\(\int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx\) [299]

Optimal result

Integrand size = 25, antiderivative size = 222 \[ \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=-\frac {\sqrt {a-b} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}+\frac {\left (a^3+2 a^2 b+8 a b^2-16 b^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{16 b^{5/2} f}-\frac {(a-2 b) (a+4 b) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{16 b^2 f}+\frac {(a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{24 b f}+\frac {\tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{6 f} \] Output:

-(a-b)^(1/2)*arctan((a-b)^(1/2)*tan(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2))/f+1/1 
6*(a^3+2*a^2*b+8*a*b^2-16*b^3)*arctanh(b^(1/2)*tan(f*x+e)/(a+b*tan(f*x+e)^ 
2)^(1/2))/b^(5/2)/f-1/16*(a-2*b)*(a+4*b)*tan(f*x+e)*(a+b*tan(f*x+e)^2)^(1/ 
2)/b^2/f+1/24*(a-6*b)*tan(f*x+e)^3*(a+b*tan(f*x+e)^2)^(1/2)/b/f+1/6*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 = 5.79 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.89 \[ \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\frac {3 \sqrt {2} a \left (a^2+2 a b-8 b^2\right ) \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \tan (e+f x)+48 \sqrt {2} a b^2 \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \operatorname {EllipticPi}\left (-\frac {b}{a-b},\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \tan (e+f x)-\left (30 a^3+62 a^2 b-224 a b^2-104 b^3+\left (45 a^3+91 a^2 b-332 a b^2+84 b^3\right ) \cos (2 (e+f x))+2 \left (9 a^3+17 a^2 b-80 a b^2-12 b^3\right ) \cos (4 (e+f x))+3 a^3 \cos (6 (e+f x))+5 a^2 b \cos (6 (e+f x))-52 a b^2 \cos (6 (e+f x))+44 b^3 \cos (6 (e+f x))\right ) \csc ^4(2 (e+f x)) \sin ^2(e+f x) \tan ^3(e+f x)}{48 \sqrt {2} b^2 f \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}} \] Input:

Integrate[Tan[e + f*x]^6*Sqrt[a + b*Tan[e + f*x]^2],x]
 

Output:

(3*Sqrt[2]*a*(a^2 + 2*a*b - 8*b^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]*Tan[e + f*x] + 48*Sqrt[2]*a*b^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]/S 
qrt[2]], 1]*Tan[e + f*x] - (30*a^3 + 62*a^2*b - 224*a*b^2 - 104*b^3 + (45* 
a^3 + 91*a^2*b - 332*a*b^2 + 84*b^3)*Cos[2*(e + f*x)] + 2*(9*a^3 + 17*a^2* 
b - 80*a*b^2 - 12*b^3)*Cos[4*(e + f*x)] + 3*a^3*Cos[6*(e + f*x)] + 5*a^2*b 
*Cos[6*(e + f*x)] - 52*a*b^2*Cos[6*(e + f*x)] + 44*b^3*Cos[6*(e + f*x)])*C 
sc[2*(e + f*x)]^4*Sin[e + f*x]^2*Tan[e + f*x]^3)/(48*Sqrt[2]*b^2*f*Sqrt[(a 
 + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 4153, 380, 444, 27, 444, 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.

\(\displaystyle \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \tan (e+f x)^6 \sqrt {a+b \tan (e+f x)^2}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle \frac {\int \frac {\tan ^6(e+f x) \sqrt {b \tan ^2(e+f x)+a}}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 380

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

\(\Big \downarrow \) 444

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 444

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 216

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

Input:

Int[Tan[e + f*x]^6*Sqrt[a + b*Tan[e + f*x]^2],x]
 

Output:

((Tan[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/6 + (((a - 6*b)*Tan[e + f*x]^ 
3*Sqrt[a + b*Tan[e + f*x]^2])/(4*b) - (3*(-1/2*(-16*Sqrt[a - b]*b^2*ArcTan 
[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]] + ((a^3 + 2*a^2*b 
+ 8*a*b^2 - 16*b^3)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x] 
^2]])/Sqrt[b])/b + ((a - 2*b)*(a + 4*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f* 
x]^2])/(2*b)))/(4*b))/6)/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 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 380

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b* 
(m + 2*(p + q) + 1))), x] - Simp[e^2/(b*(m + 2*(p + q) + 1))   Int[(e*x)^(m 
 - 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[a*c*(m - 1) + (a*d*(m - 1) - 2 
*q*(b*c - a*d))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c 
- a*d, 0] && GtQ[q, 0] && GtQ[m, 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. \(428\) vs. \(2(196)=392\).

Time = 0.84 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.93

method	result	size
derivativedivides	\(\frac {\frac {\tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{2}+\frac {a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 \sqrt {b}}+\frac {\tan \left (f x +e \right )^{3} \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {\tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{2}+\frac {a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}-\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 b}+\frac {a \left (\frac {\tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{2}+\frac {a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 \sqrt {b}}\right )}{4 b}-b \left (\frac {\ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{\sqrt {b}}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{b^{2} \left (a -b \right )}\right )-\frac {a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{b^{2} \left (a -b \right )}}{f}\)	\(429\)
default	\(\frac {\frac {\tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{2}+\frac {a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 \sqrt {b}}+\frac {\tan \left (f x +e \right )^{3} \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {\tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{2}+\frac {a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}-\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 b}+\frac {a \left (\frac {\tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{2}+\frac {a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 \sqrt {b}}\right )}{4 b}-b \left (\frac {\ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{\sqrt {b}}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{b^{2} \left (a -b \right )}\right )-\frac {a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{b^{2} \left (a -b \right )}}{f}\)	\(429\)

Input:

int(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 1.00 (sec) , antiderivative size = 810, normalized size of antiderivative = 3.65 \[ \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx =\text {Too large to display} \] Input:

integrate(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm="fricas")
 

Output:

[1/96*(48*sqrt(-a + b)*b^3*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)) - 3*(a 
^3 + 2*a^2*b + 8*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) + 2*(8*b^3*tan(f*x + e)^5 + 
 2*(a*b^2 - 6*b^3)*tan(f*x + e)^3 - 3*(a^2*b + 2*a*b^2 - 8*b^3)*tan(f*x + 
e))*sqrt(b*tan(f*x + e)^2 + a))/(b^3*f), -1/96*(96*sqrt(a - b)*b^3*arctan( 
sqrt(a - b)*tan(f*x + e)/sqrt(b*tan(f*x + e)^2 + a)) + 3*(a^3 + 2*a^2*b + 
8*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) - 2*(8*b^3*tan(f*x + e)^5 + 2*(a*b^2 - 6*b 
^3)*tan(f*x + e)^3 - 3*(a^2*b + 2*a*b^2 - 8*b^3)*tan(f*x + e))*sqrt(b*tan( 
f*x + e)^2 + a))/(b^3*f), 1/48*(24*sqrt(-a + b)*b^3*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)/(ta 
n(f*x + e)^2 + 1)) - 3*(a^3 + 2*a^2*b + 8*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*(a*b^2 - 6*b^3)*tan(f*x + e)^3 - 3*(a^2*b + 2*a*b^2 - 8*b^3)*tan(f*x + 
 e))*sqrt(b*tan(f*x + e)^2 + a))/(b^3*f), -1/48*(48*sqrt(a - b)*b^3*arctan 
(sqrt(a - b)*tan(f*x + e)/sqrt(b*tan(f*x + e)^2 + a)) + 3*(a^3 + 2*a^2*b + 
 8*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*(a*b^2 - 6*b^3)*tan(f*x + e)^3 - 3* 
(a^2*b + 2*a*b^2 - 8*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^...
 

Sympy [F]

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

integrate(tan(f*x+e)**6*(a+b*tan(f*x+e)**2)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*tan(f*x + e)^2 + a)*tan(f*x + e)^6, x)
 

Giac [F(-2)]

Exception generated. \[ \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^(1/2),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. \[ \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^6\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a} \,d x \] Input:

int(tan(e + f*x)^6*(a + b*tan(e + f*x)^2)^(1/2),x)
 

Output:

int(tan(e + f*x)^6*(a + b*tan(e + f*x)^2)^(1/2), x)
 

Reduce [F]

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

int(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^(1/2),x)
 

Output:

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