\(\int \frac {\tan ^4(e+f x)}{(a+b \tan ^2(e+f x))^3} \, dx\) [244]

Optimal result

Integrand size = 23, antiderivative size = 145 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\frac {x}{(a-b)^3}+\frac {\left (a^2-6 a b-3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 \sqrt {a} (a-b)^3 b^{3/2} f}-\frac {a \tan (e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {(a-5 b) \tan (e+f x)}{8 (a-b)^2 b f \left (a+b \tan ^2(e+f x)\right )} \] Output:

x/(a-b)^3+1/8*(a^2-6*a*b-3*b^2)*arctan(b^(1/2)*tan(f*x+e)/a^(1/2))/a^(1/2) 
/(a-b)^3/b^(3/2)/f-1/4*a*tan(f*x+e)/(a-b)/b/f/(a+b*tan(f*x+e)^2)^2+1/8*(a- 
5*b)*tan(f*x+e)/(a-b)^2/b/f/(a+b*tan(f*x+e)^2)
 

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.94 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\frac {8 (e+f x)+\frac {\left (a^2-6 a b-3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {(a-b) \left (a^2+2 a b+5 b^2+\left (a^2+4 a b-5 b^2\right ) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{b (a+b+(a-b) \cos (2 (e+f x)))^2}}{8 (a-b)^3 f} \] Input:

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

Output:

(8*(e + f*x) + ((a^2 - 6*a*b - 3*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a 
]])/(Sqrt[a]*b^(3/2)) - ((a - b)*(a^2 + 2*a*b + 5*b^2 + (a^2 + 4*a*b - 5*b 
^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(b*(a + b + (a - b)*Cos[2*(e + f*x 
)])^2))/(8*(a - b)^3*f)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 4153, 372, 402, 27, 397, 216, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\tan (e+f x)^4}{\left (a+b \tan (e+f x)^2\right )^3}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle \frac {\int \frac {\tan ^4(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^3}d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 372

\(\displaystyle \frac {\frac {\int \frac {(a-4 b) \tan ^2(e+f x)+a}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^2}d\tan (e+f x)}{4 b (a-b)}-\frac {a \tan (e+f x)}{4 b (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {\frac {\int \frac {a \left ((a-5 b) \tan ^2(e+f x)+a+3 b\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{2 a (a-b)}+\frac {(a-5 b) \tan (e+f x)}{2 (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 b (a-b)}-\frac {a \tan (e+f x)}{4 b (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {(a-5 b) \tan ^2(e+f x)+a+3 b}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{2 (a-b)}+\frac {(a-5 b) \tan (e+f x)}{2 (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 b (a-b)}-\frac {a \tan (e+f x)}{4 b (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {\frac {\frac {\left (a^2-6 a b-3 b^2\right ) \int \frac {1}{b \tan ^2(e+f x)+a}d\tan (e+f x)}{a-b}+\frac {8 b \int \frac {1}{\tan ^2(e+f x)+1}d\tan (e+f x)}{a-b}}{2 (a-b)}+\frac {(a-5 b) \tan (e+f x)}{2 (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 b (a-b)}-\frac {a \tan (e+f x)}{4 b (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {\frac {\frac {\left (a^2-6 a b-3 b^2\right ) \int \frac {1}{b \tan ^2(e+f x)+a}d\tan (e+f x)}{a-b}+\frac {8 b \arctan (\tan (e+f x))}{a-b}}{2 (a-b)}+\frac {(a-5 b) \tan (e+f x)}{2 (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 b (a-b)}-\frac {a \tan (e+f x)}{4 b (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\frac {\left (a^2-6 a b-3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} (a-b)}+\frac {8 b \arctan (\tan (e+f x))}{a-b}}{2 (a-b)}+\frac {(a-5 b) \tan (e+f x)}{2 (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 b (a-b)}-\frac {a \tan (e+f x)}{4 b (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\)

Input:

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

Output:

(-1/4*(a*Tan[e + f*x])/((a - b)*b*(a + b*Tan[e + f*x]^2)^2) + (((8*b*ArcTa 
n[Tan[e + f*x]])/(a - b) + ((a^2 - 6*a*b - 3*b^2)*ArcTan[(Sqrt[b]*Tan[e + 
f*x])/Sqrt[a]])/(Sqrt[a]*(a - b)*Sqrt[b]))/(2*(a - b)) + ((a - 5*b)*Tan[e 
+ f*x])/(2*(a - b)*(a + b*Tan[e + f*x]^2)))/(4*(a - b)*b))/f
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

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

rule 218

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

rule 372

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 
)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 
))   Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + 
 (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, 
e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a 
, b, c, d, e, m, 2, p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 402

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4153

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]}, Simp[c*(ff/f)   Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f 
f^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] && Ratio 
nalQ[n]))
 

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\frac {\arctan \left (\tan \left (f x +e \right )\right )}{\left (a -b \right )^{3}}+\frac {\frac {\left (\frac {1}{8} a^{2}-\frac {3}{4} a b +\frac {5}{8} b^{2}\right ) \tan \left (f x +e \right )^{3}-\frac {a \left (a^{2}+2 a b -3 b^{2}\right ) \tan \left (f x +e \right )}{8 b}}{\left (a +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {\left (a^{2}-6 a b -3 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{8 b \sqrt {a b}}}{\left (a -b \right )^{3}}}{f}\)	\(132\)
default	\(\frac {\frac {\arctan \left (\tan \left (f x +e \right )\right )}{\left (a -b \right )^{3}}+\frac {\frac {\left (\frac {1}{8} a^{2}-\frac {3}{4} a b +\frac {5}{8} b^{2}\right ) \tan \left (f x +e \right )^{3}-\frac {a \left (a^{2}+2 a b -3 b^{2}\right ) \tan \left (f x +e \right )}{8 b}}{\left (a +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {\left (a^{2}-6 a b -3 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{8 b \sqrt {a b}}}{\left (a -b \right )^{3}}}{f}\)	\(132\)
risch	\(\frac {x}{a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}}-\frac {i \left (a^{3} {\mathrm e}^{6 i \left (f x +e \right )}+9 a^{2} b \,{\mathrm e}^{6 i \left (f x +e \right )}-5 a \,b^{2} {\mathrm e}^{6 i \left (f x +e \right )}-5 b^{3} {\mathrm e}^{6 i \left (f x +e \right )}+3 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+17 a^{2} b \,{\mathrm e}^{4 i \left (f x +e \right )}+13 a \,b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+15 b^{3} {\mathrm e}^{4 i \left (f x +e \right )}+3 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}+11 a^{2} b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \,b^{2} {\mathrm e}^{2 i \left (f x +e \right )}-15 b^{3} {\mathrm e}^{2 i \left (f x +e \right )}+a^{3}+3 a^{2} b -9 a \,b^{2}+5 b^{3}\right )}{4 \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}-b \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a -b \right )^{2} \left (a^{2}-2 a b +b^{2}\right ) \left (a -b \right ) b f}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i a b +\sqrt {-a b}\, a +\sqrt {-a b}\, b}{\left (a -b \right ) \sqrt {-a b}}\right ) a^{2}}{16 \sqrt {-a b}\, \left (a -b \right )^{3} f b}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i a b +\sqrt {-a b}\, a +\sqrt {-a b}\, b}{\left (a -b \right ) \sqrt {-a b}}\right ) a}{8 \sqrt {-a b}\, \left (a -b \right )^{3} f}+\frac {3 b \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i a b +\sqrt {-a b}\, a +\sqrt {-a b}\, b}{\left (a -b \right ) \sqrt {-a b}}\right )}{16 \sqrt {-a b}\, \left (a -b \right )^{3} f}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i a b -\sqrt {-a b}\, a -\sqrt {-a b}\, b}{\left (a -b \right ) \sqrt {-a b}}\right ) a^{2}}{16 \sqrt {-a b}\, \left (a -b \right )^{3} f b}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i a b -\sqrt {-a b}\, a -\sqrt {-a b}\, b}{\left (a -b \right ) \sqrt {-a b}}\right ) a}{8 \sqrt {-a b}\, \left (a -b \right )^{3} f}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i a b -\sqrt {-a b}\, a -\sqrt {-a b}\, b}{\left (a -b \right ) \sqrt {-a b}}\right )}{16 \sqrt {-a b}\, \left (a -b \right )^{3} f}\)	\(718\)

Input:

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

Output:

1/f*(1/(a-b)^3*arctan(tan(f*x+e))+1/(a-b)^3*(((1/8*a^2-3/4*a*b+5/8*b^2)*ta 
n(f*x+e)^3-1/8*a*(a^2+2*a*b-3*b^2)/b*tan(f*x+e))/(a+b*tan(f*x+e)^2)^2+1/8* 
(a^2-6*a*b-3*b^2)/b/(a*b)^(1/2)*arctan(b*tan(f*x+e)/(a*b)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (131) = 262\).

Time = 0.12 (sec) , antiderivative size = 749, normalized size of antiderivative = 5.17 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^3,x, algorithm="fricas")
 

Output:

[1/32*(32*a*b^4*f*x*tan(f*x + e)^4 + 64*a^2*b^3*f*x*tan(f*x + e)^2 + 32*a^ 
3*b^2*f*x + 4*(a^3*b^2 - 6*a^2*b^3 + 5*a*b^4)*tan(f*x + e)^3 - ((a^2*b^2 - 
 6*a*b^3 - 3*b^4)*tan(f*x + e)^4 + a^4 - 6*a^3*b - 3*a^2*b^2 + 2*(a^3*b - 
6*a^2*b^2 - 3*a*b^3)*tan(f*x + e)^2)*sqrt(-a*b)*log((b^2*tan(f*x + e)^4 - 
6*a*b*tan(f*x + e)^2 + a^2 - 4*(b*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(-a 
*b))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) - 4*(a^4*b + 2*a^3 
*b^2 - 3*a^2*b^3)*tan(f*x + e))/((a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7) 
*f*tan(f*x + e)^4 + 2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*f*tan(f* 
x + e)^2 + (a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f), 1/16*(16*a*b^4* 
f*x*tan(f*x + e)^4 + 32*a^2*b^3*f*x*tan(f*x + e)^2 + 16*a^3*b^2*f*x + 2*(a 
^3*b^2 - 6*a^2*b^3 + 5*a*b^4)*tan(f*x + e)^3 + ((a^2*b^2 - 6*a*b^3 - 3*b^4 
)*tan(f*x + e)^4 + a^4 - 6*a^3*b - 3*a^2*b^2 + 2*(a^3*b - 6*a^2*b^2 - 3*a* 
b^3)*tan(f*x + e)^2)*sqrt(a*b)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(a*b) 
/(a*b*tan(f*x + e))) - 2*(a^4*b + 2*a^3*b^2 - 3*a^2*b^3)*tan(f*x + e))/((a 
^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*f*tan(f*x + e)^4 + 2*(a^5*b^3 - 3* 
a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*f*tan(f*x + e)^2 + (a^6*b^2 - 3*a^5*b^3 + 3 
*a^4*b^4 - a^3*b^5)*f)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8957 vs. \(2 (126) = 252\).

Time = 47.80 (sec) , antiderivative size = 8957, normalized size of antiderivative = 61.77 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(tan(f*x+e)**4/(a+b*tan(f*x+e)**2)**3,x)
 

Output:

Piecewise((zoo*x/tan(e)**2, Eq(a, 0) & Eq(b, 0) & Eq(f, 0)), ((x + tan(e + 
 f*x)**3/(3*f) - tan(e + f*x)/f)/a**3, Eq(b, 0)), ((-x - 1/(f*tan(e + f*x) 
))/b**3, Eq(a, 0)), (3*f*x*tan(e + f*x)**6/(48*b**3*f*tan(e + f*x)**6 + 14 
4*b**3*f*tan(e + f*x)**4 + 144*b**3*f*tan(e + f*x)**2 + 48*b**3*f) + 9*f*x 
*tan(e + f*x)**4/(48*b**3*f*tan(e + f*x)**6 + 144*b**3*f*tan(e + f*x)**4 + 
 144*b**3*f*tan(e + f*x)**2 + 48*b**3*f) + 9*f*x*tan(e + f*x)**2/(48*b**3* 
f*tan(e + f*x)**6 + 144*b**3*f*tan(e + f*x)**4 + 144*b**3*f*tan(e + f*x)** 
2 + 48*b**3*f) + 3*f*x/(48*b**3*f*tan(e + f*x)**6 + 144*b**3*f*tan(e + f*x 
)**4 + 144*b**3*f*tan(e + f*x)**2 + 48*b**3*f) + 3*tan(e + f*x)**5/(48*b** 
3*f*tan(e + f*x)**6 + 144*b**3*f*tan(e + f*x)**4 + 144*b**3*f*tan(e + f*x) 
**2 + 48*b**3*f) - 8*tan(e + f*x)**3/(48*b**3*f*tan(e + f*x)**6 + 144*b**3 
*f*tan(e + f*x)**4 + 144*b**3*f*tan(e + f*x)**2 + 48*b**3*f) - 3*tan(e + f 
*x)/(48*b**3*f*tan(e + f*x)**6 + 144*b**3*f*tan(e + f*x)**4 + 144*b**3*f*t 
an(e + f*x)**2 + 48*b**3*f), Eq(a, b)), (x*tan(e)**4/(a + b*tan(e)**2)**3, 
 Eq(f, 0)), (a**4*log(-sqrt(-a/b) + tan(e + f*x))/(16*a**5*b**2*f*sqrt(-a/ 
b) + 32*a**4*b**3*f*sqrt(-a/b)*tan(e + f*x)**2 - 48*a**4*b**3*f*sqrt(-a/b) 
 + 16*a**3*b**4*f*sqrt(-a/b)*tan(e + f*x)**4 - 96*a**3*b**4*f*sqrt(-a/b)*t 
an(e + f*x)**2 + 48*a**3*b**4*f*sqrt(-a/b) - 48*a**2*b**5*f*sqrt(-a/b)*tan 
(e + f*x)**4 + 96*a**2*b**5*f*sqrt(-a/b)*tan(e + f*x)**2 - 16*a**2*b**5*f* 
sqrt(-a/b) + 48*a*b**6*f*sqrt(-a/b)*tan(e + f*x)**4 - 32*a*b**6*f*sqrt(...
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.46 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\frac {\frac {{\left (a^{2} - 6 \, a b - 3 \, b^{2}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} \sqrt {a b}} + \frac {{\left (a b - 5 \, b^{2}\right )} \tan \left (f x + e\right )^{3} - {\left (a^{2} + 3 \, a b\right )} \tan \left (f x + e\right )}{a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )} \tan \left (f x + e\right )^{2}} + \frac {8 \, {\left (f x + e\right )}}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}}}{8 \, f} \] Input:

integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^3,x, algorithm="maxima")
 

Output:

1/8*((a^2 - 6*a*b - 3*b^2)*arctan(b*tan(f*x + e)/sqrt(a*b))/((a^3*b - 3*a^ 
2*b^2 + 3*a*b^3 - b^4)*sqrt(a*b)) + ((a*b - 5*b^2)*tan(f*x + e)^3 - (a^2 + 
 3*a*b)*tan(f*x + e))/(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^2*b^3 - 2*a*b^4 + 
b^5)*tan(f*x + e)^4 + 2*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*tan(f*x + e)^2) + 8* 
(f*x + e)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3))/f
 

Giac [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.26 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\frac {{\left (a^{2} - 6 \, a b - 3 \, b^{2}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{8 \, {\left (a^{3} b f - 3 \, a^{2} b^{2} f + 3 \, a b^{3} f - b^{4} f\right )} \sqrt {a b}} + \frac {f x + e}{a^{3} f - 3 \, a^{2} b f + 3 \, a b^{2} f - b^{3} f} + \frac {a b \tan \left (f x + e\right )^{3} - 5 \, b^{2} \tan \left (f x + e\right )^{3} - a^{2} \tan \left (f x + e\right ) - 3 \, a b \tan \left (f x + e\right )}{8 \, {\left (a^{2} b f - 2 \, a b^{2} f + b^{3} f\right )} {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{2}} \] Input:

integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^3,x, algorithm="giac")
 

Output:

1/8*(a^2 - 6*a*b - 3*b^2)*arctan(b*tan(f*x + e)/sqrt(a*b))/((a^3*b*f - 3*a 
^2*b^2*f + 3*a*b^3*f - b^4*f)*sqrt(a*b)) + (f*x + e)/(a^3*f - 3*a^2*b*f + 
3*a*b^2*f - b^3*f) + 1/8*(a*b*tan(f*x + e)^3 - 5*b^2*tan(f*x + e)^3 - a^2* 
tan(f*x + e) - 3*a*b*tan(f*x + e))/((a^2*b*f - 2*a*b^2*f + b^3*f)*(b*tan(f 
*x + e)^2 + a)^2)
 

Mupad [B] (verification not implemented)

Time = 11.53 (sec) , antiderivative size = 3667, normalized size of antiderivative = 25.29 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((tan(e + f*x)^3*(a - 5*b))/(8*(a^2 - 2*a*b + b^2)) - (a*tan(e + f*x)*(a + 
 3*b))/(8*(a^2*b - 2*a*b^2 + b^3)))/(f*(a^2 + b^2*tan(e + f*x)^4 + 2*a*b*t 
an(e + f*x)^2)) - (2*atan((((((544*a*b^8 - 96*b^9 - 1248*a^2*b^7 + 1440*a^ 
3*b^6 - 800*a^4*b^5 + 96*a^5*b^4 + 96*a^6*b^3 - 32*a^7*b^2)/(64*(a^6*b - 6 
*a*b^6 + b^7 + 15*a^2*b^5 - 20*a^3*b^4 + 15*a^4*b^3 - 6*a^5*b^2)) - (tan(e 
 + f*x)*(1280*a*b^9 - 256*b^10 - 2304*a^2*b^8 + 1280*a^3*b^7 + 1280*a^4*b^ 
6 - 2304*a^5*b^5 + 1280*a^6*b^4 - 256*a^7*b^3)*1i)/(32*(6*a*b^2 - 6*a^2*b 
+ 2*a^3 - 2*b^3)*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*1i)/(6* 
a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3) + (tan(e + f*x)*(36*a*b^3 - 12*a^3*b + a^ 
4 + 73*b^4 + 30*a^2*b^2))/(32*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b 
^2)))/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3) - ((((544*a*b^8 - 96*b^9 - 1248* 
a^2*b^7 + 1440*a^3*b^6 - 800*a^4*b^5 + 96*a^5*b^4 + 96*a^6*b^3 - 32*a^7*b^ 
2)/(64*(a^6*b - 6*a*b^6 + b^7 + 15*a^2*b^5 - 20*a^3*b^4 + 15*a^4*b^3 - 6*a 
^5*b^2)) + (tan(e + f*x)*(1280*a*b^9 - 256*b^10 - 2304*a^2*b^8 + 1280*a^3* 
b^7 + 1280*a^4*b^6 - 2304*a^5*b^5 + 1280*a^6*b^4 - 256*a^7*b^3)*1i)/(32*(6 
*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3)*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a 
^3*b^2)))*1i)/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3) - (tan(e + f*x)*(36*a*b^ 
3 - 12*a^3*b + a^4 + 73*b^4 + 30*a^2*b^2))/(32*(a^4*b - 4*a*b^4 + b^5 + 6* 
a^2*b^3 - 4*a^3*b^2)))/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3))/((((((544*a*b^ 
8 - 96*b^9 - 1248*a^2*b^7 + 1440*a^3*b^6 - 800*a^4*b^5 + 96*a^5*b^4 + 9...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 580, normalized size of antiderivative = 4.00 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) \tan \left (f x +e \right )^{4} a^{2} b^{2}-6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) \tan \left (f x +e \right )^{4} a \,b^{3}-3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) \tan \left (f x +e \right )^{4} b^{4}+2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) \tan \left (f x +e \right )^{2} a^{3} b -12 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) \tan \left (f x +e \right )^{2} a^{2} b^{2}-6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) \tan \left (f x +e \right )^{2} a \,b^{3}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) a^{4}-6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b -3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2}+8 \tan \left (f x +e \right )^{4} a \,b^{4} f x +\tan \left (f x +e \right )^{3} a^{3} b^{2}-6 \tan \left (f x +e \right )^{3} a^{2} b^{3}+5 \tan \left (f x +e \right )^{3} a \,b^{4}+16 \tan \left (f x +e \right )^{2} a^{2} b^{3} f x -\tan \left (f x +e \right ) a^{4} b -2 \tan \left (f x +e \right ) a^{3} b^{2}+3 \tan \left (f x +e \right ) a^{2} b^{3}+8 a^{3} b^{2} f x}{8 a \,b^{2} f \left (\tan \left (f x +e \right )^{4} a^{3} b^{2}-3 \tan \left (f x +e \right )^{4} a^{2} b^{3}+3 \tan \left (f x +e \right )^{4} a \,b^{4}-\tan \left (f x +e \right )^{4} b^{5}+2 \tan \left (f x +e \right )^{2} a^{4} b -6 \tan \left (f x +e \right )^{2} a^{3} b^{2}+6 \tan \left (f x +e \right )^{2} a^{2} b^{3}-2 \tan \left (f x +e \right )^{2} a \,b^{4}+a^{5}-3 a^{4} b +3 a^{3} b^{2}-a^{2} b^{3}\right )} \] Input:

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

Output:

(sqrt(b)*sqrt(a)*atan((tan(e + f*x)*b)/(sqrt(b)*sqrt(a)))*tan(e + f*x)**4* 
a**2*b**2 - 6*sqrt(b)*sqrt(a)*atan((tan(e + f*x)*b)/(sqrt(b)*sqrt(a)))*tan 
(e + f*x)**4*a*b**3 - 3*sqrt(b)*sqrt(a)*atan((tan(e + f*x)*b)/(sqrt(b)*sqr 
t(a)))*tan(e + f*x)**4*b**4 + 2*sqrt(b)*sqrt(a)*atan((tan(e + f*x)*b)/(sqr 
t(b)*sqrt(a)))*tan(e + f*x)**2*a**3*b - 12*sqrt(b)*sqrt(a)*atan((tan(e + f 
*x)*b)/(sqrt(b)*sqrt(a)))*tan(e + f*x)**2*a**2*b**2 - 6*sqrt(b)*sqrt(a)*at 
an((tan(e + f*x)*b)/(sqrt(b)*sqrt(a)))*tan(e + f*x)**2*a*b**3 + sqrt(b)*sq 
rt(a)*atan((tan(e + f*x)*b)/(sqrt(b)*sqrt(a)))*a**4 - 6*sqrt(b)*sqrt(a)*at 
an((tan(e + f*x)*b)/(sqrt(b)*sqrt(a)))*a**3*b - 3*sqrt(b)*sqrt(a)*atan((ta 
n(e + f*x)*b)/(sqrt(b)*sqrt(a)))*a**2*b**2 + 8*tan(e + f*x)**4*a*b**4*f*x 
+ tan(e + f*x)**3*a**3*b**2 - 6*tan(e + f*x)**3*a**2*b**3 + 5*tan(e + f*x) 
**3*a*b**4 + 16*tan(e + f*x)**2*a**2*b**3*f*x - tan(e + f*x)*a**4*b - 2*ta 
n(e + f*x)*a**3*b**2 + 3*tan(e + f*x)*a**2*b**3 + 8*a**3*b**2*f*x)/(8*a*b* 
*2*f*(tan(e + f*x)**4*a**3*b**2 - 3*tan(e + f*x)**4*a**2*b**3 + 3*tan(e + 
f*x)**4*a*b**4 - tan(e + f*x)**4*b**5 + 2*tan(e + f*x)**2*a**4*b - 6*tan(e 
 + f*x)**2*a**3*b**2 + 6*tan(e + f*x)**2*a**2*b**3 - 2*tan(e + f*x)**2*a*b 
**4 + a**5 - 3*a**4*b + 3*a**3*b**2 - a**2*b**3))