Integrand size = 49, antiderivative size = 370 \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=-\frac {(i A+B-i C) \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{5/2} f}-\frac {(B-i (A-C)) \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{5/2} f}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (2 a^3 b B d+a^4 C d+b^4 (3 B c+A d)+2 a b^3 (3 A c-3 c C-2 B d)-a^2 b^2 (3 B c+5 A d-7 C d)\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right )^2 (b c-a d) f \sqrt {a+b \tan (e+f x)}} \]
-(I*A+B-I*C)*arctanh((c-I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a-I*b)^(1/2)/(c +d*tan(f*x+e))^(1/2))*(c-I*d)^(1/2)/(a-I*b)^(5/2)/f-(B-I*(A-C))*arctanh((c +I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a+I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))*( c+I*d)^(1/2)/(a+I*b)^(5/2)/f-2/3*(2*a^3*b*B*d+a^4*C*d+b^4*(A*d+3*B*c)+2*a* b^3*(3*A*c-2*B*d-3*C*c)-a^2*b^2*(5*A*d+3*B*c-7*C*d))*(c+d*tan(f*x+e))^(1/2 )/b/(a^2+b^2)^2/(-a*d+b*c)/f/(a+b*tan(f*x+e))^(1/2)-2/3*(A*b^2-a*(B*b-C*a) )*(c+d*tan(f*x+e))^(1/2)/b/(a^2+b^2)/f/(a+b*tan(f*x+e))^(3/2)
Time = 7.10 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.62 \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=-\frac {C \sqrt {c+d \tan (e+f x)}}{b f (a+b \tan (e+f x))^{3/2}}-\frac {-\frac {2 \left (\frac {1}{2} b^2 (-2 A b c+3 b c C-a C d)-a \left (-b^2 (B c+(A-C) d)-\frac {1}{2} a (b c C-2 b B d-a C d)\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (-\frac {3 b (b c-a d) \left (\frac {(a+i b)^2 (i A+B-i C) \sqrt {-c+i d} \text {arctanh}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a+i b}}+\frac {(a-i b)^2 (B-i (A-C)) \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+i b}}\right )}{2 \left (a^2+b^2\right ) f}-\frac {2 \left (\frac {1}{2} b^2 (b c-a d) \left (a^2 C d+b^2 (3 B c+A d)+a b (3 A c-3 c C-B d)\right )-a \left (\frac {1}{2} a \left (2 A b^2-2 a b B-a^2 C-3 b^2 C\right ) d (b c-a d)-\frac {3}{2} b^2 (b c-a d) (A b c-a B c-b c C-a A d-b B d+a C d)\right )\right ) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}\right )}{3 \left (a^2+b^2\right ) (b c-a d)}}{b} \]
Integrate[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2 ))/(a + b*Tan[e + f*x])^(5/2),x]
-((C*Sqrt[c + d*Tan[e + f*x]])/(b*f*(a + b*Tan[e + f*x])^(3/2))) - ((-2*(( b^2*(-2*A*b*c + 3*b*c*C - a*C*d))/2 - a*(-(b^2*(B*c + (A - C)*d)) - (a*(b* c*C - 2*b*B*d - a*C*d))/2))*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)) - (2*((-3*b*(b*c - a*d)*(((a + I*b)^2 *(I*A + B - I*C)*Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[-a + I*b] + ((a - I*b)^2*(B - I*(A - C))*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Ta n[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[a + I*b]))/(2 *(a^2 + b^2)*f) - (2*((b^2*(b*c - a*d)*(a^2*C*d + b^2*(3*B*c + A*d) + a*b* (3*A*c - 3*c*C - B*d)))/2 - a*((a*(2*A*b^2 - 2*a*b*B - a^2*C - 3*b^2*C)*d* (b*c - a*d))/2 - (3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d))/2))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]])))/(3*(a^2 + b^2)*(b*c - a*d)))/b
Time = 2.59 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.245, Rules used = {3042, 4128, 27, 3042, 4132, 27, 3042, 4099, 3042, 4098, 104, 221}
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.
| \(\displaystyle \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{(a+b \tan (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {2 \int \frac {-\left (\left (-C a^2-2 b B a+2 A b^2-3 b^2 C\right ) d \tan ^2(e+f x)\right )-3 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (3 b c-a d)+A b (3 a c+b d)}{2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-\left (\left (-C a^2-2 b B a+2 A b^2-3 b^2 C\right ) d \tan ^2(e+f x)\right )-3 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (3 b c-a d)+A b (3 a c+b d)}{(a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-\left (\left (-C a^2-2 b B a+2 A b^2-3 b^2 C\right ) d \tan (e+f x)^2\right )-3 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (3 b c-a d)+A b (3 a c+b d)}{(a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {3 \left (b (b c-a d) \left ((A c-C c-B d) a^2+2 b (B c+(A-C) d) a-b^2 (A c-C c-B d)\right )-b (b c-a d) \left (-\left ((B c+(A-C) d) a^2\right )+2 b (A c-C c-B d) a+b^2 (B c+(A-C) d)\right ) \tan (e+f x)\right )}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+2 a^3 b B d-a^2 b^2 (5 A d+3 B c-7 C d)+2 a b^3 (3 A c-2 B d-3 c C)+b^4 (A d+3 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {b (b c-a d) \left ((A c-C c-B d) a^2+2 b (B c+(A-C) d) a-b^2 (A c-C c-B d)\right )-b (b c-a d) \left (-\left ((B c+(A-C) d) a^2\right )+2 b (A c-C c-B d) a+b^2 (B c+(A-C) d)\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+2 a^3 b B d-a^2 b^2 (5 A d+3 B c-7 C d)+2 a b^3 (3 A c-2 B d-3 c C)+b^4 (A d+3 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \int \frac {b (b c-a d) \left ((A c-C c-B d) a^2+2 b (B c+(A-C) d) a-b^2 (A c-C c-B d)\right )-b (b c-a d) \left (-\left ((B c+(A-C) d) a^2\right )+2 b (A c-C c-B d) a+b^2 (B c+(A-C) d)\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+2 a^3 b B d-a^2 b^2 (5 A d+3 B c-7 C d)+2 a b^3 (3 A c-2 B d-3 c C)+b^4 (A d+3 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4099 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+2 a^3 b B d-a^2 b^2 (5 A d+3 B c-7 C d)+2 a b^3 (3 A c-2 B d-3 c C)+b^4 (A d+3 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}+\frac {3 \left (\frac {1}{2} b (a-i b)^2 (c+i d) (A+i B-C) (b c-a d) \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} b (a+i b)^2 (c-i d) (A-i B-C) (b c-a d) \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+2 a^3 b B d-a^2 b^2 (5 A d+3 B c-7 C d)+2 a b^3 (3 A c-2 B d-3 c C)+b^4 (A d+3 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}+\frac {3 \left (\frac {1}{2} b (a-i b)^2 (c+i d) (A+i B-C) (b c-a d) \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} b (a+i b)^2 (c-i d) (A-i B-C) (b c-a d) \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4098 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+2 a^3 b B d-a^2 b^2 (5 A d+3 B c-7 C d)+2 a b^3 (3 A c-2 B d-3 c C)+b^4 (A d+3 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}+\frac {3 \left (\frac {b (a-i b)^2 (c+i d) (A+i B-C) (b c-a d) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}+\frac {b (a+i b)^2 (c-i d) (A-i B-C) (b c-a d) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+2 a^3 b B d-a^2 b^2 (5 A d+3 B c-7 C d)+2 a b^3 (3 A c-2 B d-3 c C)+b^4 (A d+3 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}+\frac {3 \left (\frac {b (a-i b)^2 (c+i d) (A+i B-C) (b c-a d) \int \frac {1}{-i a+b+\frac {(i c-d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}+\frac {b (a+i b)^2 (c-i d) (A-i B-C) (b c-a d) \int \frac {1}{i a+b-\frac {(i c+d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+2 a^3 b B d-a^2 b^2 (5 A d+3 B c-7 C d)+2 a b^3 (3 A c-2 B d-3 c C)+b^4 (A d+3 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}+\frac {3 \left (\frac {i b (a-i b)^2 \sqrt {c+i d} (A+i B-C) (b c-a d) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a+i b}}-\frac {i b (a+i b)^2 \sqrt {c-i d} (A-i B-C) (b c-a d) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a-i b}}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}\) |
Int[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(5/2),x]
(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*Tan[e + f*x]])/(3*b*(a^2 + b^2)*f*( a + b*Tan[e + f*x])^(3/2)) + ((3*(((-I)*(a + I*b)^2*b*(A - I*B - C)*Sqrt[c - I*d]*(b*c - a*d)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt [a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*f) + (I*(a - I*b)^2*b *(A + I*B - C)*Sqrt[c + I*d]*(b*c - a*d)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b *Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]* f)))/((a^2 + b^2)*(b*c - a*d)) - (2*(2*a^3*b*B*d + a^4*C*d + b^4*(3*B*c + A*d) + 2*a*b^3*(3*A*c - 3*c*C - 2*B*d) - a^2*b^2*(3*B*c + 5*A*d - 7*C*d))* Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f* x]]))/(3*b*(a^2 + b^2))
3.2.33.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[A^2/f Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e + f* x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 + B^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(A + I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*T an[e + f*x]), x], x] + Simp[(A - I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d *Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A , B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2 + B^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim p[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d* (n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b *(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ [a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Timed out.
\[\int \frac {\sqrt {c +d \tan \left (f x +e \right )}\, \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {5}{2}}}d x\]
Timed out. \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(5/2),x, algorithm="fricas")
\[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {c + d \tan {\left (e + f x \right )}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
integrate((c+d*tan(f*x+e))**(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*ta n(f*x+e))**(5/2),x)
Integral(sqrt(c + d*tan(e + f*x))*(A + B*tan(e + f*x) + C*tan(e + f*x)**2) /(a + b*tan(e + f*x))**(5/2), x)
Exception generated. \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: ValueError} \]
integrate((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(5/2),x, algorithm="maxima")
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(((2*b*d+2*a*c)^2>0)', see `assum e?` for mo
Timed out. \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(5/2),x, algorithm="giac")
Timed out. \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=\text {Hanged} \]