\(\int \tan ^6(e+f x) (a+b \tan ^2(e+f x))^{3/2} \, dx\) [312]

Optimal result

Integrand size = 25, antiderivative size = 294 \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=-\frac {(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}+\frac {\left (3 a^4+8 a^3 b+48 a^2 b^2-192 a b^3+128 b^4\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{128 b^{5/2} f}-\frac {\left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{128 b^2 f}+\frac {\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{192 b f}+\frac {(9 a-8 b) \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{48 f}+\frac {b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{8 f} \] Output:

-(a-b)^(3/2)*arctan((a-b)^(1/2)*tan(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2))/f+1/1 
28*(3*a^4+8*a^3*b+48*a^2*b^2-192*a*b^3+128*b^4)*arctanh(b^(1/2)*tan(f*x+e) 
/(a+b*tan(f*x+e)^2)^(1/2))/b^(5/2)/f-1/128*(3*a^3+8*a^2*b-80*a*b^2+64*b^3) 
*tan(f*x+e)*(a+b*tan(f*x+e)^2)^(1/2)/b^2/f+1/192*(3*a^2-56*a*b+48*b^2)*tan 
(f*x+e)^3*(a+b*tan(f*x+e)^2)^(1/2)/b/f+1/48*(9*a-8*b)*tan(f*x+e)^5*(a+b*ta 
n(f*x+e)^2)^(1/2)/f+1/8*b*tan(f*x+e)^7*(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 = 6.34 (sec) , antiderivative size = 908, normalized size of antiderivative = 3.09 \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {-\frac {b \left (3 a^4+8 a^3 b-16 a^2 b^2-64 a b^3+64 b^4\right ) \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \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)) \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 ) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac {4 b \left (-64 a^2 b^2+128 a b^3-64 b^4\right ) \sqrt {1+\cos (2 (e+f x))} \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \left (\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \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)) \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 ) \sin ^4(e+f x)}{4 a \sqrt {1+\cos (2 (e+f x))} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}-\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \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)) \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 ) \sin ^4(e+f x)}{2 (a-b) \sqrt {1+\cos (2 (e+f x))} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}\right )}{\sqrt {a+b+(a-b) \cos (2 (e+f x))}}}{64 b^2 f}+\frac {\sqrt {\frac {a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \left (\frac {1}{48} \sec ^5(e+f x) (9 a \sin (e+f x)-26 b \sin (e+f x))+\frac {\sec ^3(e+f x) \left (3 a^2 \sin (e+f x)-128 a b \sin (e+f x)+184 b^2 \sin (e+f x)\right )}{192 b}+\frac {\sec (e+f x) \left (-9 a^3 \sin (e+f x)-30 a^2 b \sin (e+f x)+424 a b^2 \sin (e+f x)-400 b^3 \sin (e+f x)\right )}{384 b^2}+\frac {1}{8} b \sec ^6(e+f x) \tan (e+f x)\right )}{f} \] Input:

Integrate[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^(3/2),x]
 

Output:

(-((b*(3*a^4 + 8*a^3*b - 16*a^2*b^2 - 64*a*b^3 + 64*b^4)*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[ArcSi 
n[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*(-64*a^2*b 
^2 + 128*a*b^3 - 64*b^4)*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[S 
qrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Si 
n[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)]])/(64*b^2*f) + (Sqrt[(a + b + 
a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sec[e + 
 f*x]^5*(9*a*Sin[e + f*x] - 26*b*Sin[e + f*x]))/48 + (Sec[e + f*x]^3*(3...
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 4153, 379, 444, 27, 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.

\(\displaystyle \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \tan (e+f x)^6 \left (a+b \tan (e+f x)^2\right )^{3/2}dx\)

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 379

\(\displaystyle \frac {\frac {1}{8} \int \frac {\tan ^6(e+f x) \left ((9 a-8 b) b \tan ^2(e+f x)+a (8 a-7 b)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)+\frac {1}{8} b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} (9 a-8 b) \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}-\frac {\int \frac {b \tan ^4(e+f x) \left (5 a (9 a-8 b)-\left (3 a^2-56 b a+48 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{6 b}\right )+\frac {1}{8} b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} (9 a-8 b) \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}-\frac {1}{6} \int \frac {\tan ^4(e+f x) \left (5 a (9 a-8 b)-\left (3 a^2-56 b a+48 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)\right )+\frac {1}{8} b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {\int -\frac {3 \tan ^2(e+f x) \left (\left (3 a^3+8 b a^2-80 b^2 a+64 b^3\right ) \tan ^2(e+f x)+a \left (3 a^2-56 b a+48 b^2\right )\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{4 b}+\frac {\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 b}\right )+\frac {1}{6} (9 a-8 b) \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )+\frac {1}{8} b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 b}-\frac {3 \int \frac {\tan ^2(e+f x) \left (\left (3 a^3+8 b a^2-80 b^2 a+64 b^3\right ) \tan ^2(e+f x)+a \left (3 a^2-56 b a+48 b^2\right )\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{4 b}\right )+\frac {1}{6} (9 a-8 b) \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )+\frac {1}{8} b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 b}-\frac {3 \left (\frac {\left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}-\frac {\int \frac {\left (3 a^4+8 b a^3+48 b^2 a^2-192 b^3 a+128 b^4\right ) \tan ^2(e+f x)+a \left (3 a^3+8 b a^2-80 b^2 a+64 b^3\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{2 b}\right )}{4 b}\right )+\frac {1}{6} (9 a-8 b) \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )+\frac {1}{8} b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 b}-\frac {3 \left (\frac {\left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}-\frac {\left (3 a^4+8 a^3 b+48 a^2 b^2-192 a b^3+128 b^4\right ) \int \frac {1}{\sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)-128 b^2 (a-b)^2 \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{2 b}\right )}{4 b}\right )+\frac {1}{6} (9 a-8 b) \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )+\frac {1}{8} b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 b}-\frac {3 \left (\frac {\left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}-\frac {\left (3 a^4+8 a^3 b+48 a^2 b^2-192 a b^3+128 b^4\right ) \int \frac {1}{1-\frac {b \tan ^2(e+f x)}{b \tan ^2(e+f x)+a}}d\frac {\tan (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}-128 b^2 (a-b)^2 \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{2 b}\right )}{4 b}\right )+\frac {1}{6} (9 a-8 b) \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )+\frac {1}{8} b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 b}-\frac {3 \left (\frac {\left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}-\frac {\frac {\left (3 a^4+8 a^3 b+48 a^2 b^2-192 a b^3+128 b^4\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{\sqrt {b}}-128 b^2 (a-b)^2 \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{2 b}\right )}{4 b}\right )+\frac {1}{6} (9 a-8 b) \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )+\frac {1}{8} b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 b}-\frac {3 \left (\frac {\left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}-\frac {\frac {\left (3 a^4+8 a^3 b+48 a^2 b^2-192 a b^3+128 b^4\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{\sqrt {b}}-128 b^2 (a-b)^2 \int \frac {1}{1-\frac {(b-a) \tan ^2(e+f x)}{b \tan ^2(e+f x)+a}}d\frac {\tan (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}}{2 b}\right )}{4 b}\right )+\frac {1}{6} (9 a-8 b) \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )+\frac {1}{8} b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 b}-\frac {3 \left (\frac {\left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}-\frac {\frac {\left (3 a^4+8 a^3 b+48 a^2 b^2-192 a b^3+128 b^4\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{\sqrt {b}}-128 b^2 (a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{2 b}\right )}{4 b}\right )+\frac {1}{6} (9 a-8 b) \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )+\frac {1}{8} b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\)

Input:

Int[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^(3/2),x]
 

Output:

((b*Tan[e + f*x]^7*Sqrt[a + b*Tan[e + f*x]^2])/8 + (((9*a - 8*b)*Tan[e + f 
*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/6 + (((3*a^2 - 56*a*b + 48*b^2)*Tan[e + 
f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(4*b) - (3*(-1/2*(-128*(a - b)^(3/2)*b^ 
2*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]] + ((3*a^4 
+ 8*a^3*b + 48*a^2*b^2 - 192*a*b^3 + 128*b^4)*ArcTanh[(Sqrt[b]*Tan[e + f*x 
])/Sqrt[a + b*Tan[e + f*x]^2]])/Sqrt[b])/b + ((3*a^3 + 8*a^2*b - 80*a*b^2 
+ 64*b^3)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*b)))/(4*b))/6)/8)/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 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. \(668\) vs. \(2(264)=528\).

Time = 0.76 (sec) , antiderivative size = 669, 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 {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}}+\frac {\tan \left (f x +e \right )^{3} \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{8 f b}-\frac {a \tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{16 f \,b^{2}}+\frac {a^{2} \tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{64 f \,b^{2}}+\frac {3 a^{3} \tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{128 f \,b^{2}}+\frac {3 a^{4} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{128 f \,b^{\frac {5}{2}}}-\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 )}\)	\(669\)
default	\(\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}}+\frac {\tan \left (f x +e \right )^{3} \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{8 f b}-\frac {a \tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{16 f \,b^{2}}+\frac {a^{2} \tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{64 f \,b^{2}}+\frac {3 a^{3} \tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{128 f \,b^{2}}+\frac {3 a^{4} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{128 f \,b^{\frac {5}{2}}}-\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 )}\)	\(669\)

Input:

int(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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))+1/8/f*tan(f*x+e)^3*(a+b*tan(f*x+e)^2)^(5/2)/b-1/16/f*a/b^2*tan(f*x+e)* 
(a+b*tan(f*x+e)^2)^(5/2)+1/64/f*a^2/b^2*tan(f*x+e)*(a+b*tan(f*x+e)^2)^(3/2 
)+3/128/f*a^3/b^2*tan(f*x+e)*(a+b*tan(f*x+e)^2)^(1/2)+3/128/f*a^4/b^(5/2)* 
ln(b^(1/2)*tan(f*x+e)+(a+b*tan(f*x+e)^2)^(1/2))-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*tan(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)*arc 
tan(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 = 2.52 (sec) , antiderivative size = 1043, normalized size of antiderivative = 3.55 \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="fricas")
 

Output:

[1/768*(3*(3*a^4 + 8*a^3*b + 48*a^2*b^2 - 192*a*b^3 + 128*b^4)*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) - 384*(a*b^3 - b^4)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqr 
t(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1 
)) + 2*(48*b^4*tan(f*x + e)^7 + 8*(9*a*b^3 - 8*b^4)*tan(f*x + e)^5 + 2*(3* 
a^2*b^2 - 56*a*b^3 + 48*b^4)*tan(f*x + e)^3 - 3*(3*a^3*b + 8*a^2*b^2 - 80* 
a*b^3 + 64*b^4)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^3*f), -1/384* 
(3*(3*a^4 + 8*a^3*b + 48*a^2*b^2 - 192*a*b^3 + 128*b^4)*sqrt(-b)*arctan(sq 
rt(-b)*tan(f*x + e)/sqrt(b*tan(f*x + e)^2 + a)) + 192*(a*b^3 - b^4)*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)) - (48*b^4*tan(f*x + e)^7 + 
 8*(9*a*b^3 - 8*b^4)*tan(f*x + e)^5 + 2*(3*a^2*b^2 - 56*a*b^3 + 48*b^4)*ta 
n(f*x + e)^3 - 3*(3*a^3*b + 8*a^2*b^2 - 80*a*b^3 + 64*b^4)*tan(f*x + e))*s 
qrt(b*tan(f*x + e)^2 + a))/(b^3*f), -1/768*(768*(a*b^3 - b^4)*sqrt(a - b)* 
arctan(sqrt(a - b)*tan(f*x + e)/sqrt(b*tan(f*x + e)^2 + a)) - 3*(3*a^4 + 8 
*a^3*b + 48*a^2*b^2 - 192*a*b^3 + 128*b^4)*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*(48*b^4*tan(f 
*x + e)^7 + 8*(9*a*b^3 - 8*b^4)*tan(f*x + e)^5 + 2*(3*a^2*b^2 - 56*a*b^3 + 
 48*b^4)*tan(f*x + e)^3 - 3*(3*a^3*b + 8*a^2*b^2 - 80*a*b^3 + 64*b^4)*tan( 
f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^3*f), -1/384*(384*(a*b^3 - b^4...
 

Sympy [F]

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

integrate(tan(f*x+e)**6*(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)**6, x)
 

Maxima [F]

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

integrate(tan(f*x+e)^6*(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)^6, x)
 

Giac [F(-2)]

Exception generated. \[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

int(tan(e + f*x)^6*(a + b*tan(e + f*x)^2)^(3/2),x)
 

Output:

int(tan(e + f*x)^6*(a + b*tan(e + f*x)^2)^(3/2), x)
 

Reduce [F]

\[ \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\left (\int \sqrt {\tan \left (f x +e \right )^{2} b +a}\, \tan \left (f x +e \right )^{8}d x \right ) b +\left (\int \sqrt {\tan \left (f x +e \right )^{2} b +a}\, \tan \left (f x +e \right )^{6}d x \right ) a \] Input:

int(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^(3/2),x)
 

Output:

int(sqrt(tan(e + f*x)**2*b + a)*tan(e + f*x)**8,x)*b + int(sqrt(tan(e + f* 
x)**2*b + a)*tan(e + f*x)**6,x)*a