\(\int \tan ^4(e+f x) (a+b \tan ^2(e+f x))^{3/2} \, dx\) [313]

Optimal result

Integrand size = 25, antiderivative size = 224 \[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}-\frac {\left (a^3+6 a^2 b-24 a b^2+16 b^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{16 b^{3/2} f}+\frac {\left (a^2-10 a b+8 b^2\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{16 b f}+\frac {(7 a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{24 f}+\frac {b \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{6 f} \] Output:

(a-b)^(3/2)*arctan((a-b)^(1/2)*tan(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2))/f-1/16 
*(a^3+6*a^2*b-24*a*b^2+16*b^3)*arctanh(b^(1/2)*tan(f*x+e)/(a+b*tan(f*x+e)^ 
2)^(1/2))/b^(3/2)/f+1/16*(a^2-10*a*b+8*b^2)*tan(f*x+e)*(a+b*tan(f*x+e)^2)^ 
(1/2)/b/f+1/24*(7*a-6*b)*tan(f*x+e)^3*(a+b*tan(f*x+e)^2)^(1/2)/f+1/6*b*tan 
(f*x+e)^5*(a+b*tan(f*x+e)^2)^(1/2)/f
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 4.85 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.97 \[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {\left (\left (30 a^3-266 a^2 b+200 a b^2+104 b^3+\left (45 a^3-433 a^2 b+296 a b^2-84 b^3\right ) \cos (2 (e+f x))+2 \left (9 a^3-107 a^2 b+92 a b^2+12 b^3\right ) \cos (4 (e+f x))+3 a^3 \cos (6 (e+f x))-47 a^2 b \cos (6 (e+f x))+88 a b^2 \cos (6 (e+f x))-44 b^3 \cos (6 (e+f x))\right ) \csc ^4(2 (e+f x))-3 \sqrt {2} a \left (a^2-10 a b+8 b^2\right ) \cot ^2(e+f x) \csc ^2(e+f x) \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right )+48 \sqrt {2} a b (-a+b) \cot ^2(e+f x) \csc ^2(e+f x) \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \operatorname {EllipticPi}\left (-\frac {b}{a-b},\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right )\right ) \sin ^2(e+f x) \tan ^3(e+f x)}{48 \sqrt {2} b f \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}} \] Input:

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

Output:

(((30*a^3 - 266*a^2*b + 200*a*b^2 + 104*b^3 + (45*a^3 - 433*a^2*b + 296*a* 
b^2 - 84*b^3)*Cos[2*(e + f*x)] + 2*(9*a^3 - 107*a^2*b + 92*a*b^2 + 12*b^3) 
*Cos[4*(e + f*x)] + 3*a^3*Cos[6*(e + f*x)] - 47*a^2*b*Cos[6*(e + f*x)] + 8 
8*a*b^2*Cos[6*(e + f*x)] - 44*b^3*Cos[6*(e + f*x)])*Csc[2*(e + f*x)]^4 - 3 
*Sqrt[2]*a*(a^2 - 10*a*b + 8*b^2)*Cot[e + f*x]^2*Csc[e + f*x]^2*Sqrt[((a + 
 b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[(( 
a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1] + 48*Sqr 
t[2]*a*b*(-a + b)*Cot[e + f*x]^2*Csc[e + f*x]^2*Sqrt[((a + b + (a - b)*Cos 
[2*(e + f*x)])*Csc[e + f*x]^2)/b]*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]^2*Tan[e + f*x]^3)/(48*Sqrt[2]*b*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f* 
x)])*Sec[e + f*x]^2])
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 4153, 379, 444, 27, 444, 25, 398, 224, 219, 291, 216}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 379

\(\displaystyle \frac {\frac {1}{6} \int \frac {\tan ^4(e+f x) \left ((7 a-6 b) b \tan ^2(e+f x)+a (6 a-5 b)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)+\frac {1}{6} b \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} (7 a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}-\frac {\int \frac {3 b \tan ^2(e+f x) \left (a (7 a-6 b)-\left (a^2-10 b a+8 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{4 b}\right )+\frac {1}{6} b \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} (7 a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}-\frac {3}{4} \int \frac {\tan ^2(e+f x) \left (a (7 a-6 b)-\left (a^2-10 b a+8 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)\right )+\frac {1}{6} b \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} (7 a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}-\frac {3}{4} \left (-\frac {\int -\frac {(a-2 b) \left (a^2+8 b a-8 b^2\right ) \tan ^2(e+f x)+a \left (a^2-10 b a+8 b^2\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{2 b}-\frac {\left (a^2-10 a b+8 b^2\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}\right )\right )+\frac {1}{6} b \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} (7 a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}-\frac {3}{4} \left (\frac {\int \frac {(a-2 b) \left (a^2+8 b a-8 b^2\right ) \tan ^2(e+f x)+a \left (a^2-10 b a+8 b^2\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{2 b}-\frac {\left (a^2-10 a b+8 b^2\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}\right )\right )+\frac {1}{6} b \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} (7 a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}-\frac {3}{4} \left (\frac {(a-2 b) \left (a^2+8 a b-8 b^2\right ) \int \frac {1}{\sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)-16 b (a-b)^2 \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{2 b}-\frac {\left (a^2-10 a b+8 b^2\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}\right )\right )+\frac {1}{6} b \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} (7 a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}-\frac {3}{4} \left (\frac {(a-2 b) \left (a^2+8 a b-8 b^2\right ) \int \frac {1}{1-\frac {b \tan ^2(e+f x)}{b \tan ^2(e+f x)+a}}d\frac {\tan (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}-16 b (a-b)^2 \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{2 b}-\frac {\left (a^2-10 a b+8 b^2\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}\right )\right )+\frac {1}{6} b \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} (7 a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}-\frac {3}{4} \left (\frac {\frac {(a-2 b) \left (a^2+8 a b-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{\sqrt {b}}-16 b (a-b)^2 \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{2 b}-\frac {\left (a^2-10 a b+8 b^2\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}\right )\right )+\frac {1}{6} b \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} (7 a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}-\frac {3}{4} \left (\frac {\frac {(a-2 b) \left (a^2+8 a b-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{\sqrt {b}}-16 b (a-b)^2 \int \frac {1}{1-\frac {(b-a) \tan ^2(e+f x)}{b \tan ^2(e+f x)+a}}d\frac {\tan (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}}{2 b}-\frac {\left (a^2-10 a b+8 b^2\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}\right )\right )+\frac {1}{6} b \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} (7 a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}-\frac {3}{4} \left (\frac {\frac {(a-2 b) \left (a^2+8 a b-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{\sqrt {b}}-16 b (a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{2 b}-\frac {\left (a^2-10 a b+8 b^2\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}\right )\right )+\frac {1}{6} b \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

Input:

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

Output:

((b*Tan[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/6 + (((7*a - 6*b)*Tan[e + f 
*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/4 - (3*((-16*(a - b)^(3/2)*b*ArcTan[(Sqr 
t[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]] + ((a - 2*b)*(a^2 + 8*a 
*b - 8*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/Sq 
rt[b])/(2*b) - ((a^2 - 10*a*b + 8*b^2)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x 
]^2])/(2*b)))/4)/6)/f
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 219

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

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 379

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

rule 398

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

rule 444

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

Leaf count of result is larger than twice the leaf count of optimal. \(509\) vs. \(2(198)=396\).

Time = 0.69 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.28

method	result	size
derivativedivides	\(\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{6 f b}-\frac {a \tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{24 f b}-\frac {a^{2} \tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{16 f b}-\frac {a^{3} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{16 f \,b^{\frac {3}{2}}}+\frac {b \tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{2 f}+\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 f}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f \left (a -b \right )}-\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{f}-\frac {2 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f b \left (a -b \right )}+\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f \,b^{2} \left (a -b \right )}-\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 f}-\frac {3 a \tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{8 f}-\frac {3 a^{2} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{8 f \sqrt {b}}\)	\(510\)
default	\(\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{6 f b}-\frac {a \tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{24 f b}-\frac {a^{2} \tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{16 f b}-\frac {a^{3} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{16 f \,b^{\frac {3}{2}}}+\frac {b \tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{2 f}+\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 f}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f \left (a -b \right )}-\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{f}-\frac {2 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f b \left (a -b \right )}+\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f \,b^{2} \left (a -b \right )}-\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 f}-\frac {3 a \tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{8 f}-\frac {3 a^{2} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{8 f \sqrt {b}}\)	\(510\)

Input:

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

Output:

1/6/f*tan(f*x+e)*(a+b*tan(f*x+e)^2)^(5/2)/b-1/24/f*a/b*tan(f*x+e)*(a+b*tan 
(f*x+e)^2)^(3/2)-1/16/f*a^2/b*tan(f*x+e)*(a+b*tan(f*x+e)^2)^(1/2)-1/16/f*a 
^3/b^(3/2)*ln(b^(1/2)*tan(f*x+e)+(a+b*tan(f*x+e)^2)^(1/2))+1/2*b*tan(f*x+e 
)*(a+b*tan(f*x+e)^2)^(1/2)/f+3/2/f*b^(1/2)*a*ln(b^(1/2)*tan(f*x+e)+(a+b*ta 
n(f*x+e)^2)^(1/2))+1/f*(b^4*(a-b))^(1/2)/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b) 
)^(1/2)/(a+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e))-1/f*b^(3/2)*ln(b^(1/2)*tan(f* 
x+e)+(a+b*tan(f*x+e)^2)^(1/2))-2/f*a/b*(b^4*(a-b))^(1/2)/(a-b)*arctan(b^2* 
(a-b)/(b^4*(a-b))^(1/2)/(a+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e))+1/f*a^2*(b^4* 
(a-b))^(1/2)/b^2/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*tan(f*x+e)^ 
2)^(1/2)*tan(f*x+e))-1/4/f*tan(f*x+e)*(a+b*tan(f*x+e)^2)^(3/2)-3/8/f*a*tan 
(f*x+e)*(a+b*tan(f*x+e)^2)^(1/2)-3/8/f*a^2/b^(1/2)*ln(b^(1/2)*tan(f*x+e)+( 
a+b*tan(f*x+e)^2)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 1.50 (sec) , antiderivative size = 845, normalized size of antiderivative = 3.77 \[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/96*(3*(a^3 + 6*a^2*b - 24*a*b^2 + 16*b^3)*sqrt(b)*log(2*b*tan(f*x + e)^ 
2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 48*(a*b^2 - b 
^3)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 
+ a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + 2*(8*b^3*tan(f 
*x + e)^5 + 2*(7*a*b^2 - 6*b^3)*tan(f*x + e)^3 + 3*(a^2*b - 10*a*b^2 + 8*b 
^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^2*f), 1/48*(3*(a^3 + 6*a^ 
2*b - 24*a*b^2 + 16*b^3)*sqrt(-b)*arctan(sqrt(-b)*tan(f*x + e)/sqrt(b*tan( 
f*x + e)^2 + a)) - 24*(a*b^2 - b^3)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + 
 e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f 
*x + e)^2 + 1)) + (8*b^3*tan(f*x + e)^5 + 2*(7*a*b^2 - 6*b^3)*tan(f*x + e) 
^3 + 3*(a^2*b - 10*a*b^2 + 8*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a) 
)/(b^2*f), 1/96*(96*(a*b^2 - b^3)*sqrt(a - b)*arctan(sqrt(a - b)*tan(f*x + 
 e)/sqrt(b*tan(f*x + e)^2 + a)) + 3*(a^3 + 6*a^2*b - 24*a*b^2 + 16*b^3)*sq 
rt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f* 
x + e) + a) + 2*(8*b^3*tan(f*x + e)^5 + 2*(7*a*b^2 - 6*b^3)*tan(f*x + e)^3 
 + 3*(a^2*b - 10*a*b^2 + 8*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/ 
(b^2*f), 1/48*(48*(a*b^2 - b^3)*sqrt(a - b)*arctan(sqrt(a - b)*tan(f*x + e 
)/sqrt(b*tan(f*x + e)^2 + a)) + 3*(a^3 + 6*a^2*b - 24*a*b^2 + 16*b^3)*sqrt 
(-b)*arctan(sqrt(-b)*tan(f*x + e)/sqrt(b*tan(f*x + e)^2 + a)) + (8*b^3*tan 
(f*x + e)^5 + 2*(7*a*b^2 - 6*b^3)*tan(f*x + e)^3 + 3*(a^2*b - 10*a*b^2 ...
 

Sympy [F]

\[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{4}{\left (e + f x \right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{4} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\left (\int \sqrt {\tan \left (f x +e \right )^{2} b +a}\, \tan \left (f x +e \right )^{6}d x \right ) b +\left (\int \sqrt {\tan \left (f x +e \right )^{2} b +a}\, \tan \left (f x +e \right )^{4}d x \right ) a \] Input:

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

Output:

int(sqrt(tan(e + f*x)**2*b + a)*tan(e + f*x)**6,x)*b + int(sqrt(tan(e + f* 
x)**2*b + a)*tan(e + f*x)**4,x)*a