Integrand size = 49, antiderivative size = 535 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx=-\frac {(i A+B-i C) (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{3/2} f}-\frac {(B-i (A-C)) (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{3/2} f}+\frac {\sqrt {d} \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 b^{7/2} f}-\frac {d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac {\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}} \]
-(I*A+B-I*C)*(c-I*d)^(5/2)*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))/(a-I*b)^(3/2)/f-(B-I*(A-C))*(c+I*d)^(5 /2)*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))/(a+I*b)^(3/2)/f+1/4*(15*a^2*C*d^2-6*a*b*d*(2*B*d+5*C*c)+b^2*( 15*c^2*C+20*B*c*d+8*(A-C)*d^2))*arctanh(d^(1/2)*(a+b*tan(f*x+e))^(1/2)/b^( 1/2)/(c+d*tan(f*x+e))^(1/2))*d^(1/2)/b^(7/2)/f-1/4*d*(15*a^3*C*d-8*A*b^2*( -a*d+b*c)-3*a^2*b*(4*B*d+5*C*c)-b^3*(4*B*d+7*C*c)+a*b^2*(8*B*c+7*C*d))*(a+ b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/b^3/(a^2+b^2)/f+1/2*(4*A*b^2-4* B*a*b+5*C*a^2+C*b^2)*d*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2)/b^2/( a^2+b^2)/f-2*(A*b^2-a*(B*b-C*a))*(c+d*tan(f*x+e))^(5/2)/b/(a^2+b^2)/f/(a+b *tan(f*x+e))^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1774\) vs. \(2(535)=1070\).
Time = 8.66 (sec) , antiderivative size = 1774, normalized size of antiderivative = 3.32 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx =\text {Too large to display} \]
Integrate[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x] ^2))/(a + b*Tan[e + f*x])^(3/2),x]
(C*(c + d*Tan[e + f*x])^(5/2))/(2*b*f*Sqrt[a + b*Tan[e + f*x]]) + (((5*b*c *C + 4*b*B*d - 5*a*C*d)*(c + d*Tan[e + f*x])^(3/2))/(2*b*f*Sqrt[a + b*Tan[ e + f*x]]) + ((8*b^2*(I*A + B - I*C)*(-c + I*d)^(5/2)*ArcTanh[(Sqrt[-c + I *d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/ ((-a + I*b)^(3/2)*f) - (8*b^2*(B - I*(A - C))*(c + I*d)^(5/2)*ArcTanh[(Sqr t[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x ]])])/((a + I*b)^(3/2)*f) + (8*b^2*(I*A + B - I*C)*(c - I*d)^2*Sqrt[c + d* Tan[e + f*x]])/((a - I*b)*f*Sqrt[a + b*Tan[e + f*x]]) + (8*b^2*(A + I*B - C)*(c + I*d)^2*Sqrt[c + d*Tan[e + f*x]])/((I*a - b)*f*Sqrt[a + b*Tan[e + f *x]]) + (30*a^2*C*d^2*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f* x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)*(1 - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt [b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*T an[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a *d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*f*Sqrt[a + b*Tan[e + f*x]]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))) - (12*a*d*(5*c*C + 2*B*d)*Sqrt[c + d*Tan[e + f*x] ]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (...
Time = 4.48 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {3042, 4128, 27, 3042, 4130, 27, 3042, 4130, 27, 3042, 4138, 2348, 2009}
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.
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{(a+b \tan (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {2 \int \frac {(c+d \tan (e+f x))^{3/2} \left (\left (5 C a^2-4 b B a+4 A b^2+b^2 C\right ) d \tan ^2(e+f x)-b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (b c-5 a d)+A b (a c+5 b d)\right )}{2 \sqrt {a+b \tan (e+f x)}}dx}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (\left (5 C a^2-4 b B a+4 A b^2+b^2 C\right ) d \tan ^2(e+f x)-b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (b c-5 a d)+A b (a c+5 b d)\right )}{\sqrt {a+b \tan (e+f x)}}dx}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (\left (5 C a^2-4 b B a+4 A b^2+b^2 C\right ) d \tan (e+f x)^2-b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (b c-5 a d)+A b (a c+5 b d)\right )}{\sqrt {a+b \tan (e+f x)}}dx}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\int -\frac {\sqrt {c+d \tan (e+f x)} \left (-4 \left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )+b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2-4 c ((b B-a C) (b c-5 a d)+A b (a c+5 b d)) b+d \left (15 C d a^3-3 b (5 c C+4 B d) a^2+b^2 (8 B c+7 C d) a-8 A b^2 (b c-a d)-b^3 (7 c C+4 B d)\right ) \tan ^2(e+f x)+\left (5 C a^2-4 b B a+4 A b^2+b^2 C\right ) d (b c+3 a d)\right )}{2 \sqrt {a+b \tan (e+f x)}}dx}{2 b}+\frac {d \left (5 a^2 C-4 a b B+4 A b^2+b^2 C\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \left (5 a^2 C-4 a b B+4 A b^2+b^2 C\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}-\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-4 \left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )+b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2-4 c ((b B-a C) (b c-5 a d)+A b (a c+5 b d)) b+d \left (15 C d a^3-3 b (5 c C+4 B d) a^2+b^2 (8 B c+7 C d) a-8 A b^2 (b c-a d)-b^3 (7 c C+4 B d)\right ) \tan ^2(e+f x)+\left (5 C a^2-4 b B a+4 A b^2+b^2 C\right ) d (b c+3 a d)\right )}{\sqrt {a+b \tan (e+f x)}}dx}{4 b}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {d \left (5 a^2 C-4 a b B+4 A b^2+b^2 C\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}-\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-4 \left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )+b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2-4 c ((b B-a C) (b c-5 a d)+A b (a c+5 b d)) b+d \left (15 C d a^3-3 b (5 c C+4 B d) a^2+b^2 (8 B c+7 C d) a-8 A b^2 (b c-a d)-b^3 (7 c C+4 B d)\right ) \tan (e+f x)^2+\left (5 C a^2-4 b B a+4 A b^2+b^2 C\right ) d (b c+3 a d)\right )}{\sqrt {a+b \tan (e+f x)}}dx}{4 b}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {d \left (5 a^2 C-4 a b B+4 A b^2+b^2 C\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}-\frac {\frac {\int -\frac {15 C d^3 a^4-6 b d^2 (5 c C+2 B d) a^3+b^2 d \left (15 C c^2+20 B d c+(8 A+7 C) d^2\right ) a^2-2 b^3 \left (4 C c^3+12 B d c^2+3 C d^2 c+2 B d^3-4 A \left (c^3-3 c d^2\right )\right ) a+\left (a^2+b^2\right ) d \left (\left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-6 a d (5 c C+2 B d) b+15 a^2 C d^2\right ) \tan ^2(e+f x)+b^4 c \left (8 B c^2+24 A d c-9 C d c-4 B d^2\right )+8 b^3 \left (b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{b}+\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (15 a^3 C d-3 a^2 b (4 B d+5 c C)-8 A b^2 (b c-a d)+a b^2 (8 B c+7 C d)-b^3 (4 B d+7 c C)\right )}{b f}}{4 b}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \left (5 a^2 C-4 a b B+4 A b^2+b^2 C\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}-\frac {\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (15 a^3 C d-3 a^2 b (4 B d+5 c C)-8 A b^2 (b c-a d)+a b^2 (8 B c+7 C d)-b^3 (4 B d+7 c C)\right )}{b f}-\frac {\int \frac {15 C d^3 a^4-6 b d^2 (5 c C+2 B d) a^3+b^2 d \left (15 C c^2+20 B d c+(8 A+7 C) d^2\right ) a^2-2 b^3 \left (4 C c^3+12 B d c^2+3 C d^2 c+2 B d^3-4 A \left (c^3-3 c d^2\right )\right ) a+\left (a^2+b^2\right ) d \left (\left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-6 a d (5 c C+2 B d) b+15 a^2 C d^2\right ) \tan ^2(e+f x)+b^4 c \left (8 B c^2+24 A d c-9 C d c-4 B d^2\right )+8 b^3 \left (b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 b}}{4 b}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {d \left (5 a^2 C-4 a b B+4 A b^2+b^2 C\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}-\frac {\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (15 a^3 C d-3 a^2 b (4 B d+5 c C)-8 A b^2 (b c-a d)+a b^2 (8 B c+7 C d)-b^3 (4 B d+7 c C)\right )}{b f}-\frac {\int \frac {15 C d^3 a^4-6 b d^2 (5 c C+2 B d) a^3+b^2 d \left (15 C c^2+20 B d c+(8 A+7 C) d^2\right ) a^2-2 b^3 \left (4 C c^3+12 B d c^2+3 C d^2 c+2 B d^3-4 A \left (c^3-3 c d^2\right )\right ) a+\left (a^2+b^2\right ) d \left (\left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-6 a d (5 c C+2 B d) b+15 a^2 C d^2\right ) \tan (e+f x)^2+b^4 c \left (8 B c^2+24 A d c-9 C d c-4 B d^2\right )+8 b^3 \left (b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 b}}{4 b}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4138 |
| \(\displaystyle \frac {\frac {d \left (5 a^2 C-4 a b B+4 A b^2+b^2 C\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}-\frac {\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (15 a^3 C d-3 a^2 b (4 B d+5 c C)-8 A b^2 (b c-a d)+a b^2 (8 B c+7 C d)-b^3 (4 B d+7 c C)\right )}{b f}-\frac {\int \frac {15 C d^3 a^4-6 b d^2 (5 c C+2 B d) a^3+b^2 d \left (15 C c^2+20 B d c+(8 A+7 C) d^2\right ) a^2-2 b^3 \left (4 C c^3+12 B d c^2+3 C d^2 c+2 B d^3-4 A \left (c^3-3 c d^2\right )\right ) a+\left (a^2+b^2\right ) d \left (\left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-6 a d (5 c C+2 B d) b+15 a^2 C d^2\right ) \tan ^2(e+f x)+b^4 c \left (8 B c^2+24 A d c-9 C d c-4 B d^2\right )+8 b^3 \left (b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{2 b f}}{4 b}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}\) |
| \(\Big \downarrow \) 2348 |
| \(\displaystyle \frac {\frac {\left (5 C a^2-4 b B a+4 A b^2+b^2 C\right ) d \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}-\frac {\frac {d \left (15 C d a^3-3 b (5 c C+4 B d) a^2+b^2 (8 B c+7 C d) a-8 A b^2 (b c-a d)-b^3 (7 c C+4 B d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}-\frac {\int \left (\frac {\left (a^2+b^2\right ) d \left (\left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-6 a d (5 c C+2 B d) b+15 a^2 C d^2\right )}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {8 A c^3 b^4+8 B d^3 b^4-24 A c d^2 b^4+24 c C d^2 b^4-8 c^3 C b^4-24 B c^2 d b^4-8 a B c^3 b^3+8 a A d^3 b^3-8 a C d^3 b^3+24 a B c d^2 b^3-24 a A c^2 d b^3+24 a c^2 C d b^3+i \left (8 B c^3 b^4-8 A d^3 b^4+8 C d^3 b^4-24 B c d^2 b^4+24 A c^2 d b^4-24 c^2 C d b^4+8 a A c^3 b^3+8 a B d^3 b^3-24 a A c d^2 b^3+24 a c C d^2 b^3-8 a c^3 C b^3-24 a B c^2 d b^3\right )}{2 (i-\tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {-8 A c^3 b^4-8 B d^3 b^4+24 A c d^2 b^4-24 c C d^2 b^4+8 c^3 C b^4+24 B c^2 d b^4+8 a B c^3 b^3-8 a A d^3 b^3+8 a C d^3 b^3-24 a B c d^2 b^3+24 a A c^2 d b^3-24 a c^2 C d b^3+i \left (8 B c^3 b^4-8 A d^3 b^4+8 C d^3 b^4-24 B c d^2 b^4+24 A c^2 d b^4-24 c^2 C d b^4+8 a A c^3 b^3+8 a B d^3 b^3-24 a A c d^2 b^3+24 a c C d^2 b^3-8 a c^3 C b^3-24 a B c^2 d b^3\right )}{2 (\tan (e+f x)+i) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}\right )d\tan (e+f x)}{2 b f}}{4 b}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {\frac {d \left (5 a^2 C-4 a b B+4 A b^2+b^2 C\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}-\frac {\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (15 a^3 C d-3 a^2 b (4 B d+5 c C)-8 A b^2 (b c-a d)+a b^2 (8 B c+7 C d)-b^3 (4 B d+7 c C)\right )}{b f}-\frac {\frac {2 \sqrt {d} \left (a^2+b^2\right ) \left (15 a^2 C d^2-6 a b d (2 B d+5 c C)+b^2 \left (8 d^2 (A-C)+20 B c d+15 c^2 C\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b}}-\frac {8 b^3 (a+i b) (c-i d)^{5/2} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a-i b}}+\frac {8 b^3 (b+i a) (c+i d)^{5/2} (A+i B-C) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+i b}}}{2 b f}}{4 b}}{b \left (a^2+b^2\right )}\) |
Int[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/( a + b*Tan[e + f*x])^(3/2),x]
(-2*(A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(5/2))/(b*(a^2 + b^2)*f*S qrt[a + b*Tan[e + f*x]]) + (((4*A*b^2 - 4*a*b*B + 5*a^2*C + b^2*C)*d*Sqrt[ a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*b*f) - (-1/2*((-8*(a + I*b)*b^3*(I*A + B - I*C)*(c - I*d)^(5/2)*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] + (8*b^3*(I*a + b)*(A + I*B - C)*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqr t[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[a + I*b] + (2*(a^2 + b^2)*Sqrt[d]*(15*a^2*C*d^2 - 6*a*b*d*(5*c*C + 2*B*d) + b ^2*(15*c^2*C + 20*B*c*d + 8*(A - C)*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[ e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[b])/(b*f) + (d*(15*a^ 3*C*d - 8*A*b^2*(b*c - a*d) - 3*a^2*b*(5*c*C + 4*B*d) - b^3*(7*c*C + 4*B*d ) + a*b^2*(8*B*c + 7*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x ]])/(b*f))/(4*b))/(b*(a^2 + b^2))
3.2.43.3.1 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[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. )*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(c + d*x)^m*(e + f*x)^ n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[P x, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0])) && !(IGtQ[m, 0] && IGtQ[n, 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[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[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] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 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] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S imp[ff/f Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n*((A + B*ff*x + C*ff^2*x^ 2)/(1 + ff^2*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f , A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Timed out.
\[\int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(3/2),x, algorithm="fricas")
\[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx=\int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
integrate((c+d*tan(f*x+e))**(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*ta n(f*x+e))**(3/2),x)
Integral((c + d*tan(e + f*x))**(5/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)* *2)/(a + b*tan(e + f*x))**(3/2), x)
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(3/2),x, algorithm="maxima")
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(3/2),x, algorithm="giac")
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx=\text {Hanged} \]