Integrand size = 33, antiderivative size = 975 \[ \int \tan ^5(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^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \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 \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 )}{2 \sqrt {c} e}-\frac {b \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 )}{16 c^{5/2} e}+\frac {b \left (7 b^2-12 a c\right ) \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 )}{256 c^{9/2} 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^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \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 {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}+\frac {b (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{8 c^2 e}-\frac {b \left (7 b^2-12 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{128 c^4 e}-\frac {\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c e}+\frac {\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c e}+\frac {\left (35 b^2-32 a c-42 b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{240 c^3 e} \]
-1/16*b*(-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^(5/2)/e+1/256*b*(-12*a*c+7*b^2)*(-4*a*c+b^2)*ar ctanh(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^(9/2)/e+1/2*b*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))/e/c^(1/2)-1/2*arctanh(1/2*(b^2+(a-c)*(a-c+(a^2-2*a*c +b^2+c^2)^(1/2))+b*(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*arctan(1/2*(b^2+(a-c)*(a-c-(a^2-2*a* c+b^2+c^2)^(1/2))-b*(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)+(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)/e +1/8*b*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*(b+2*c*tan(e*x+d))/c^2/e-1/12 8*b*(-12*a*c+7*b^2)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*(b+2*c*tan(e*x+d ))/c^4/e-1/3*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2)/c/e+1/5*tan(e*x+d)^2*(a +b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2)/c/e+1/240*(35*b^2-32*a*c-42*b*c*tan(e* x+d))*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2)/c^3/e
Result contains complex when optimal does not.
Time = 6.28 (sec) , antiderivative size = 623, normalized size of antiderivative = 0.64 \[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {-\frac {1}{2} \sqrt {a-i b-c} \text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )-\frac {1}{2} \sqrt {a+i b-c} \text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt {c}}+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}-\frac {\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c}+\frac {\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c}+\frac {b \left (-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2}}+\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c}\right )}{2 c}+\frac {\frac {\left (\frac {35 b^2}{4}-8 a c-\frac {21}{2} b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{12 c^2}+\frac {\left (-\frac {35 b^3}{4}+15 a b c\right ) \left (-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2}}+\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c}\right )}{8 c^2}}{5 c}}{e} \]
(-1/2*(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])]) - (Sqr t[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])])/2 + (b*ArcTanh[( 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]) + Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2] - (a + b*T an[d + e*x] + c*Tan[d + e*x]^2)^(3/2)/(3*c) + (Tan[d + e*x]^2*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(5*c) + (b*(-1/8*((b^2 - 4*a*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])])/c^(3/2) + ((b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[ d + e*x]^2])/(4*c)))/(2*c) + ((((35*b^2)/4 - 8*a*c - (21*b*c*Tan[d + e*x]) /2)*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(12*c^2) + (((-35*b^3)/ 4 + 15*a*b*c)*(-1/8*((b^2 - 4*a*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])])/c^(3/2) + ((b + 2*c*Tan[ d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*c)))/(8*c^2))/(5 *c))/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.
| \(\displaystyle \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (d+e x)^5 \sqrt {a+b \tan (d+e x)+c \tan (d+e x)^2}dx\) |
| \(\Big \downarrow \) 4183 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan ^3(d+e x)+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)}{\tan ^2(d+e x)+1}-\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \tan (d+e x)\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\tan ^5(d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
3.1.1.3.1 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 = 3.57 (sec) , antiderivative size = 17768518, normalized size of antiderivative = 18224.12
\[\text {output too large to display}\]
Leaf count of result is larger than twice the leaf count of optimal. 2511 vs. \(2 (876) = 1752\).
Time = 1.25 (sec) , antiderivative size = 5023, normalized size of antiderivative = 5.15 \[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Too large to display} \]
\[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int \sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}} \tan ^{5}{\left (d + e x \right )}\, dx \]
\[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a} \tan \left (e x + d\right )^{5} \,d x } \]
Timed out. \[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Hanged} \]