Integrand size = 25, antiderivative size = 224 \[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}-\frac {\left (a^3+6 a^2 b-24 a b^2+16 b^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{16 b^{3/2} f}+\frac {\left (a^2-10 a b+8 b^2\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{16 b f}+\frac {(7 a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{24 f}+\frac {b \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{6 f} \]
(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)*(a+b*tan(f*x+e)^2)^(1/2)*tan(f *x+e)/b/f+1/24*(7*a-6*b)*(a+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)^3/f+1/6*b*(a+ b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)^5/f
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 6.43 (sec) , antiderivative size = 833, normalized size of antiderivative = 3.72 \[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=-\frac {-\frac {b \left (a^3-2 a^2 b-8 a b^2+8 b^3\right ) \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac {4 b \left (-8 a^2 b+16 a b^2-8 b^3\right ) \sqrt {1+\cos (2 (e+f x))} \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \left (\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \sin ^4(e+f x)}{4 a \sqrt {1+\cos (2 (e+f x))} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}-\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \operatorname {EllipticPi}\left (-\frac {b}{a-b},\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \sin ^4(e+f x)}{2 (a-b) \sqrt {1+\cos (2 (e+f x))} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}\right )}{\sqrt {a+b+(a-b) \cos (2 (e+f x))}}}{8 b f}+\frac {\sqrt {\frac {a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \left (\frac {7}{24} \sec ^3(e+f x) (a \sin (e+f x)-2 b \sin (e+f x))+\frac {\sec (e+f x) \left (3 a^2 \sin (e+f x)-44 a b \sin (e+f x)+44 b^2 \sin (e+f x)\right )}{48 b}+\frac {1}{6} b \sec ^4(e+f x) \tan (e+f x)\right )}{f} \]
-1/8*(-((b*(a^3 - 2*a^2*b - 8*a*b^2 + 8*b^3)*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*(-8*a^2*b + 16*a*b^2 - 8*b^3)*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 + Co s[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)]]) - (Sq rt[-((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)]])/(b*f) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b *Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((7*Sec[e + f*x]^3*(a*Sin[e + f *x] - 2*b*Sin[e + f*x]))/24 + (Sec[e + f*x]*(3*a^2*Sin[e + f*x] - 44*a*...
Time = 0.51 (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 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} (7 a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}-\frac {3}{4} \left (\frac {\frac {(a-2 b) \left (a^2+8 a b-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{\sqrt {b}}-16 b (a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{2 b}-\frac {\left (a^2-10 a b+8 b^2\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 b}\right )\right )+\frac {1}{6} b \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\) |
((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
3.4.13.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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])
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]
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]
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]
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]
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]
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]))
Leaf count of result is larger than twice the leaf count of optimal. \(509\) vs. \(2(198)=396\).
Time = 0.07 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.28
| method | result | size |
| derivativedivides | \(-\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 f}-\frac {3 a \tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{8 f}-\frac {3 a^{2} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{8 f \sqrt {b}}+\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{6 f b}-\frac {a \tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{24 f b}-\frac {a^{2} \tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{16 f b}-\frac {a^{3} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{16 f \,b^{\frac {3}{2}}}-\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{f}+\frac {b \sqrt {a +b \tan \left (f x +e \right )^{2}}\, \tan \left (f x +e \right )}{2 f}+\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 f}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f \left (a -b \right )}-\frac {2 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f b \left (a -b \right )}+\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f \,b^{2} \left (a -b \right )}\) | \(510\) |
| default | \(-\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 f}-\frac {3 a \tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{8 f}-\frac {3 a^{2} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{8 f \sqrt {b}}+\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{6 f b}-\frac {a \tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{24 f b}-\frac {a^{2} \tan \left (f x +e \right ) \sqrt {a +b \tan \left (f x +e \right )^{2}}}{16 f b}-\frac {a^{3} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{16 f \,b^{\frac {3}{2}}}-\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{f}+\frac {b \sqrt {a +b \tan \left (f x +e \right )^{2}}\, \tan \left (f x +e \right )}{2 f}+\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 f}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f \left (a -b \right )}-\frac {2 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f b \left (a -b \right )}+\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f \,b^{2} \left (a -b \right )}\) | \(510\) |
-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))+1/6/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)-1/1 6/f*a^3/b^(3/2)*ln(b^(1/2)*tan(f*x+e)+(a+b*tan(f*x+e)^2)^(1/2))-1/f*b^(3/2 )*ln(b^(1/2)*tan(f*x+e)+(a+b*tan(f*x+e)^2)^(1/2))+1/2*b*(a+b*tan(f*x+e)^2) ^(1/2)*tan(f*x+e)/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))-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))
Time = 1.68 (sec) , antiderivative size = 861, normalized size of antiderivative = 3.84 \[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\left [\frac {3 \, {\left (a^{3} + 6 \, a^{2} b - 24 \, a b^{2} + 16 \, b^{3}\right )} \sqrt {b} \log \left (2 \, b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {b} \tan \left (f x + e\right ) + a\right ) - 48 \, {\left (a b^{2} - b^{3}\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b} \tan \left (f x + e\right ) - a}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \, {\left (8 \, b^{3} \tan \left (f x + e\right )^{5} + 2 \, {\left (7 \, a b^{2} - 6 \, b^{3}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (a^{2} b - 10 \, a b^{2} + 8 \, b^{3}\right )} \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{96 \, b^{2} f}, \frac {3 \, {\left (a^{3} + 6 \, a^{2} b - 24 \, a b^{2} + 16 \, b^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-b}}{b \tan \left (f x + e\right )}\right ) - 24 \, {\left (a b^{2} - b^{3}\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b} \tan \left (f x + e\right ) - a}{\tan \left (f x + e\right )^{2} + 1}\right ) + {\left (8 \, b^{3} \tan \left (f x + e\right )^{5} + 2 \, {\left (7 \, a b^{2} - 6 \, b^{3}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (a^{2} b - 10 \, a b^{2} + 8 \, b^{3}\right )} \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{48 \, b^{2} f}, \frac {96 \, {\left (a b^{2} - b^{3}\right )} \sqrt {a - b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{\sqrt {a - b} \tan \left (f x + e\right )}\right ) + 3 \, {\left (a^{3} + 6 \, a^{2} b - 24 \, a b^{2} + 16 \, b^{3}\right )} \sqrt {b} \log \left (2 \, b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {b} \tan \left (f x + e\right ) + a\right ) + 2 \, {\left (8 \, b^{3} \tan \left (f x + e\right )^{5} + 2 \, {\left (7 \, a b^{2} - 6 \, b^{3}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (a^{2} b - 10 \, a b^{2} + 8 \, b^{3}\right )} \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{96 \, b^{2} f}, \frac {48 \, {\left (a b^{2} - b^{3}\right )} \sqrt {a - b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{\sqrt {a - b} \tan \left (f x + e\right )}\right ) + 3 \, {\left (a^{3} + 6 \, a^{2} b - 24 \, a b^{2} + 16 \, b^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-b}}{b \tan \left (f x + e\right )}\right ) + {\left (8 \, b^{3} \tan \left (f x + e\right )^{5} + 2 \, {\left (7 \, a b^{2} - 6 \, b^{3}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (a^{2} b - 10 \, a b^{2} + 8 \, b^{3}\right )} \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{48 \, b^{2} f}\right ] \]
[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)^2 + a)*sqrt(- b)/(b*tan(f*x + e))) - 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)/(t an(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(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + 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) + 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(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + 3*(a^3 + 6*a^2*b - 24*a*b^2 + 16 *b^3)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e)) ) + (8*b^3*tan(f*x + e)^5 + 2*(7*a*b^2 - 6*b^3)*tan(f*x + e)^3 + 3*(a^2...
\[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{4}{\left (e + f x \right )}\, dx \]
\[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{4} \,d x } \]
Timed out. \[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \tan ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]