3.374 \(\int \frac{\cot ^4(e+f x)}{(a+b \sec ^2(e+f x))^3} \, dx\)

Optimal. Leaf size=230 \[ -\frac{b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{8 a^3 f (a+b)^{9/2}}-\frac{\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{24 a^2 f (a+b)^3}+\frac{\left (32 a^2 b+8 a^3-15 a b^2-4 b^3\right ) \cot (e+f x)}{8 a^2 f (a+b)^4}-\frac{b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 f (a+b)^2 \left (a+b \tan ^2(e+f x)+b\right )}+\frac{x}{a^3}-\frac{b \cot ^3(e+f x)}{4 a f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2} \]

[Out]

x/a^3 - (b^(5/2)*(63*a^2 + 36*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(9/2)*f)
+ ((8*a^3 + 32*a^2*b - 15*a*b^2 - 4*b^3)*Cot[e + f*x])/(8*a^2*(a + b)^4*f) - ((8*a^2 - 39*a*b - 12*b^2)*Cot[e
+ f*x]^3)/(24*a^2*(a + b)^3*f) - (b*Cot[e + f*x]^3)/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(11*a +
4*b)*Cot[e + f*x]^3)/(8*a^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))

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Rubi [A]  time = 0.46112, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {4141, 1975, 472, 579, 583, 522, 203, 205} \[ -\frac{b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{8 a^3 f (a+b)^{9/2}}-\frac{\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{24 a^2 f (a+b)^3}+\frac{\left (32 a^2 b+8 a^3-15 a b^2-4 b^3\right ) \cot (e+f x)}{8 a^2 f (a+b)^4}-\frac{b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 f (a+b)^2 \left (a+b \tan ^2(e+f x)+b\right )}+\frac{x}{a^3}-\frac{b \cot ^3(e+f x)}{4 a f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]

[Out]

x/a^3 - (b^(5/2)*(63*a^2 + 36*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(9/2)*f)
+ ((8*a^3 + 32*a^2*b - 15*a*b^2 - 4*b^3)*Cot[e + f*x])/(8*a^2*(a + b)^4*f) - ((8*a^2 - 39*a*b - 12*b^2)*Cot[e
+ f*x]^3)/(24*a^2*(a + b)^3*f) - (b*Cot[e + f*x]^3)/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(11*a +
4*b)*Cot[e + f*x]^3)/(8*a^2*(a + b)^2*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 579

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

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 )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+x^2\right ) \left (a+b \left (1+x^2\right )\right )^3} \, 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 )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{4 a-3 b-7 b x^2}{x^4 \left (1+x^2\right ) \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{4 a (a+b) f}\\ &=-\frac{b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac{b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{8 a^2-39 a b-12 b^2-5 b (11 a+4 b) x^2}{x^4 \left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{8 a^2 (a+b)^2 f}\\ &=-\frac{\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{24 a^2 (a+b)^3 f}-\frac{b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac{b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{3 \left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right )+3 b \left (8 a^2-39 a b-12 b^2\right ) x^2}{x^2 \left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{24 a^2 (a+b)^3 f}\\ &=\frac{\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \cot (e+f x)}{8 a^2 (a+b)^4 f}-\frac{\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{24 a^2 (a+b)^3 f}-\frac{b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac{b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{3 \left (8 a^4+40 a^3 b+80 a^2 b^2+17 a b^3+4 b^4\right )+3 b \left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{24 a^2 (a+b)^4 f}\\ &=\frac{\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \cot (e+f x)}{8 a^2 (a+b)^4 f}-\frac{\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{24 a^2 (a+b)^3 f}-\frac{b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac{b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 (a+b)^2 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^3 f}-\frac{\left (b^3 \left (63 a^2+36 a b+8 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a^3 (a+b)^4 f}\\ &=\frac{x}{a^3}-\frac{b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{8 a^3 (a+b)^{9/2} f}+\frac{\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \cot (e+f x)}{8 a^2 (a+b)^4 f}-\frac{\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{24 a^2 (a+b)^3 f}-\frac{b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac{b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}\\ \end{align*}

Mathematica [C]  time = 7.61893, size = 3340, normalized size = 14.52 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]

[Out]

((63*a^2 + 36*a*b + 8*b^2)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((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])]*Cos[2*e])/(64*a^3*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[
4*e]]) - ((I/64)*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^3*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]])))/((a + b)^4*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*C
os[2*e + 2*f*x])*Csc[e]*Csc[e + f*x]^3*Sec[2*e]*Sec[e + f*x]^6*(-36*a^6*f*x*Cos[f*x] - 336*a^5*b*f*x*Cos[f*x]
- 1560*a^4*b^2*f*x*Cos[f*x] - 3600*a^3*b^3*f*x*Cos[f*x] - 4260*a^2*b^4*f*x*Cos[f*x] - 2496*a*b^5*f*x*Cos[f*x]
- 576*b^6*f*x*Cos[f*x] + 36*a^6*f*x*Cos[3*f*x] + 240*a^5*b*f*x*Cos[3*f*x] + 408*a^4*b^2*f*x*Cos[3*f*x] - 48*a^
3*b^3*f*x*Cos[3*f*x] - 732*a^2*b^4*f*x*Cos[3*f*x] - 672*a*b^5*f*x*Cos[3*f*x] - 192*b^6*f*x*Cos[3*f*x] + 36*a^6
*f*x*Cos[2*e - f*x] + 336*a^5*b*f*x*Cos[2*e - f*x] + 1560*a^4*b^2*f*x*Cos[2*e - f*x] + 3600*a^3*b^3*f*x*Cos[2*
e - f*x] + 4260*a^2*b^4*f*x*Cos[2*e - f*x] + 2496*a*b^5*f*x*Cos[2*e - f*x] + 576*b^6*f*x*Cos[2*e - f*x] + 36*a
^6*f*x*Cos[2*e + f*x] + 336*a^5*b*f*x*Cos[2*e + f*x] + 1560*a^4*b^2*f*x*Cos[2*e + f*x] + 3600*a^3*b^3*f*x*Cos[
2*e + f*x] + 4260*a^2*b^4*f*x*Cos[2*e + f*x] + 2496*a*b^5*f*x*Cos[2*e + f*x] + 576*b^6*f*x*Cos[2*e + f*x] - 36
*a^6*f*x*Cos[4*e + f*x] - 336*a^5*b*f*x*Cos[4*e + f*x] - 1560*a^4*b^2*f*x*Cos[4*e + f*x] - 3600*a^3*b^3*f*x*Co
s[4*e + f*x] - 4260*a^2*b^4*f*x*Cos[4*e + f*x] - 2496*a*b^5*f*x*Cos[4*e + f*x] - 576*b^6*f*x*Cos[4*e + f*x] -
36*a^6*f*x*Cos[2*e + 3*f*x] - 240*a^5*b*f*x*Cos[2*e + 3*f*x] - 408*a^4*b^2*f*x*Cos[2*e + 3*f*x] + 48*a^3*b^3*f
*x*Cos[2*e + 3*f*x] + 732*a^2*b^4*f*x*Cos[2*e + 3*f*x] + 672*a*b^5*f*x*Cos[2*e + 3*f*x] + 192*b^6*f*x*Cos[2*e
+ 3*f*x] + 36*a^6*f*x*Cos[4*e + 3*f*x] + 240*a^5*b*f*x*Cos[4*e + 3*f*x] + 408*a^4*b^2*f*x*Cos[4*e + 3*f*x] - 4
8*a^3*b^3*f*x*Cos[4*e + 3*f*x] - 732*a^2*b^4*f*x*Cos[4*e + 3*f*x] - 672*a*b^5*f*x*Cos[4*e + 3*f*x] - 192*b^6*f
*x*Cos[4*e + 3*f*x] - 36*a^6*f*x*Cos[6*e + 3*f*x] - 240*a^5*b*f*x*Cos[6*e + 3*f*x] - 408*a^4*b^2*f*x*Cos[6*e +
3*f*x] + 48*a^3*b^3*f*x*Cos[6*e + 3*f*x] + 732*a^2*b^4*f*x*Cos[6*e + 3*f*x] + 672*a*b^5*f*x*Cos[6*e + 3*f*x]
+ 192*b^6*f*x*Cos[6*e + 3*f*x] - 12*a^6*f*x*Cos[2*e + 5*f*x] - 144*a^5*b*f*x*Cos[2*e + 5*f*x] - 456*a^4*b^2*f*
x*Cos[2*e + 5*f*x] - 624*a^3*b^3*f*x*Cos[2*e + 5*f*x] - 396*a^2*b^4*f*x*Cos[2*e + 5*f*x] - 96*a*b^5*f*x*Cos[2*
e + 5*f*x] + 12*a^6*f*x*Cos[4*e + 5*f*x] + 144*a^5*b*f*x*Cos[4*e + 5*f*x] + 456*a^4*b^2*f*x*Cos[4*e + 5*f*x] +
624*a^3*b^3*f*x*Cos[4*e + 5*f*x] + 396*a^2*b^4*f*x*Cos[4*e + 5*f*x] + 96*a*b^5*f*x*Cos[4*e + 5*f*x] - 12*a^6*
f*x*Cos[6*e + 5*f*x] - 144*a^5*b*f*x*Cos[6*e + 5*f*x] - 456*a^4*b^2*f*x*Cos[6*e + 5*f*x] - 624*a^3*b^3*f*x*Cos
[6*e + 5*f*x] - 396*a^2*b^4*f*x*Cos[6*e + 5*f*x] - 96*a*b^5*f*x*Cos[6*e + 5*f*x] + 12*a^6*f*x*Cos[8*e + 5*f*x]
+ 144*a^5*b*f*x*Cos[8*e + 5*f*x] + 456*a^4*b^2*f*x*Cos[8*e + 5*f*x] + 624*a^3*b^3*f*x*Cos[8*e + 5*f*x] + 396*
a^2*b^4*f*x*Cos[8*e + 5*f*x] + 96*a*b^5*f*x*Cos[8*e + 5*f*x] - 12*a^6*f*x*Cos[4*e + 7*f*x] - 48*a^5*b*f*x*Cos[
4*e + 7*f*x] - 72*a^4*b^2*f*x*Cos[4*e + 7*f*x] - 48*a^3*b^3*f*x*Cos[4*e + 7*f*x] - 12*a^2*b^4*f*x*Cos[4*e + 7*
f*x] + 12*a^6*f*x*Cos[6*e + 7*f*x] + 48*a^5*b*f*x*Cos[6*e + 7*f*x] + 72*a^4*b^2*f*x*Cos[6*e + 7*f*x] + 48*a^3*
b^3*f*x*Cos[6*e + 7*f*x] + 12*a^2*b^4*f*x*Cos[6*e + 7*f*x] - 12*a^6*f*x*Cos[8*e + 7*f*x] - 48*a^5*b*f*x*Cos[8*
e + 7*f*x] - 72*a^4*b^2*f*x*Cos[8*e + 7*f*x] - 48*a^3*b^3*f*x*Cos[8*e + 7*f*x] - 12*a^2*b^4*f*x*Cos[8*e + 7*f*
x] + 12*a^6*f*x*Cos[10*e + 7*f*x] + 48*a^5*b*f*x*Cos[10*e + 7*f*x] + 72*a^4*b^2*f*x*Cos[10*e + 7*f*x] + 48*a^3
*b^3*f*x*Cos[10*e + 7*f*x] + 12*a^2*b^4*f*x*Cos[10*e + 7*f*x] - 128*a^6*Sin[f*x] - 440*a^5*b*Sin[f*x] - 1152*a
^4*b^2*Sin[f*x] - 1920*a^3*b^3*Sin[f*x] + 228*a^2*b^4*Sin[f*x] + 1320*a*b^5*Sin[f*x] + 432*b^6*Sin[f*x] + 48*a
^6*Sin[3*f*x] + 104*a^5*b*Sin[3*f*x] + 640*a^4*b^2*Sin[3*f*x] + 1511*a^3*b^3*Sin[3*f*x] - 528*a^2*b^4*Sin[3*f*
x] + 264*a*b^5*Sin[3*f*x] + 144*b^6*Sin[3*f*x] - 32*a^6*Sin[2*e - f*x] + 384*a^5*b*Sin[2*e - f*x] + 2048*a^4*b
^2*Sin[2*e - f*x] + 3072*a^3*b^3*Sin[2*e - f*x] + 228*a^2*b^4*Sin[2*e - f*x] + 1320*a*b^5*Sin[2*e - f*x] + 432
*b^6*Sin[2*e - f*x] + 32*a^6*Sin[2*e + f*x] - 384*a^5*b*Sin[2*e + f*x] - 2048*a^4*b^2*Sin[2*e + f*x] - 2919*a^
3*b^3*Sin[2*e + f*x] + 642*a^2*b^4*Sin[2*e + f*x] + 1416*a*b^5*Sin[2*e + f*x] + 432*b^6*Sin[2*e + f*x] - 128*a
^6*Sin[4*e + f*x] - 440*a^5*b*Sin[4*e + f*x] - 1152*a^4*b^2*Sin[4*e + f*x] - 2073*a^3*b^3*Sin[4*e + f*x] - 642
*a^2*b^4*Sin[4*e + f*x] - 1416*a*b^5*Sin[4*e + f*x] - 432*b^6*Sin[4*e + f*x] - 144*a^6*Sin[2*e + 3*f*x] - 672*
a^5*b*Sin[2*e + 3*f*x] - 960*a^4*b^2*Sin[2*e + 3*f*x] + 153*a^3*b^3*Sin[2*e + 3*f*x] + 528*a^2*b^4*Sin[2*e + 3
*f*x] - 264*a*b^5*Sin[2*e + 3*f*x] - 144*b^6*Sin[2*e + 3*f*x] + 48*a^6*Sin[4*e + 3*f*x] + 104*a^5*b*Sin[4*e +
3*f*x] + 640*a^4*b^2*Sin[4*e + 3*f*x] + 1664*a^3*b^3*Sin[4*e + 3*f*x] - 66*a^2*b^4*Sin[4*e + 3*f*x] - 408*a*b^
5*Sin[4*e + 3*f*x] - 144*b^6*Sin[4*e + 3*f*x] - 144*a^6*Sin[6*e + 3*f*x] - 672*a^5*b*Sin[6*e + 3*f*x] - 960*a^
4*b^2*Sin[6*e + 3*f*x] + 66*a^2*b^4*Sin[6*e + 3*f*x] + 408*a*b^5*Sin[6*e + 3*f*x] + 144*b^6*Sin[6*e + 3*f*x] +
80*a^6*Sin[2*e + 5*f*x] + 480*a^5*b*Sin[2*e + 5*f*x] + 832*a^4*b^2*Sin[2*e + 5*f*x] + 294*a^2*b^4*Sin[2*e + 5
*f*x] + 96*a*b^5*Sin[2*e + 5*f*x] - 48*a^6*Sin[4*e + 5*f*x] - 120*a^5*b*Sin[4*e + 5*f*x] - 294*a^2*b^4*Sin[4*e
+ 5*f*x] - 96*a*b^5*Sin[4*e + 5*f*x] + 80*a^6*Sin[6*e + 5*f*x] + 480*a^5*b*Sin[6*e + 5*f*x] + 832*a^4*b^2*Sin
[6*e + 5*f*x] - 51*a^3*b^3*Sin[6*e + 5*f*x] - 132*a^2*b^4*Sin[6*e + 5*f*x] - 48*a*b^5*Sin[6*e + 5*f*x] - 48*a^
6*Sin[8*e + 5*f*x] - 120*a^5*b*Sin[8*e + 5*f*x] + 51*a^3*b^3*Sin[8*e + 5*f*x] + 132*a^2*b^4*Sin[8*e + 5*f*x] +
48*a*b^5*Sin[8*e + 5*f*x] + 32*a^6*Sin[4*e + 7*f*x] + 104*a^5*b*Sin[4*e + 7*f*x] + 51*a^3*b^3*Sin[4*e + 7*f*x
] + 18*a^2*b^4*Sin[4*e + 7*f*x] - 51*a^3*b^3*Sin[6*e + 7*f*x] - 18*a^2*b^4*Sin[6*e + 7*f*x] + 32*a^6*Sin[8*e +
7*f*x] + 104*a^5*b*Sin[8*e + 7*f*x]))/(6144*a^3*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^3)

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Maple [A]  time = 0.122, size = 374, normalized size = 1.6 \begin{align*}{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) }{f{a}^{3}}}-{\frac{1}{3\,f \left ( a+b \right ) ^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}}+{\frac{a}{f \left ( a+b \right ) ^{4}\tan \left ( fx+e \right ) }}+4\,{\frac{b}{f \left ( a+b \right ) ^{4}\tan \left ( fx+e \right ) }}-{\frac{15\,{b}^{4} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{8\,fa \left ( a+b \right ) ^{4} \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{{b}^{5} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{2\,f{a}^{2} \left ( a+b \right ) ^{4} \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{17\,{b}^{3}\tan \left ( fx+e \right ) }{8\,f \left ( a+b \right ) ^{4} \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{21\,{b}^{4}\tan \left ( fx+e \right ) }{8\,fa \left ( a+b \right ) ^{4} \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{{b}^{5}\tan \left ( fx+e \right ) }{2\,f{a}^{2} \left ( a+b \right ) ^{4} \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{63\,{b}^{3}}{8\,fa \left ( a+b \right ) ^{4}}\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{9\,{b}^{4}}{2\,f{a}^{2} \left ( a+b \right ) ^{4}}\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}^{5}}{f{a}^{3} \left ( a+b \right ) ^{4}}\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)^3,x)

[Out]

1/f/a^3*arctan(tan(f*x+e))-1/3/f/(a+b)^3/tan(f*x+e)^3+1/f/(a+b)^4/tan(f*x+e)*a+4/f/(a+b)^4/tan(f*x+e)*b-15/8/f
*b^4/a/(a+b)^4/(a+b+b*tan(f*x+e)^2)^2*tan(f*x+e)^3-1/2/f*b^5/a^2/(a+b)^4/(a+b+b*tan(f*x+e)^2)^2*tan(f*x+e)^3-1
7/8/f*b^3/(a+b)^4/(a+b+b*tan(f*x+e)^2)^2*tan(f*x+e)-21/8/f*b^4/a/(a+b)^4/(a+b+b*tan(f*x+e)^2)^2*tan(f*x+e)-1/2
/f*b^5/a^2/(a+b)^4/(a+b+b*tan(f*x+e)^2)^2*tan(f*x+e)-63/8/f*b^3/a/(a+b)^4/((a+b)*b)^(1/2)*arctan(tan(f*x+e)*b/
((a+b)*b)^(1/2))-9/2/f*b^4/a^2/(a+b)^4/((a+b)*b)^(1/2)*arctan(tan(f*x+e)*b/((a+b)*b)^(1/2))-1/f*b^5/a^3/(a+b)^
4/((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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.949497, size = 3640, normalized size = 15.83 \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)^3,x, algorithm="fricas")

[Out]

[1/96*(4*(32*a^6 + 104*a^5*b + 51*a^3*b^3 + 18*a^2*b^4)*cos(f*x + e)^7 - 4*(24*a^6 + 32*a^5*b - 208*a^4*b^2 +
102*a^3*b^3 - 9*a^2*b^4 - 12*a*b^5)*cos(f*x + e)^5 - 4*(48*a^5*b + 160*a^4*b^2 - 155*a^3*b^3 + 72*a^2*b^4 + 24
*a*b^5)*cos(f*x + e)^3 + 3*((63*a^4*b^2 + 36*a^3*b^3 + 8*a^2*b^4)*cos(f*x + e)^6 - 63*a^2*b^4 - 36*a*b^5 - 8*b
^6 - (63*a^4*b^2 - 90*a^3*b^3 - 64*a^2*b^4 - 16*a*b^5)*cos(f*x + e)^4 - (126*a^3*b^3 + 9*a^2*b^4 - 20*a*b^5 -
8*b^6)*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*(8*a^4*b^2 + 32*a^3*b^3 - 15*a^2*b^4
- 4*a*b^5)*cos(f*x + e) + 96*((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*f*x*cos(f*x + e)^6 - (a^6 + 2*
a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*f*x*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2
*a^2*b^4 - 2*a*b^5 - b^6)*f*x*cos(f*x + e)^2 - (a^4*b^2 + 4*a^3*b^3 + 6*a^2*b^4 + 4*a*b^5 + b^6)*f*x)*sin(f*x
+ e))/(((a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*f*cos(f*x + e)^6 - (a^9 + 2*a^8*b - 2*a^7*b^2 - 8*a^
6*b^3 - 7*a^5*b^4 - 2*a^4*b^5)*f*cos(f*x + e)^4 - (2*a^8*b + 7*a^7*b^2 + 8*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - a
^3*b^6)*f*cos(f*x + e)^2 - (a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 + 4*a^4*b^5 + a^3*b^6)*f)*sin(f*x + e)), 1/48*(2*(
32*a^6 + 104*a^5*b + 51*a^3*b^3 + 18*a^2*b^4)*cos(f*x + e)^7 - 2*(24*a^6 + 32*a^5*b - 208*a^4*b^2 + 102*a^3*b^
3 - 9*a^2*b^4 - 12*a*b^5)*cos(f*x + e)^5 - 2*(48*a^5*b + 160*a^4*b^2 - 155*a^3*b^3 + 72*a^2*b^4 + 24*a*b^5)*co
s(f*x + e)^3 + 3*((63*a^4*b^2 + 36*a^3*b^3 + 8*a^2*b^4)*cos(f*x + e)^6 - 63*a^2*b^4 - 36*a*b^5 - 8*b^6 - (63*a
^4*b^2 - 90*a^3*b^3 - 64*a^2*b^4 - 16*a*b^5)*cos(f*x + e)^4 - (126*a^3*b^3 + 9*a^2*b^4 - 20*a*b^5 - 8*b^6)*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*(8*a^4*b^2 + 32*a^3*b^3 - 15*a^2*b^4 - 4*a*b^5)*cos(f*x + e) + 48*((a^6 + 4*a^5*b + 6
*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*f*x*cos(f*x + e)^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*
b^5)*f*x*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*f*x*cos(f*x + e)^2 - (
a^4*b^2 + 4*a^3*b^3 + 6*a^2*b^4 + 4*a*b^5 + b^6)*f*x)*sin(f*x + e))/(((a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 +
a^5*b^4)*f*cos(f*x + e)^6 - (a^9 + 2*a^8*b - 2*a^7*b^2 - 8*a^6*b^3 - 7*a^5*b^4 - 2*a^4*b^5)*f*cos(f*x + e)^4
- (2*a^8*b + 7*a^7*b^2 + 8*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - a^3*b^6)*f*cos(f*x + e)^2 - (a^7*b^2 + 4*a^6*b^3
+ 6*a^5*b^4 + 4*a^4*b^5 + a^3*b^6)*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)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.4433, size = 424, normalized size = 1.84 \begin{align*} -\frac{\frac{3 \,{\left (63 \, a^{2} b^{3} + 36 \, a b^{4} + 8 \, b^{5}\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^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \sqrt{a b + b^{2}}} + \frac{3 \,{\left (15 \, a b^{4} \tan \left (f x + e\right )^{3} + 4 \, b^{5} \tan \left (f x + e\right )^{3} + 17 \, a^{2} b^{3} \tan \left (f x + e\right ) + 21 \, a b^{4} \tan \left (f x + e\right ) + 4 \, b^{5} \tan \left (f x + e\right )\right )}}{{\left (a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right )}{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{2}} - \frac{24 \,{\left (f x + e\right )}}{a^{3}} - \frac{8 \,{\left (3 \, a \tan \left (f x + e\right )^{2} + 12 \, b \tan \left (f x + e\right )^{2} - a - b\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \tan \left (f x + e\right )^{3}}}{24 \, 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)^3,x, algorithm="giac")

[Out]

-1/24*(3*(63*a^2*b^3 + 36*a*b^4 + 8*b^5)*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b
+ b^2)))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*sqrt(a*b + b^2)) + 3*(15*a*b^4*tan(f*x + e)^3 + 4
*b^5*tan(f*x + e)^3 + 17*a^2*b^3*tan(f*x + e) + 21*a*b^4*tan(f*x + e) + 4*b^5*tan(f*x + e))/((a^6 + 4*a^5*b +
6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*(b*tan(f*x + e)^2 + a + b)^2) - 24*(f*x + e)/a^3 - 8*(3*a*tan(f*x + e)^2 + 12
*b*tan(f*x + e)^2 - a - b)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*tan(f*x + e)^3))/f