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

Optimal. Leaf size=295 \[ -\frac{b^2 \log \left (a \cos ^3(e+f x)+b\right )}{3 a f \left (a^2-b^2\right )}+\frac{b^{2/3} \left (a^{2/3}+b^{2/3}\right ) \log \left (a^{2/3} \cos ^2(e+f x)-\sqrt [3]{a} \sqrt [3]{b} \cos (e+f x)+b^{2/3}\right )}{6 \sqrt [3]{a} f \left (a^2-b^2\right )}-\frac{b^{2/3} \left (a^{2/3}+b^{2/3}\right ) \log \left (\sqrt [3]{a} \cos (e+f x)+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} f \left (a^2-b^2\right )}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} \cos (e+f x)}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} \sqrt [3]{a} f \left (a^{2/3} b^{2/3}+a^{4/3}+b^{4/3}\right )}+\frac{\log (1-\cos (e+f x))}{2 f (a+b)}+\frac{\log (\cos (e+f x)+1)}{2 f (a-b)} \]

[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)

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Rubi [A]  time = 0.517365, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {4138, 6725, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{b^2 \log \left (a \cos ^3(e+f x)+b\right )}{3 a f \left (a^2-b^2\right )}+\frac{b^{2/3} \left (a^{2/3}+b^{2/3}\right ) \log \left (a^{2/3} \cos ^2(e+f x)-\sqrt [3]{a} \sqrt [3]{b} \cos (e+f x)+b^{2/3}\right )}{6 \sqrt [3]{a} f \left (a^2-b^2\right )}-\frac{b^{2/3} \left (a^{2/3}+b^{2/3}\right ) \log \left (\sqrt [3]{a} \cos (e+f x)+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} f \left (a^2-b^2\right )}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} \cos (e+f x)}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} \sqrt [3]{a} f \left (a^{2/3} b^{2/3}+a^{4/3}+b^{4/3}\right )}+\frac{\log (1-\cos (e+f x))}{2 f (a+b)}+\frac{\log (\cos (e+f x)+1)}{2 f (a-b)} \]

Antiderivative was successfully verified.

[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 4138

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 6725

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]

Rule 1871

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 1860

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 31

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

Rule 634

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 617

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 204

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

Rule 628

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 260

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]

Rubi steps

\begin{align*} \int \frac{\cot (e+f x)}{a+b \sec ^3(e+f x)} \, dx &=-\frac{\operatorname{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{\operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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} \operatorname{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 ) \operatorname{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 \operatorname{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 ) \operatorname{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} \operatorname{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} \tan ^{-1}\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]  time = 0.388984, size = 290, normalized size = 0.98 \[ \frac{3 \left (a (a-b) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )\right )+a (a+b) \log \left (\cos \left (\frac{1}{2} (e+f x)\right )\right )+b^2 \log \left (\sec ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )-b \text{RootSum}\left [\text{$\#$1}^3 a-6 \text{$\#$1}^2 a-\text{$\#$1}^3 b+12 \text{$\#$1} a-8 a\& ,\frac{\text{$\#$1}^2 a b \log \left (-\text{$\#$1}+\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )-\text{$\#$1}^2 b^2 \log \left (-\text{$\#$1}+\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )-4 a^2 \log \left (-\text{$\#$1}+\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )+2 \text{$\#$1} a^2 \log \left (-\text{$\#$1}+\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )+4 a b \log \left (-\text{$\#$1}+\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )-2 \text{$\#$1} a b \log \left (-\text{$\#$1}+\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )}{\text{$\#$1}^2 a-\text{$\#$1}^2 b-4 \text{$\#$1} a+4 a}\& \right ]}{3 a f (a-b) (a+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.081, size = 393, normalized size = 1.3 \begin{align*} -{\frac{{b}^{2}}{3\,f \left ( a-b \right ) \left ( a+b \right ) a}\ln \left ( \cos \left ( fx+e \right ) +\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{b}^{2}}{6\,f \left ( a-b \right ) \left ( a+b \right ) a}\ln \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}-\sqrt [3]{{\frac{b}{a}}}\cos \left ( fx+e \right ) + \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{b}^{2}\sqrt{3}}{3\,f \left ( a-b \right ) \left ( a+b \right ) a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\cos \left ( fx+e \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{3\,f \left ( a-b \right ) \left ( a+b \right ) }\ln \left ( \cos \left ( fx+e \right ) +\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{b}{6\,f \left ( a-b \right ) \left ( a+b \right ) }\ln \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}-\sqrt [3]{{\frac{b}{a}}}\cos \left ( fx+e \right ) + \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{b\sqrt{3}}{3\,f \left ( a-b \right ) \left ( a+b \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\cos \left ( fx+e \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-{\frac{{b}^{2}\ln \left ( b+a \left ( \cos \left ( fx+e \right ) \right ) ^{3} \right ) }{3\,f \left ( a-b \right ) \left ( a+b \right ) a}}+{\frac{\ln \left ( 1+\cos \left ( fx+e \right ) \right ) }{f \left ( 2\,a-2\,b \right ) }}+{\frac{\ln \left ( -1+\cos \left ( fx+e \right ) \right ) }{f \left ( 2\,a+2\,b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 3.76252, size = 12783, normalized size = 43.33 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*(2*(a^3 - a*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a
^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) +
1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(
a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) +
6*b^2/(a^3*f - a*b^2*f))*f*log(1/2*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/
(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^
3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a
^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*s
qrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*a*b^2*f - 1/36*(a^4 - a^2*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2
- a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f -
a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a
*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2
/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))^2*f^2 - a*b*cos(f*x + e) + 2*b^2) - (
(a^3 - a*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*
b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/
((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a
*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a
^3*f - a*b^2*f))*f + 3*sqrt(1/3)*(a^3 - a*b^2)*f*sqrt(-((a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^3*f - a*b^2*f)^2
+ b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*
f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*
b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f
^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))^2*f^2 - 144*a^2*b^2 + 3
6*b^4 - 12*(a^3*b^2 - a*b^4)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*
b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f
^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*
f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3)
+ 1) + 6*b^2/(a^3*f - a*b^2*f))*f)/((a^6 - 2*a^4*b^2 + a^2*b^4)*f^2)) - 18*b^2)*log(1/2*((b^4/(a^3*f - a*b^2*f
)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*
b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1
/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b
^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*a*b^2*f - 1/36*(a^4
- a^2*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^
2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((
a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b
^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3
*f - a*b^2*f))^2*f^2 + 1/12*sqrt(1/3)*(a^4 - a^2*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))
*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54
*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18
*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*
a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f^2*sqrt(-((a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^3*f -
a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^
2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)
+ 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3
- a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))^2*f^2 - 144
*a^2*b^2 + 36*b^4 - 12*(a^3*b^2 - a*b^4)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3)
+ 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3
- a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^
2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)
*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f)/((a^6 - 2*a^4*b^2 + a^2*b^4)*f^2)) + 2*a*b*cos(f*x + e) + 2*b^2
) - ((a^3 - a*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f
- a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54
*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*
f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b
^2/(a^3*f - a*b^2*f))*f - 3*sqrt(1/3)*(a^3 - a*b^2)*f*sqrt(-((a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^3*f - a*b^2*
f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2
*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-
1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*
b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))^2*f^2 - 144*a^2*b^
2 + 36*b^4 - 12*(a^3*b^2 - a*b^4)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-
1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*
b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2
*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqr
t(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f)/((a^6 - 2*a^4*b^2 + a^2*b^4)*f^2)) - 18*b^2)*log(-1/2*((b^4/(a^3*f - a
*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2
- a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) +
9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 -
a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*a*b^2*f + 1/3
6*(a^4 - a^2*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f
- a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*
b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f
- a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^
2/(a^3*f - a*b^2*f))^2*f^2 + 1/12*sqrt(1/3)*(a^4 - a^2*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2
*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f))
- 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3
- 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b
^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f^2*sqrt(-((a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a
^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((
a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))
^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a
^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))^2*f^2
- 144*a^2*b^2 + 36*b^4 - 12*(a^3*b^2 - a*b^4)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sq
rt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a
^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((
a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))
^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f)/((a^6 - 2*a^4*b^2 + a^2*b^4)*f^2)) - 2*a*b*cos(f*x + e) -
2*b^2) - 18*(a^2 + a*b)*log(1/2*cos(f*x + e) + 1/2) - 18*(a^2 - a*b)*log(-1/2*cos(f*x + e) + 1/2))/((a^3 - a*
b^2)*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(cot(f*x+e)/(a+b*sec(f*x+e)**3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (f x + e\right )}{b \sec \left (f x + e\right )^{3} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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