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

Optimal. Leaf size=393 \[ -\frac{b^2 \left (2 a^2+b^2\right ) \log \left (a \cos ^3(e+f x)+b\right )}{3 a f \left (a^2-b^2\right )^2}+\frac{b^{4/3} \left (3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \log \left (a^{2/3} \cos ^2(e+f x)-\sqrt [3]{a} \sqrt [3]{b} \cos (e+f x)+b^{2/3}\right )}{6 \sqrt [3]{a} f \left (a^2-b^2\right )^2}-\frac{b^{4/3} \left (3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \log \left (\sqrt [3]{a} \cos (e+f x)+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} f \left (a^2-b^2\right )^2}+\frac{b^{4/3} \left (-3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} \cos (e+f x)}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} \sqrt [3]{a} f \left (a^2-b^2\right )^2}-\frac{1}{4 f (a+b) (1-\cos (e+f x))}-\frac{1}{4 f (a-b) (\cos (e+f x)+1)}-\frac{(2 a+5 b) \log (1-\cos (e+f x))}{4 f (a+b)^2}-\frac{(2 a-5 b) \log (\cos (e+f x)+1)}{4 f (a-b)^2} \]

[Out]

(b^(4/3)*(a^2 - 3*a^(2/3)*b^(4/3) + 2*b^2)*ArcTan[(b^(1/3) - 2*a^(1/3)*Cos[e + f*x])/(Sqrt[3]*b^(1/3))])/(Sqrt
[3]*a^(1/3)*(a^2 - b^2)^2*f) - 1/(4*(a + b)*f*(1 - Cos[e + f*x])) - 1/(4*(a - b)*f*(1 + Cos[e + f*x])) - ((2*a
+ 5*b)*Log[1 - Cos[e + f*x]])/(4*(a + b)^2*f) - ((2*a - 5*b)*Log[1 + Cos[e + f*x]])/(4*(a - b)^2*f) - (b^(4/3
)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[b^(1/3) + a^(1/3)*Cos[e + f*x]])/(3*a^(1/3)*(a^2 - b^2)^2*f) + (b^(4/3
)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*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)^2*f) - (b^2*(2*a^2 + b^2)*Log[b + a*Cos[e + f*x]^3])/(3*a*(a^2 - b^2)^2*f)

________________________________________________________________________________________

Rubi [A]  time = 0.631667, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {4138, 6725, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{b^2 \left (2 a^2+b^2\right ) \log \left (a \cos ^3(e+f x)+b\right )}{3 a f \left (a^2-b^2\right )^2}+\frac{b^{4/3} \left (3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \log \left (a^{2/3} \cos ^2(e+f x)-\sqrt [3]{a} \sqrt [3]{b} \cos (e+f x)+b^{2/3}\right )}{6 \sqrt [3]{a} f \left (a^2-b^2\right )^2}-\frac{b^{4/3} \left (3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \log \left (\sqrt [3]{a} \cos (e+f x)+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} f \left (a^2-b^2\right )^2}+\frac{b^{4/3} \left (-3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} \cos (e+f x)}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} \sqrt [3]{a} f \left (a^2-b^2\right )^2}-\frac{1}{4 f (a+b) (1-\cos (e+f x))}-\frac{1}{4 f (a-b) (\cos (e+f x)+1)}-\frac{(2 a+5 b) \log (1-\cos (e+f x))}{4 f (a+b)^2}-\frac{(2 a-5 b) \log (\cos (e+f x)+1)}{4 f (a-b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^(4/3)*(a^2 - 3*a^(2/3)*b^(4/3) + 2*b^2)*ArcTan[(b^(1/3) - 2*a^(1/3)*Cos[e + f*x])/(Sqrt[3]*b^(1/3))])/(Sqrt
[3]*a^(1/3)*(a^2 - b^2)^2*f) - 1/(4*(a + b)*f*(1 - Cos[e + f*x])) - 1/(4*(a - b)*f*(1 + Cos[e + f*x])) - ((2*a
+ 5*b)*Log[1 - Cos[e + f*x]])/(4*(a + b)^2*f) - ((2*a - 5*b)*Log[1 + Cos[e + f*x]])/(4*(a - b)^2*f) - (b^(4/3
)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[b^(1/3) + a^(1/3)*Cos[e + f*x]])/(3*a^(1/3)*(a^2 - b^2)^2*f) + (b^(4/3
)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*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)^2*f) - (b^2*(2*a^2 + b^2)*Log[b + a*Cos[e + f*x]^3])/(3*a*(a^2 - b^2)^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 ^3(e+f x)}{a+b \sec ^3(e+f x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^6}{\left (1-x^2\right )^2 \left (b+a x^3\right )} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{4 (a+b) (-1+x)^2}+\frac{2 a+5 b}{4 (a+b)^2 (-1+x)}-\frac{1}{4 (a-b) (1+x)^2}+\frac{2 a-5 b}{4 (a-b)^2 (1+x)}+\frac{b^2 \left (a^2+2 b^2-3 a b x+\left (2 a^2+b^2\right ) x^2\right )}{\left (a^2-b^2\right )^2 \left (b+a x^3\right )}\right ) \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{1}{4 (a+b) f (1-\cos (e+f x))}-\frac{1}{4 (a-b) f (1+\cos (e+f x))}-\frac{(2 a+5 b) \log (1-\cos (e+f x))}{4 (a+b)^2 f}-\frac{(2 a-5 b) \log (1+\cos (e+f x))}{4 (a-b)^2 f}-\frac{b^2 \operatorname{Subst}\left (\int \frac{a^2+2 b^2-3 a b x+\left (2 a^2+b^2\right ) x^2}{b+a x^3} \, dx,x,\cos (e+f x)\right )}{\left (a^2-b^2\right )^2 f}\\ &=-\frac{1}{4 (a+b) f (1-\cos (e+f x))}-\frac{1}{4 (a-b) f (1+\cos (e+f x))}-\frac{(2 a+5 b) \log (1-\cos (e+f x))}{4 (a+b)^2 f}-\frac{(2 a-5 b) \log (1+\cos (e+f x))}{4 (a-b)^2 f}-\frac{b^2 \operatorname{Subst}\left (\int \frac{a^2+2 b^2-3 a b x}{b+a x^3} \, dx,x,\cos (e+f x)\right )}{\left (a^2-b^2\right )^2 f}-\frac{\left (b^2 \left (2 a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{b+a x^3} \, dx,x,\cos (e+f x)\right )}{\left (a^2-b^2\right )^2 f}\\ &=-\frac{1}{4 (a+b) f (1-\cos (e+f x))}-\frac{1}{4 (a-b) f (1+\cos (e+f x))}-\frac{(2 a+5 b) \log (1-\cos (e+f x))}{4 (a+b)^2 f}-\frac{(2 a-5 b) \log (1+\cos (e+f x))}{4 (a-b)^2 f}-\frac{b^2 \left (2 a^2+b^2\right ) \log \left (b+a \cos ^3(e+f x)\right )}{3 a \left (a^2-b^2\right )^2 f}-\frac{b^{4/3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{b} \left (-3 a b^{4/3}+2 \sqrt [3]{a} \left (a^2+2 b^2\right )\right )+\sqrt [3]{a} \left (-3 a b^{4/3}-\sqrt [3]{a} \left (a^2+2 b^2\right )\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 )^2 f}-\frac{\left (b^{4/3} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right )\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 )^2 f}\\ &=-\frac{1}{4 (a+b) f (1-\cos (e+f x))}-\frac{1}{4 (a-b) f (1+\cos (e+f x))}-\frac{(2 a+5 b) \log (1-\cos (e+f x))}{4 (a+b)^2 f}-\frac{(2 a-5 b) \log (1+\cos (e+f x))}{4 (a-b)^2 f}-\frac{b^{4/3} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \log \left (\sqrt [3]{b}+\sqrt [3]{a} \cos (e+f x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^2 f}-\frac{b^2 \left (2 a^2+b^2\right ) \log \left (b+a \cos ^3(e+f x)\right )}{3 a \left (a^2-b^2\right )^2 f}-\frac{\left (b^{5/3} \left (a^2-3 a^{2/3} b^{4/3}+2 b^2\right )\right ) \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^2-b^2\right )^2 f}+\frac{\left (b^{4/3} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right )\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 )^2 f}\\ &=-\frac{1}{4 (a+b) f (1-\cos (e+f x))}-\frac{1}{4 (a-b) f (1+\cos (e+f x))}-\frac{(2 a+5 b) \log (1-\cos (e+f x))}{4 (a+b)^2 f}-\frac{(2 a-5 b) \log (1+\cos (e+f x))}{4 (a-b)^2 f}-\frac{b^{4/3} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \log \left (\sqrt [3]{b}+\sqrt [3]{a} \cos (e+f x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^2 f}+\frac{b^{4/3} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \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 )^2 f}-\frac{b^2 \left (2 a^2+b^2\right ) \log \left (b+a \cos ^3(e+f x)\right )}{3 a \left (a^2-b^2\right )^2 f}-\frac{\left (b^{4/3} \left (a^2-3 a^{2/3} b^{4/3}+2 b^2\right )\right ) \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^2-b^2\right )^2 f}\\ &=\frac{b^{4/3} \left (a^2-3 a^{2/3} b^{4/3}+2 b^2\right ) \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^2-b^2\right )^2 f}-\frac{1}{4 (a+b) f (1-\cos (e+f x))}-\frac{1}{4 (a-b) f (1+\cos (e+f x))}-\frac{(2 a+5 b) \log (1-\cos (e+f x))}{4 (a+b)^2 f}-\frac{(2 a-5 b) \log (1+\cos (e+f x))}{4 (a-b)^2 f}-\frac{b^{4/3} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \log \left (\sqrt [3]{b}+\sqrt [3]{a} \cos (e+f x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^2 f}+\frac{b^{4/3} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \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 )^2 f}-\frac{b^2 \left (2 a^2+b^2\right ) \log \left (b+a \cos ^3(e+f x)\right )}{3 a \left (a^2-b^2\right )^2 f}\\ \end{align*}

Mathematica [C]  time = 2.11565, size = 336, normalized size = 0.85 \[ \frac{\frac{8 b^2 \left ((b-a) \text{RootSum}\left [\text{$\#$1}^3 a-6 \text{$\#$1}^2 a-\text{$\#$1}^3 b+12 \text{$\#$1} a-8 a\& ,\frac{2 \text{$\#$1}^2 a^2 \log \left (-\text{$\#$1}+\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )+\text{$\#$1}^2 b^2 \log \left (-\text{$\#$1}+\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )+8 a^2 \log \left (-\text{$\#$1}+\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )-6 \text{$\#$1} a^2 \log \left (-\text{$\#$1}+\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )-4 a b \log \left (-\text{$\#$1}+\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )}{\text{$\#$1}^2 a-\text{$\#$1}^2 b-4 \text{$\#$1} a+4 a}\& \right ]+3 \left (2 a^2+b^2\right ) \log \left (\sec ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )}{a \left (a^2-b^2\right )^2}-\frac{3 \csc ^2\left (\frac{1}{2} (e+f x)\right )}{a+b}-\frac{3 \sec ^2\left (\frac{1}{2} (e+f x)\right )}{a-b}-\frac{12 (2 a+5 b) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )\right )}{(a+b)^2}+\frac{12 (5 b-2 a) \log \left (\cos \left (\frac{1}{2} (e+f x)\right )\right )}{(a-b)^2}}{24 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*Csc[(e + f*x)/2]^2)/(a + b) + (12*(-2*a + 5*b)*Log[Cos[(e + f*x)/2]])/(a - b)^2 - (12*(2*a + 5*b)*Log[Sin
[(e + f*x)/2]])/(a + b)^2 + (8*b^2*(3*(2*a^2 + b^2)*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 & , (8*a^2*Log[1 - #1 + Tan[(e + f*x)/2]^2] - 4*a*b*Log[1 - #1 + Tan[(e + f*x)/2]
^2] - 6*a^2*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1 + 2*a^2*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) & ]))/(a*(a^2 - b^2)^2) - (3*Sec[(e + f*x)/2]^2)
/(a - b))/(24*f)

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Maple [B]  time = 0.098, size = 676, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/f*b^2/(a-b)^2/(a+b)^2*a/(b/a)^(2/3)*ln(cos(f*x+e)+(b/a)^(1/3))-2/3/f*b^4/(a-b)^2/(a+b)^2/a/(b/a)^(2/3)*ln
(cos(f*x+e)+(b/a)^(1/3))+1/6/f*b^2/(a-b)^2/(a+b)^2*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^4/(a-b)^2/(a+b)^2/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)^2/(a+b)^2*a/(b/a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(b/a)^(1/3)*cos(f*x+e)-1))-2/3/f*b^4/(a-b)^2/(a+b)^
2/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/f*b^3/(a-b)^2/(a+b)^2/(b/a)^(1/3)*l
n(cos(f*x+e)+(b/a)^(1/3))+1/2/f*b^3/(a-b)^2/(a+b)^2/(b/a)^(1/3)*ln(cos(f*x+e)^2-(b/a)^(1/3)*cos(f*x+e)+(b/a)^(
2/3))+1/f*b^3/(a-b)^2/(a+b)^2*3^(1/2)/(b/a)^(1/3)*arctan(1/3*3^(1/2)*(2/(b/a)^(1/3)*cos(f*x+e)-1))-2/3/f*b^2/(
a-b)^2/(a+b)^2*a*ln(b+a*cos(f*x+e)^3)-1/3/f*b^4/(a-b)^2/(a+b)^2/a*ln(b+a*cos(f*x+e)^3)-1/f/(4*a-4*b)/(1+cos(f*
x+e))-1/2/f/(a-b)^2*ln(1+cos(f*x+e))*a+5/4/f/(a-b)^2*ln(1+cos(f*x+e))*b+1/f/(4*a+4*b)/(-1+cos(f*x+e))-1/2/f/(a
+b)^2*ln(-1+cos(f*x+e))*a-5/4/f/(a+b)^2*ln(-1+cos(f*x+e))*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(cot(f*x+e)^3/(a+b*sec(f*x+e)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 7.89577, size = 21913, normalized size = 55.76 \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)^3/(a+b*sec(f*x+e)^3),x, algorithm="fricas")

[Out]

1/36*(18*a^4 - 18*a^2*b^2 + 2*((a^5 - 2*a^3*b^2 + a*b^4)*f*cos(f*x + e)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*f)*((b^4
/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3)
+ 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2
+ a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3
+ 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) +
1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(
2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*
sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*log(1/12*(a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4
/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3)
+ 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2
+ a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3
+ 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) +
1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(
2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*
sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 + 2*a^2*b^2 + 7*b^4 + 1/6*(a^5 + 16*
a^3*b^2 + 10*a*b^4)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f +
a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/
((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f -
2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5
*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b
^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 -
b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f - (a^3*b + 8*a*
b^3)*cos(f*x + e)) - 18*(a^3*b - a*b^3)*cos(f*x + e) + (36*a^2*b^2 + 18*b^4 - 18*(2*a^2*b^2 + b^4)*cos(f*x + e
)^2 - ((a^5 - 2*a^3*b^2 + a*b^4)*f*cos(f*x + e)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*f)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^
2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3
- 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f -
2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4
/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*
b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*
f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*
b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f)) + 3*sqrt(1/3)*((a^5 - 2*a^3*b^2 + a*b^4)*f*cos(f*x + e)^2 - (a^5 -
2*a^3*b^2 + a*b^4)*f)*sqrt((288*a^4*b^4 + 720*a^2*b^6 - 36*b^8 - (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 +
a^2*b^8)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2
)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 -
2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f
+ a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 +
a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^
4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f
^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 - 12*(2*a^7*b^2 - 3*a^
5*b^4 + a*b^8)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^
4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6
*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3
*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*
f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f
+ a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)
^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f)/((a^10 - 4*a^8*b^2
+ 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*f^2)))*log(1/12*(a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2
+ a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 -
2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2
*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/(
(a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^
4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f
- 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^
2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 + 2*a^2*b^2 + 7*b^4 + 1/6*(a^5 + 16*a^3*b^2 + 10*a*b^4)*((b^4/
(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) +
1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2
+ a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 +
1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) +
1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2
*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*s
qrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f + 1/4*sqrt(1/3)*((a^6 - 2*a^4*b^2 + a^2*b
^4)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I
*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^
4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*
b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b
^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f))
- 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^
(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f^2 - 2*(a^5 - 2*a^3*b^2 + a*b^4)
*f)*sqrt((288*a^4*b^4 + 720*a^2*b^6 - 36*b^8 - (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*((b^4/(a^6
*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/
(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^
2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/5
4*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18
*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2
*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(
3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 - 12*(2*a^7*b^2 - 3*a^5*b^4 + a*b^8)*((b^
4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3)
+ 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^
2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3
+ 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3)
+ 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*
(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I
*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f)/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4
*b^6 + a^2*b^8)*f^2)) + 2*(a^3*b + 8*a*b^3)*cos(f*x + e)) + (36*a^2*b^2 + 18*b^4 - 18*(2*a^2*b^2 + b^4)*cos(f*
x + e)^2 - ((a^5 - 2*a^3*b^2 + a*b^4)*f*cos(f*x + e)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*f)*((b^4/(a^6*f^2 - 2*a^4*b
^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^
7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^
5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2
)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 +
b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/
(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2
*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 3*sqrt(1/3)*((a^5 - 2*a^3*b^2 + a*b^4)*f*cos(f*x + e)^2 - (
a^5 - 2*a^3*b^2 + a*b^4)*f)*sqrt((288*a^4*b^4 + 720*a^2*b^6 - 36*b^8 - (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b
^6 + a^2*b^8)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4
*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*
f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*
b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f
^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f +
a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^
4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 - 12*(2*a^7*b^2 -
3*a^5*b^4 + a*b^8)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f +
a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/
((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f -
2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5
*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b
^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 -
b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f)/((a^10 - 4*a^8
*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*f^2)))*log(-1/12*(a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^6*f^2 - 2*a^4*b^
2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7
*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5
*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)
*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b
^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(
a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*
a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 - 2*a^2*b^2 - 7*b^4 - 1/6*(a^5 + 16*a^3*b^2 + 10*a*b^4)*
((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqr
t(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^
2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*
f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f
^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1
/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3
)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f + 1/4*sqrt(1/3)*((a^6 - 2*a^4*b^2 +
a^2*b^4)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^
2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2
- 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*
f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 +
a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b
^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*
f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f^2 - 2*(a^5 - 2*a^3*b^2 +
a*b^4)*f)*sqrt((288*a^4*b^4 + 720*a^2*b^6 - 36*b^8 - (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*((b^
4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3)
+ 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^
2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3
+ 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3)
+ 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*
(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I
*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 - 12*(2*a^7*b^2 - 3*a^5*b^4 + a*b^8
)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*s
qrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*
b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^
4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4
*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) -
1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1
/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f)/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 -
4*a^4*b^6 + a^2*b^8)*f^2)) - 2*(a^3*b + 8*a*b^3)*cos(f*x + e)) + 9*(2*a^4 - a^3*b - 8*a^2*b^2 - 5*a*b^3 - (2*
a^4 - a^3*b - 8*a^2*b^2 - 5*a*b^3)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 9*(2*a^4 + a^3*b - 8*a^2*b^2
+ 5*a*b^3 - (2*a^4 + a^3*b - 8*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^5 - 2*a^3*
b^2 + a*b^4)*f*cos(f*x + e)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*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)**3/(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 )^{3}}{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)^3/(a+b*sec(f*x+e)^3),x, algorithm="giac")

[Out]

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