1352916*root(27*a^3*b^2*
z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^7*b^5*cos(e + f*x) + 1122336*root(27*a^3*b^2*z^
3 - 27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^8*b^4*cos(e + f*x) - 229176*root(27*a^3*b^2*z^3 -
27*a^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^9*b^3*cos(e + f*x) + 3402*root(27*a^3*b^2*z^3 - 27*a
^5*z^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^6*a^10*b^2*cos(e + f*x) + 7452*root(27*a^3*b^2*z^3 - 27*a^5*z
^3 - 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^5*b^8*cos(e + f*x) - 139482*root(27*a^3*b^2*z^3 - 27*a^5*z^3
- 27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^6*b^7*cos(e + f*x) + 729810*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 2
7*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^7*b^6*cos(e + f*x) - 1208844*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*
a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^8*b^5*cos(e + f*x) + 752328*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2
*b^2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^9*b^4*cos(e + f*x) - 144666*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^
2*z^2 + 9*a*b^2*z - b^2, z, k)^7*a^10*b^3*cos(e + f*x) + 3402*root(27*a^3*b^2*z^3 - 27*a^5*z^3 - 27*a^2*b^2*z^
2 + 9*a*b^2*z - b^2, z, k)^7*a^11*b^2*cos(e + f*x)))/cos(e/2 + (f*x)/2)^2)*root(27*a^3*b^2*z^3 - 27*a^5*z^3 -
27*a^2*b^2*z^2 + 9*a*b^2*z - b^2, z, k), k, 1, 3))/(f*(a + b)) - (b*log(1/cos(e/2 + (f*x)/2)^2))/(a*f*(a + b))