3.357 \(\int \frac{\cot ^4(e+f x)}{(a+b \tan ^2(e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 f (a-b)^2}+\frac{(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^4 f (a-b)^2}-\frac{b (3 a-2 b) \cot ^3(e+f x)}{a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f (a-b)^{5/2}}-\frac{b \cot ^3(e+f x)}{3 a f (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}} \]

[Out]

ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f) - (b*Cot[e + f*x]^3)/(3*a*(a -
b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((3*a - 2*b)*b*Cot[e + f*x]^3)/(a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^
2]) + ((a - 2*b)*(3*a^2 + 8*a*b - 8*b^2)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^4*(a - b)^2*f) - ((a^2
- 12*a*b + 8*b^2)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^3*(a - b)^2*f)

________________________________________________________________________________________

Rubi [A]  time = 0.377712, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3670, 472, 579, 583, 12, 377, 203} \[ -\frac{\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 f (a-b)^2}+\frac{(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^4 f (a-b)^2}-\frac{b (3 a-2 b) \cot ^3(e+f x)}{a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f (a-b)^{5/2}}-\frac{b \cot ^3(e+f x)}{3 a f (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2),x]

[Out]

ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f) - (b*Cot[e + f*x]^3)/(3*a*(a -
b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((3*a - 2*b)*b*Cot[e + f*x]^3)/(a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^
2]) + ((a - 2*b)*(3*a^2 + 8*a*b - 8*b^2)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^4*(a - b)^2*f) - ((a^2
- 12*a*b + 8*b^2)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^3*(a - b)^2*f)

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

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])

Rubi steps

\begin{align*} \int \frac{\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{3 (a-2 b)-6 b x^2}{x^4 \left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 a (a-b) f}\\ &=-\frac{b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac{(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt{a+b \tan ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{3 \left (a^2-12 a b+8 b^2\right )-12 (3 a-2 b) b x^2}{x^4 \left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{3 a^2 (a-b)^2 f}\\ &=-\frac{b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac{(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 (a-b)^2 f}-\frac{\operatorname{Subst}\left (\int \frac{3 (a-2 b) \left (3 a^2+8 a b-8 b^2\right )+6 b \left (a^2-12 a b+8 b^2\right ) x^2}{x^2 \left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{9 a^3 (a-b)^2 f}\\ &=-\frac{b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac{(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt{a+b \tan ^2(e+f x)}}+\frac{(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^4 (a-b)^2 f}-\frac{\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 (a-b)^2 f}+\frac{\operatorname{Subst}\left (\int \frac{9 a^4}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{9 a^4 (a-b)^2 f}\\ &=-\frac{b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac{(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt{a+b \tan ^2(e+f x)}}+\frac{(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^4 (a-b)^2 f}-\frac{\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 (a-b)^2 f}+\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{(a-b)^2 f}\\ &=-\frac{b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac{(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt{a+b \tan ^2(e+f x)}}+\frac{(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^4 (a-b)^2 f}-\frac{\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 (a-b)^2 f}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-(-a+b) x^2} \, dx,x,\frac{\tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{(a-b)^2 f}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{(a-b)^{5/2} f}-\frac{b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac{(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt{a+b \tan ^2(e+f x)}}+\frac{(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^4 (a-b)^2 f}-\frac{\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 (a-b)^2 f}\\ \end{align*}

Mathematica [C]  time = 16.506, size = 871, normalized size = 3.5 \[ \frac{-\frac{b \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right ),1\right ) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac{4 b \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right ),1\right ) \sin ^4(e+f x)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}-\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \Pi \left (-\frac{b}{a-b};\left .\sin ^{-1}\left (\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right )\right |1\right ) \sin ^4(e+f x)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}\right )}{\sqrt{a+b+(a-b) \cos (2 (e+f x))}}}{(a-b)^2 f}+\frac{\sqrt{\frac{\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\frac{2 \sin (2 (e+f x)) b^4}{3 a^3 (a-b)^2 (\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x)))^2}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a^3}+\frac{4 (a \cos (e+f x)+2 b \cos (e+f x)) \csc (e+f x)}{3 a^4}-\frac{4 \left (3 a b^3 \sin (2 (e+f x))-2 b^4 \sin (2 (e+f x))\right )}{3 a^4 (a-b)^2 (\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x)))}\right )}{f} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2),x]

[Out]

(-((b*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*
(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(
e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*
x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x)]))) - (4*b*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e
+ f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x
]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((
a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x
)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*
Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(
b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2
*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a + b + (a - b)*Cos[2*(e +
f*x)]])/((a - b)^2*f) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((4*(a
*Cos[e + f*x] + 2*b*Cos[e + f*x])*Csc[e + f*x])/(3*a^4) - (Cot[e + f*x]*Csc[e + f*x]^2)/(3*a^3) + (2*b^4*Sin[2
*(e + f*x)])/(3*a^3*(a - b)^2*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])^2) - (4*(3*a*b^3*Sin[2*(e + f*
x)] - 2*b^4*Sin[2*(e + f*x)]))/(3*a^4*(a - b)^2*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)]))))/f

________________________________________________________________________________________

Maple [F]  time = 0.23, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cot \left ( fx+e \right ) \right ) ^{4} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{-{\frac{5}{2}}}}\, dx \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)^(5/2),x)

[Out]

int(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(5/2),x)

________________________________________________________________________________________

Maxima [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)^(5/2),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [A]  time = 3.21615, size = 1937, normalized size = 7.78 \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)^(5/2),x, algorithm="fricas")

[Out]

[-1/12*(3*(a^4*b^2*tan(f*x + e)^7 + 2*a^5*b*tan(f*x + e)^5 + a^6*tan(f*x + e)^3)*sqrt(-a + b)*log(-((a^2 - 8*a
*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 - 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x
+ e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) - 4*((3*a^4*b^2 - a^3*
b^3 - 26*a^2*b^4 + 40*a*b^5 - 16*b^6)*tan(f*x + e)^6 - a^6 + 3*a^5*b - 3*a^4*b^2 + a^3*b^3 + 3*(2*a^5*b - a^4*
b^2 - 13*a^3*b^3 + 20*a^2*b^4 - 8*a*b^5)*tan(f*x + e)^4 + 3*(a^6 - a^5*b - 3*a^4*b^2 + 5*a^3*b^3 - 2*a^2*b^4)*
tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((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), 1/6*(3*(a^4*b^2*tan(f*x + e)^7 + 2*a^5*b*tan(f*x + e)^5 + a^6*tan(f*x + e)^3)*sqrt(a - b)*arctan(-2*sq
rt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a)) + 2*((3*a^4*b^2 - a^3*b^3 -
26*a^2*b^4 + 40*a*b^5 - 16*b^6)*tan(f*x + e)^6 - a^6 + 3*a^5*b - 3*a^4*b^2 + a^3*b^3 + 3*(2*a^5*b - a^4*b^2 -
13*a^3*b^3 + 20*a^2*b^4 - 8*a*b^5)*tan(f*x + e)^4 + 3*(a^6 - a^5*b - 3*a^4*b^2 + 5*a^3*b^3 - 2*a^2*b^4)*tan(f*
x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{4}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac{5}{2}}}\, dx \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)**(5/2),x)

[Out]

Integral(cot(e + f*x)**4/(a + b*tan(e + f*x)**2)**(5/2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (f x + e\right )^{4}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \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)^(5/2),x, algorithm="giac")

[Out]

integrate(cot(f*x + e)^4/(b*tan(f*x + e)^2 + a)^(5/2), x)