Integrand size = 33, antiderivative size = 889 \[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2} e}-\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{128 c^{7/2} e}-\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}+\left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}+\frac {\left (5 b^2-4 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{64 c^3 e}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{24 c^2 e}+\frac {\tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{4 c e} \]
1/8*(-4*a*c+b^2)*arctanh(1/2*(b+2*c*tan(e*x+d))/c^(1/2)/(a+b*tan(e*x+d)+c* tan(e*x+d)^2)^(1/2))/c^(3/2)/e-1/128*(-4*a*c+b^2)*(-4*a*c+5*b^2)*arctanh(1 /2*(b+2*c*tan(e*x+d))/c^(1/2)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2))/c^(7/ 2)/e+arctanh(1/2*(b+2*c*tan(e*x+d))/c^(1/2)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2 )^(1/2))*c^(1/2)/e-1/2*arctan(1/2*(b*(a^2-2*a*c+b^2+c^2)^(1/2)-(b^2+(a-c)* (a-c+(a^2-2*a*c+b^2+c^2)^(1/2)))*tan(e*x+d))/(a^2-2*a*c+b^2+c^2)^(1/4)*2^( 1/2)/(a^2+b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+b^2+c^2)^( 1/2)))^(1/2)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2))*(a^2+b^2+c*(c-(a^2-2*a *c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/2)/(a^2-2*a*c+b^2 +c^2)^(1/4)/e*2^(1/2)-1/2*arctanh(1/2*(b*(a^2-2*a*c+b^2+c^2)^(1/2)+(b^2+(a -c)*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2)))*tan(e*x+d))/(a^2-2*a*c+b^2+c^2)^(1/4) *2^(1/2)/(a^2+b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*c+b^2+c^ 2)^(1/2)))^(1/2)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2))*(a^2+b^2+c*(c+(a^2 -2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/2)/(a^2-2*a*c +b^2+c^2)^(1/4)/e*2^(1/2)-1/4*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*(b+2*c *tan(e*x+d))/c/e+1/64*(-4*a*c+5*b^2)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2) *(b+2*c*tan(e*x+d))/c^3/e-5/24*b*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2)/c^2 /e+1/4*tan(e*x+d)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2)/c/e
Result contains complex when optimal does not.
Time = 5.62 (sec) , antiderivative size = 511, normalized size of antiderivative = 0.57 \[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {-4 i \sqrt {a-i b-c} \text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+4 i \sqrt {a+i b-c} \text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+8 \sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{c^{3/2}}-\frac {2 (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c^2}+\frac {2 \tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{c}-\frac {\left (-5 b^2+4 a c\right ) \left (-\left (\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )\right )+2 \sqrt {c} (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}\right )}{16 c^{7/2}}}{8 e} \]
((-4*I)*Sqrt[a - I*b - c]*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x]) /(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + (4*I )*Sqrt[a + I*b - c]*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sq rt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])] + 8*Sqrt[c]* ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[ d + e*x]^2])] + ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*S qrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/c^(3/2) - (2*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/c - (5*b*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(3*c^2) + (2*Tan[d + e*x]*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/c - ((-5*b^2 + 4*a*c)*(-((b^2 - 4*a*c)*A rcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]) + 2*Sqrt[c]*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]))/(16*c^(7/2)))/(8*e)
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.
| \(\displaystyle \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (d+e x)^4 \sqrt {a+b \tan (d+e x)+c \tan (d+e x)^2}dx\) |
| \(\Big \downarrow \) 4183 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^2(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^4(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
3.1.2.3.1 Defintions of rubi rules used
Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*( x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n2_.))^(p_), x_Symbol] :> Simp[f/e Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)), x ], x, f*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n 2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 2.22 (sec) , antiderivative size = 17248526, normalized size of antiderivative = 19402.17
\[\text {output too large to display}\]
Leaf count of result is larger than twice the leaf count of optimal. 2427 vs. \(2 (795) = 1590\).
Time = 1.13 (sec) , antiderivative size = 4855, normalized size of antiderivative = 5.46 \[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Too large to display} \]
[1/768*(192*c^4*e*sqrt((e^2*sqrt(-b^2/e^4) - a + c)/e^2)*log((2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) + (2*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^2*tan(e*x + d) + (4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt(-b^2/e^4))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a) + ((b^5 + 2*(4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 5*a*b^3)*c)*e*tan(e*x + d)^2 - 4*(2*a^3*b^2 + a*b^4 + b^4*c - 2*a*b^2*c^2) *e*tan(e*x + d) - (8*a^4*b + 6*a^2*b^3 + b^5 - 2*(4*a^3*b + a*b^3)*c)*e - ((4*a^3*b + 3*a*b^3 - 8*a*b*c^2 - (4*a^2*b - 3*b^3)*c)*e^3*tan(e*x + d)^2 + 2*(4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)* e^3*tan(e*x + d) - (4*a^3*b + a*b^3 + (4*a^2*b + b^3)*c)*e^3)*sqrt(-b^2/e^ 4))*sqrt((e^2*sqrt(-b^2/e^4) - a + c)/e^2))/(tan(e*x + d)^2 + 1)) - 192*c^ 4*e*sqrt((e^2*sqrt(-b^2/e^4) - a + c)/e^2)*log((2*(4*a^3*b^2 + 2*a*b^4 + 2 *b^4*c - 4*a*b^2*c^2 - (4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) + (2*(2*a^3*b + a*b^3 + b^3*c - 2*a* b*c^2)*e^2*tan(e*x + d) + (4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2 *(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt(-b^2/e^4))*sqrt(c*tan(e*x + d)^2 + b*tan(e *x + d) + a) - ((b^5 + 2*(4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 5*a*b^3)*c)*e* tan(e*x + d)^2 - 4*(2*a^3*b^2 + a*b^4 + b^4*c - 2*a*b^2*c^2)*e*tan(e*x + d ) - (8*a^4*b + 6*a^2*b^3 + b^5 - 2*(4*a^3*b + a*b^3)*c)*e - ((4*a^3*b +...
\[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int \sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}} \tan ^{4}{\left (d + e x \right )}\, dx \]
\[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a} \tan \left (e x + d\right )^{4} \,d x } \]
Timed out. \[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Hanged} \]