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

Optimal. Leaf size=240 \[ -\frac{b^{5/2} \left (63 a^2-90 a b+35 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{8 a^{9/2} f (a-b)^3}-\frac{\left (8 a^2-55 a b+35 b^2\right ) \cot ^3(e+f x)}{24 a^3 f (a-b)^2}+\frac{\left (8 a^2 b+8 a^3-55 a b^2+35 b^3\right ) \cot (e+f x)}{8 a^4 f (a-b)^2}-\frac{b (11 a-7 b) \cot ^3(e+f x)}{8 a^2 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}-\frac{b \cot ^3(e+f x)}{4 a f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}+\frac{x}{(a-b)^3} \]

[Out]

x/(a - b)^3 - (b^(5/2)*(63*a^2 - 90*a*b + 35*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(9/2)*(a - b)^3
*f) + ((8*a^3 + 8*a^2*b - 55*a*b^2 + 35*b^3)*Cot[e + f*x])/(8*a^4*(a - b)^2*f) - ((8*a^2 - 55*a*b + 35*b^2)*Co
t[e + f*x]^3)/(24*a^3*(a - b)^2*f) - (b*Cot[e + f*x]^3)/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((11*a - 7*
b)*b*Cot[e + f*x]^3)/(8*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))

________________________________________________________________________________________

Rubi [A]  time = 0.364925, antiderivative size = 240, 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 = {3670, 472, 579, 583, 522, 203, 205} \[ -\frac{b^{5/2} \left (63 a^2-90 a b+35 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{8 a^{9/2} f (a-b)^3}-\frac{\left (8 a^2-55 a b+35 b^2\right ) \cot ^3(e+f x)}{24 a^3 f (a-b)^2}+\frac{\left (8 a^2 b+8 a^3-55 a b^2+35 b^3\right ) \cot (e+f x)}{8 a^4 f (a-b)^2}-\frac{b (11 a-7 b) \cot ^3(e+f x)}{8 a^2 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}-\frac{b \cot ^3(e+f x)}{4 a f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}+\frac{x}{(a-b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(a - b)^3 - (b^(5/2)*(63*a^2 - 90*a*b + 35*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(9/2)*(a - b)^3
*f) + ((8*a^3 + 8*a^2*b - 55*a*b^2 + 35*b^3)*Cot[e + f*x])/(8*a^4*(a - b)^2*f) - ((8*a^2 - 55*a*b + 35*b^2)*Co
t[e + f*x]^3)/(24*a^3*(a - b)^2*f) - (b*Cot[e + f*x]^3)/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((11*a - 7*
b)*b*Cot[e + f*x]^3)/(8*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))

Rule 3670

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

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 \tan ^2(e+f x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+x^2\right ) \left (a+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 \tan ^2(e+f x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{4 a-7 b-7 b x^2}{x^4 \left (1+x^2\right ) \left (a+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 \tan ^2(e+f x)\right )^2}-\frac{(11 a-7 b) b \cot ^3(e+f x)}{8 a^2 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{8 a^2-55 a b+35 b^2-5 (11 a-7 b) b x^2}{x^4 \left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{8 a^2 (a-b)^2 f}\\ &=-\frac{\left (8 a^2-55 a b+35 b^2\right ) \cot ^3(e+f x)}{24 a^3 (a-b)^2 f}-\frac{b \cot ^3(e+f x)}{4 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac{(11 a-7 b) b \cot ^3(e+f x)}{8 a^2 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{3 \left (8 a^3+8 a^2 b-55 a b^2+35 b^3\right )+3 b \left (8 a^2-55 a b+35 b^2\right ) x^2}{x^2 \left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{24 a^3 (a-b)^2 f}\\ &=\frac{\left (8 a^3+8 a^2 b-55 a b^2+35 b^3\right ) \cot (e+f x)}{8 a^4 (a-b)^2 f}-\frac{\left (8 a^2-55 a b+35 b^2\right ) \cot ^3(e+f x)}{24 a^3 (a-b)^2 f}-\frac{b \cot ^3(e+f x)}{4 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac{(11 a-7 b) b \cot ^3(e+f x)}{8 a^2 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{3 \left (8 a^4+8 a^3 b+8 a^2 b^2-55 a b^3+35 b^4\right )+3 b \left (8 a^3+8 a^2 b-55 a b^2+35 b^3\right ) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{24 a^4 (a-b)^2 f}\\ &=\frac{\left (8 a^3+8 a^2 b-55 a b^2+35 b^3\right ) \cot (e+f x)}{8 a^4 (a-b)^2 f}-\frac{\left (8 a^2-55 a b+35 b^2\right ) \cot ^3(e+f x)}{24 a^3 (a-b)^2 f}-\frac{b \cot ^3(e+f x)}{4 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac{(11 a-7 b) b \cot ^3(e+f x)}{8 a^2 (a-b)^2 f \left (a+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-b)^3 f}-\frac{\left (b^3 \left (63 a^2-90 a b+35 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a^4 (a-b)^3 f}\\ &=\frac{x}{(a-b)^3}-\frac{b^{5/2} \left (63 a^2-90 a b+35 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{8 a^{9/2} (a-b)^3 f}+\frac{\left (8 a^3+8 a^2 b-55 a b^2+35 b^3\right ) \cot (e+f x)}{8 a^4 (a-b)^2 f}-\frac{\left (8 a^2-55 a b+35 b^2\right ) \cot ^3(e+f x)}{24 a^3 (a-b)^2 f}-\frac{b \cot ^3(e+f x)}{4 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac{(11 a-7 b) b \cot ^3(e+f x)}{8 a^2 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )}\\ \end{align*}

Mathematica [A]  time = 4.25921, size = 184, normalized size = 0.77 \[ \frac{-\frac{3 b^{5/2} \left (63 a^2-90 a b+35 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{a^{9/2} (a-b)^3}+\frac{3 \left (8 (e+f x)-\frac{b^3 (a-b) \sin (2 (e+f x)) \left (\left (17 a^2-28 a b+11 b^2\right ) \cos (2 (e+f x))+17 a^2+2 a b-11 b^2\right )}{a^4 ((a-b) \cos (2 (e+f x))+a+b)^2}\right )}{(a-b)^3}-\frac{8 \cot (e+f x) \left (a \csc ^2(e+f x)-4 a-9 b\right )}{a^4}}{24 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*b^(5/2)*(63*a^2 - 90*a*b + 35*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(9/2)*(a - b)^3) - (8*Cot[e
+ f*x]*(-4*a - 9*b + a*Csc[e + f*x]^2))/a^4 + (3*(8*(e + f*x) - ((a - b)*b^3*(17*a^2 + 2*a*b - 11*b^2 + (17*a
^2 - 28*a*b + 11*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(a^4*(a + b + (a - b)*Cos[2*(e + f*x)])^2)))/(a - b)
^3)/(24*f)

________________________________________________________________________________________

Maple [A]  time = 0.101, size = 413, normalized size = 1.7 \begin{align*} -{\frac{1}{3\,f{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}}+{\frac{1}{f{a}^{3}\tan \left ( fx+e \right ) }}+3\,{\frac{b}{f{a}^{4}\tan \left ( fx+e \right ) }}-{\frac{15\,{b}^{4} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{8\,f \left ( a-b \right ) ^{3} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}{a}^{2}}}+{\frac{13\,{b}^{5} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{4\,f{a}^{3} \left ( a-b \right ) ^{3} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{11\,{b}^{6} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{8\,f{a}^{4} \left ( a-b \right ) ^{3} \left ( a+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 ) ^{3} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}a}}+{\frac{15\,{b}^{4}\tan \left ( fx+e \right ) }{4\,f \left ( a-b \right ) ^{3} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}{a}^{2}}}-{\frac{13\,{b}^{5}\tan \left ( fx+e \right ) }{8\,f{a}^{3} \left ( a-b \right ) ^{3} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{63\,{b}^{3}}{8\,f{a}^{2} \left ( a-b \right ) ^{3}}\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{45\,{b}^{4}}{4\,f{a}^{3} \left ( a-b \right ) ^{3}}\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{35\,{b}^{5}}{8\,f{a}^{4} \left ( a-b \right ) ^{3}}\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) }{f \left ( a-b \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^3,x)

[Out]

-1/3/f/a^3/tan(f*x+e)^3+1/f/a^3/tan(f*x+e)+3/f/a^4/tan(f*x+e)*b-15/8/f*b^4/(a-b)^3/(a+b*tan(f*x+e)^2)^2/a^2*ta
n(f*x+e)^3+13/4/f*b^5/a^3/(a-b)^3/(a+b*tan(f*x+e)^2)^2*tan(f*x+e)^3-11/8/f*b^6/a^4/(a-b)^3/(a+b*tan(f*x+e)^2)^
2*tan(f*x+e)^3-17/8/f*b^3/(a-b)^3/(a+b*tan(f*x+e)^2)^2/a*tan(f*x+e)+15/4/f*b^4/a^2/(a-b)^3/(a+b*tan(f*x+e)^2)^
2*tan(f*x+e)-13/8/f*b^5/a^3/(a-b)^3/(a+b*tan(f*x+e)^2)^2*tan(f*x+e)-63/8/f*b^3/(a-b)^3/a^2/(a*b)^(1/2)*arctan(
b*tan(f*x+e)/(a*b)^(1/2))+45/4/f*b^4/a^3/(a-b)^3/(a*b)^(1/2)*arctan(b*tan(f*x+e)/(a*b)^(1/2))-35/8/f*b^5/a^4/(
a-b)^3/(a*b)^(1/2)*arctan(b*tan(f*x+e)/(a*b)^(1/2))+1/f/(a-b)^3*arctan(tan(f*x+e))

________________________________________________________________________________________

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*tan(f*x+e)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.12939, size = 2261, normalized size = 9.42 \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*tan(f*x+e)^2)^3,x, algorithm="fricas")

[Out]

[1/96*(96*a^4*b^2*f*x*tan(f*x + e)^7 + 192*a^5*b*f*x*tan(f*x + e)^5 + 96*a^6*f*x*tan(f*x + e)^3 + 12*(8*a^4*b^
2 - 63*a^2*b^4 + 90*a*b^5 - 35*b^6)*tan(f*x + e)^6 - 32*a^6 + 96*a^5*b - 96*a^4*b^2 + 32*a^3*b^3 + 4*(48*a^5*b
- 8*a^4*b^2 - 315*a^3*b^3 + 450*a^2*b^4 - 175*a*b^5)*tan(f*x + e)^4 + 32*(3*a^6 - 2*a^5*b - 12*a^4*b^2 + 18*a
^3*b^3 - 7*a^2*b^4)*tan(f*x + e)^2 - 3*((63*a^2*b^4 - 90*a*b^5 + 35*b^6)*tan(f*x + e)^7 + 2*(63*a^3*b^3 - 90*a
^2*b^4 + 35*a*b^5)*tan(f*x + e)^5 + (63*a^4*b^2 - 90*a^3*b^3 + 35*a^2*b^4)*tan(f*x + e)^3)*sqrt(-b/a)*log((b^2
*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 + 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(
f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*f*tan(f*x + e)^7 + 2*(
a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4)*f*tan(f*x + e)^5 + (a^9 - 3*a^8*b + 3*a^7*b^2 - a^6*b^3)*f*tan(f*x +
e)^3), 1/48*(48*a^4*b^2*f*x*tan(f*x + e)^7 + 96*a^5*b*f*x*tan(f*x + e)^5 + 48*a^6*f*x*tan(f*x + e)^3 + 6*(8*a^
4*b^2 - 63*a^2*b^4 + 90*a*b^5 - 35*b^6)*tan(f*x + e)^6 - 16*a^6 + 48*a^5*b - 48*a^4*b^2 + 16*a^3*b^3 + 2*(48*a
^5*b - 8*a^4*b^2 - 315*a^3*b^3 + 450*a^2*b^4 - 175*a*b^5)*tan(f*x + e)^4 + 16*(3*a^6 - 2*a^5*b - 12*a^4*b^2 +
18*a^3*b^3 - 7*a^2*b^4)*tan(f*x + e)^2 - 3*((63*a^2*b^4 - 90*a*b^5 + 35*b^6)*tan(f*x + e)^7 + 2*(63*a^3*b^3 -
90*a^2*b^4 + 35*a*b^5)*tan(f*x + e)^5 + (63*a^4*b^2 - 90*a^3*b^3 + 35*a^2*b^4)*tan(f*x + e)^3)*sqrt(b/a)*arcta
n(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*f*tan(f
*x + e)^7 + 2*(a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4)*f*tan(f*x + e)^5 + (a^9 - 3*a^8*b + 3*a^7*b^2 - a^6*b^
3)*f*tan(f*x + e)^3)]

________________________________________________________________________________________

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*tan(f*x+e)**2)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.63909, size = 352, normalized size = 1.47 \begin{align*} -\frac{\frac{3 \,{\left (63 \, a^{2} b^{3} - 90 \, a b^{4} + 35 \, 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}}\right )\right )}}{{\left (a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right )} \sqrt{a b}} - \frac{24 \,{\left (f x + e\right )}}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac{3 \,{\left (15 \, a b^{4} \tan \left (f x + e\right )^{3} - 11 \, b^{5} \tan \left (f x + e\right )^{3} + 17 \, a^{2} b^{3} \tan \left (f x + e\right ) - 13 \, a b^{4} \tan \left (f x + e\right )\right )}}{{\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )}{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{2}} - \frac{8 \,{\left (3 \, a \tan \left (f x + e\right )^{2} + 9 \, b \tan \left (f x + e\right )^{2} - a\right )}}{a^{4} \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*tan(f*x+e)^2)^3,x, algorithm="giac")

[Out]

-1/24*(3*(63*a^2*b^3 - 90*a*b^4 + 35*b^5)*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*
b)))/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*sqrt(a*b)) - 24*(f*x + e)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 3*(15*
a*b^4*tan(f*x + e)^3 - 11*b^5*tan(f*x + e)^3 + 17*a^2*b^3*tan(f*x + e) - 13*a*b^4*tan(f*x + e))/((a^6 - 2*a^5*
b + a^4*b^2)*(b*tan(f*x + e)^2 + a)^2) - 8*(3*a*tan(f*x + e)^2 + 9*b*tan(f*x + e)^2 - a)/(a^4*tan(f*x + e)^3))
/f