3.361 \(\int \frac{\cot ^4(e+f x)}{(a+b \sec ^2(e+f x))^2} \, dx\)

Optimal. Leaf size=160 \[ -\frac{b^{5/2} (7 a+2 b) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{2 a^2 f (a+b)^{7/2}}+\frac{\left (2 a^2+6 a b-b^2\right ) \cot (e+f x)}{2 a f (a+b)^3}+\frac{x}{a^2}-\frac{(2 a-3 b) \cot ^3(e+f x)}{6 a f (a+b)^2}-\frac{b \cot ^3(e+f x)}{2 a f (a+b) \left (a+b \tan ^2(e+f x)+b\right )} \]

[Out]

x/a^2 - (b^(5/2)*(7*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(7/2)*f) + ((2*a^2 + 6
*a*b - b^2)*Cot[e + f*x])/(2*a*(a + b)^3*f) - ((2*a - 3*b)*Cot[e + f*x]^3)/(6*a*(a + b)^2*f) - (b*Cot[e + f*x]
^3)/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))

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Rubi [A]  time = 0.353714, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4141, 1975, 472, 583, 522, 203, 205} \[ -\frac{b^{5/2} (7 a+2 b) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{2 a^2 f (a+b)^{7/2}}+\frac{\left (2 a^2+6 a b-b^2\right ) \cot (e+f x)}{2 a f (a+b)^3}+\frac{x}{a^2}-\frac{(2 a-3 b) \cot ^3(e+f x)}{6 a f (a+b)^2}-\frac{b \cot ^3(e+f x)}{2 a f (a+b) \left (a+b \tan ^2(e+f x)+b\right )} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]

[Out]

x/a^2 - (b^(5/2)*(7*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(7/2)*f) + ((2*a^2 + 6
*a*b - b^2)*Cot[e + f*x])/(2*a*(a + b)^3*f) - ((2*a - 3*b)*Cot[e + f*x]^3)/(6*a*(a + b)^2*f) - (b*Cot[e + f*x]
^3)/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))

Rule 4141

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

Rule 1975

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&&  !BinomialMatchQ[{u, v}, x]

Rule 472

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 522

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

Rule 203

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

Rule 205

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

Rubi steps

\begin{align*} \int \frac{\cot ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+x^2\right ) \left (a+b \left (1+x^2\right )\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+x^2\right ) \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{b \cot ^3(e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{2 a-3 b-5 b x^2}{x^4 \left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{2 a (a+b) f}\\ &=-\frac{(2 a-3 b) \cot ^3(e+f x)}{6 a (a+b)^2 f}-\frac{b \cot ^3(e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{3 \left (2 a^2+6 a b-b^2\right )+3 (2 a-3 b) b x^2}{x^2 \left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{6 a (a+b)^2 f}\\ &=\frac{\left (2 a^2+6 a b-b^2\right ) \cot (e+f x)}{2 a (a+b)^3 f}-\frac{(2 a-3 b) \cot ^3(e+f x)}{6 a (a+b)^2 f}-\frac{b \cot ^3(e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{3 \left (2 a^3+8 a^2 b+12 a b^2+b^3\right )+3 b \left (2 a^2+6 a b-b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{6 a (a+b)^3 f}\\ &=\frac{\left (2 a^2+6 a b-b^2\right ) \cot (e+f x)}{2 a (a+b)^3 f}-\frac{(2 a-3 b) \cot ^3(e+f x)}{6 a (a+b)^2 f}-\frac{b \cot ^3(e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{a^2 f}-\frac{\left (b^3 (7 a+2 b)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{2 a^2 (a+b)^3 f}\\ &=\frac{x}{a^2}-\frac{b^{5/2} (7 a+2 b) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{2 a^2 (a+b)^{7/2} f}+\frac{\left (2 a^2+6 a b-b^2\right ) \cot (e+f x)}{2 a (a+b)^3 f}-\frac{(2 a-3 b) \cot ^3(e+f x)}{6 a (a+b)^2 f}-\frac{b \cot ^3(e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}\\ \end{align*}

Mathematica [C]  time = 6.93164, size = 1896, normalized size = 11.85 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]

[Out]

((7*a + 2*b)*(a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*((b^3*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sq
rt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]
) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Cos[2*e])/(8*a^2*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/8)
*b^3*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]
*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Sin[2*e])/(a^2*Sqrt[a +
b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]])))/((a + b)^3*(a + b*Sec[e + f*x]^2)^2) + ((a + 2*b + a*Cos[2*e + 2*f*x])
*Csc[e]*Csc[e + f*x]^3*Sec[2*e]*Sec[e + f*x]^4*(-6*a^4*f*x*Cos[f*x] - 54*a^3*b*f*x*Cos[f*x] - 126*a^2*b^2*f*x*
Cos[f*x] - 114*a*b^3*f*x*Cos[f*x] - 36*b^4*f*x*Cos[f*x] + 3*a^4*f*x*Cos[3*f*x] - 3*a^3*b*f*x*Cos[3*f*x] - 27*a
^2*b^2*f*x*Cos[3*f*x] - 33*a*b^3*f*x*Cos[3*f*x] - 12*b^4*f*x*Cos[3*f*x] + 6*a^4*f*x*Cos[2*e - f*x] + 54*a^3*b*
f*x*Cos[2*e - f*x] + 126*a^2*b^2*f*x*Cos[2*e - f*x] + 114*a*b^3*f*x*Cos[2*e - f*x] + 36*b^4*f*x*Cos[2*e - f*x]
+ 6*a^4*f*x*Cos[2*e + f*x] + 54*a^3*b*f*x*Cos[2*e + f*x] + 126*a^2*b^2*f*x*Cos[2*e + f*x] + 114*a*b^3*f*x*Cos
[2*e + f*x] + 36*b^4*f*x*Cos[2*e + f*x] - 6*a^4*f*x*Cos[4*e + f*x] - 54*a^3*b*f*x*Cos[4*e + f*x] - 126*a^2*b^2
*f*x*Cos[4*e + f*x] - 114*a*b^3*f*x*Cos[4*e + f*x] - 36*b^4*f*x*Cos[4*e + f*x] - 3*a^4*f*x*Cos[2*e + 3*f*x] +
3*a^3*b*f*x*Cos[2*e + 3*f*x] + 27*a^2*b^2*f*x*Cos[2*e + 3*f*x] + 33*a*b^3*f*x*Cos[2*e + 3*f*x] + 12*b^4*f*x*Co
s[2*e + 3*f*x] + 3*a^4*f*x*Cos[4*e + 3*f*x] - 3*a^3*b*f*x*Cos[4*e + 3*f*x] - 27*a^2*b^2*f*x*Cos[4*e + 3*f*x] -
33*a*b^3*f*x*Cos[4*e + 3*f*x] - 12*b^4*f*x*Cos[4*e + 3*f*x] - 3*a^4*f*x*Cos[6*e + 3*f*x] + 3*a^3*b*f*x*Cos[6*
e + 3*f*x] + 27*a^2*b^2*f*x*Cos[6*e + 3*f*x] + 33*a*b^3*f*x*Cos[6*e + 3*f*x] + 12*b^4*f*x*Cos[6*e + 3*f*x] - 3
*a^4*f*x*Cos[2*e + 5*f*x] - 9*a^3*b*f*x*Cos[2*e + 5*f*x] - 9*a^2*b^2*f*x*Cos[2*e + 5*f*x] - 3*a*b^3*f*x*Cos[2*
e + 5*f*x] + 3*a^4*f*x*Cos[4*e + 5*f*x] + 9*a^3*b*f*x*Cos[4*e + 5*f*x] + 9*a^2*b^2*f*x*Cos[4*e + 5*f*x] + 3*a*
b^3*f*x*Cos[4*e + 5*f*x] - 3*a^4*f*x*Cos[6*e + 5*f*x] - 9*a^3*b*f*x*Cos[6*e + 5*f*x] - 9*a^2*b^2*f*x*Cos[6*e +
5*f*x] - 3*a*b^3*f*x*Cos[6*e + 5*f*x] + 3*a^4*f*x*Cos[8*e + 5*f*x] + 9*a^3*b*f*x*Cos[8*e + 5*f*x] + 9*a^2*b^2
*f*x*Cos[8*e + 5*f*x] + 3*a*b^3*f*x*Cos[8*e + 5*f*x] - 12*a^4*Sin[f*x] - 60*a^3*b*Sin[f*x] - 96*a^2*b^2*Sin[f*
x] + 18*b^4*Sin[f*x] + 4*a^4*Sin[3*f*x] + 36*a^3*b*Sin[3*f*x] + 80*a^2*b^2*Sin[3*f*x] - 6*a*b^3*Sin[3*f*x] + 6
*b^4*Sin[3*f*x] + 4*a^4*Sin[2*e - f*x] + 76*a^3*b*Sin[2*e - f*x] + 144*a^2*b^2*Sin[2*e - f*x] + 18*b^4*Sin[2*e
- f*x] - 4*a^4*Sin[2*e + f*x] - 76*a^3*b*Sin[2*e + f*x] - 144*a^2*b^2*Sin[2*e + f*x] + 6*a*b^3*Sin[2*e + f*x]
+ 18*b^4*Sin[2*e + f*x] - 12*a^4*Sin[4*e + f*x] - 60*a^3*b*Sin[4*e + f*x] - 96*a^2*b^2*Sin[4*e + f*x] - 6*a*b
^3*Sin[4*e + f*x] - 18*b^4*Sin[4*e + f*x] - 12*a^4*Sin[2*e + 3*f*x] - 24*a^3*b*Sin[2*e + 3*f*x] + 6*a*b^3*Sin[
2*e + 3*f*x] - 6*b^4*Sin[2*e + 3*f*x] + 4*a^4*Sin[4*e + 3*f*x] + 36*a^3*b*Sin[4*e + 3*f*x] + 80*a^2*b^2*Sin[4*
e + 3*f*x] - 3*a*b^3*Sin[4*e + 3*f*x] - 6*b^4*Sin[4*e + 3*f*x] - 12*a^4*Sin[6*e + 3*f*x] - 24*a^3*b*Sin[6*e +
3*f*x] + 3*a*b^3*Sin[6*e + 3*f*x] + 6*b^4*Sin[6*e + 3*f*x] + 8*a^4*Sin[2*e + 5*f*x] + 20*a^3*b*Sin[2*e + 5*f*x
] + 3*a*b^3*Sin[2*e + 5*f*x] - 3*a*b^3*Sin[4*e + 5*f*x] + 8*a^4*Sin[6*e + 5*f*x] + 20*a^3*b*Sin[6*e + 5*f*x]))
/(384*a^2*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^2)

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Maple [A]  time = 0.118, size = 186, normalized size = 1.2 \begin{align*}{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) }{f{a}^{2}}}-{\frac{1}{3\,f \left ( a+b \right ) ^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}}+{\frac{a}{f \left ( a+b \right ) ^{3}\tan \left ( fx+e \right ) }}+3\,{\frac{b}{f \left ( a+b \right ) ^{3}\tan \left ( fx+e \right ) }}-{\frac{{b}^{3}\tan \left ( fx+e \right ) }{2\,fa \left ( a+b \right ) ^{3} \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }}-{\frac{7\,{b}^{3}}{2\,fa \left ( a+b \right ) ^{3}}\arctan \left ({\tan \left ( fx+e \right ) b{\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}}-{\frac{{b}^{4}}{f{a}^{2} \left ( a+b \right ) ^{3}}\arctan \left ({\tan \left ( fx+e \right ) b{\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^4/(a+b*sec(f*x+e)^2)^2,x)

[Out]

1/f/a^2*arctan(tan(f*x+e))-1/3/f/(a+b)^2/tan(f*x+e)^3+1/f/(a+b)^3/tan(f*x+e)*a+3/f/(a+b)^3/tan(f*x+e)*b-1/2/f*
b^3/a/(a+b)^3*tan(f*x+e)/(a+b+b*tan(f*x+e)^2)-7/2/f*b^3/a/(a+b)^3/((a+b)*b)^(1/2)*arctan(tan(f*x+e)*b/((a+b)*b
)^(1/2))-1/f*b^4/a^2/(a+b)^3/((a+b)*b)^(1/2)*arctan(tan(f*x+e)*b/((a+b)*b)^(1/2))

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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)^4/(a+b*sec(f*x+e)^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.760028, size = 2182, normalized size = 13.64 \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)^4/(a+b*sec(f*x+e)^2)^2,x, algorithm="fricas")

[Out]

[1/24*(4*(8*a^4 + 20*a^3*b + 3*a*b^3)*cos(f*x + e)^5 - 8*(3*a^4 + 5*a^3*b - 10*a^2*b^2 + 3*a*b^3)*cos(f*x + e)
^3 + 3*((7*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^4 - 7*a*b^3 - 2*b^4 - (7*a^2*b^2 - 5*a*b^3 - 2*b^4)*cos(f*x + e)^2)
*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a
*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^
4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 12*(2*a^3*b + 6*a^2*b^2 - a*b^3)*cos(f*x + e) + 24*((a^4 + 3*a
^3*b + 3*a^2*b^2 + a*b^3)*f*x*cos(f*x + e)^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*f*x*cos(f*x + e)^2 - (a^3*b + 3
*a^2*b^2 + 3*a*b^3 + b^4)*f*x)*sin(f*x + e))/(((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 - (a^6 +
2*a^5*b - 2*a^3*b^3 - a^2*b^4)*f*cos(f*x + e)^2 - (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f)*sin(f*x + e)),
1/12*(2*(8*a^4 + 20*a^3*b + 3*a*b^3)*cos(f*x + e)^5 - 4*(3*a^4 + 5*a^3*b - 10*a^2*b^2 + 3*a*b^3)*cos(f*x + e)
^3 + 3*((7*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^4 - 7*a*b^3 - 2*b^4 - (7*a^2*b^2 - 5*a*b^3 - 2*b^4)*cos(f*x + e)^2)
*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(
f*x + e) - 6*(2*a^3*b + 6*a^2*b^2 - a*b^3)*cos(f*x + e) + 12*((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*x*cos(f*x
+ e)^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*f*x*cos(f*x + e)^2 - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f*x)*sin(f*x
+ e))/(((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 - (a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*f*cos(
f*x + e)^2 - (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f)*sin(f*x + e))]

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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)**4/(a+b*sec(f*x+e)**2)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.3606, size = 297, normalized size = 1.86 \begin{align*} -\frac{\frac{3 \, b^{3} \tan \left (f x + e\right )}{{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )}{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}} + \frac{3 \,{\left (7 \, a b^{3} + 2 \, b^{4}\right )}{\left (\pi \left \lfloor \frac{f x + e}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (f x + e\right )}{\sqrt{a b + b^{2}}}\right )\right )}}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} \sqrt{a b + b^{2}}} - \frac{6 \,{\left (f x + e\right )}}{a^{2}} - \frac{2 \,{\left (3 \, a \tan \left (f x + e\right )^{2} + 9 \, b \tan \left (f x + e\right )^{2} - a - b\right )}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{3}}}{6 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4/(a+b*sec(f*x+e)^2)^2,x, algorithm="giac")

[Out]

-1/6*(3*b^3*tan(f*x + e)/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*(b*tan(f*x + e)^2 + a + b)) + 3*(7*a*b^3 + 2*b^4
)*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b + b^2)))/((a^5 + 3*a^4*b + 3*a^3*b^2 +
a^2*b^3)*sqrt(a*b + b^2)) - 6*(f*x + e)/a^2 - 2*(3*a*tan(f*x + e)^2 + 9*b*tan(f*x + e)^2 - a - b)/((a^3 + 3*a
^2*b + 3*a*b^2 + b^3)*tan(f*x + e)^3))/f