Integrand size = 47, antiderivative size = 511 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {(a-i b)^3 (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{3/2} f}-\frac {(i a-b)^3 (A+i B-C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{3/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 b \left (6 a^2 d^2 \left (12 c^2 C-5 B c d+(5 A+7 C) d^2\right )-15 a b d \left (8 c^3 C-6 B c^2 d+c (3 A+5 C) d^2-3 B d^3\right )+b^2 \left (48 c^4 C-40 B c^3 d+6 c^2 (5 A+3 C) d^2-25 B c d^3+15 (A-C) d^4\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 d^4 \left (c^2+d^2\right ) f}-\frac {2 b^2 \left (4 (b c-a d) \left (6 c^2 C-5 B c d+(5 A+C) d^2\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{15 d^3 \left (c^2+d^2\right ) f}+\frac {2 b \left (6 c^2 C-5 B c d+(5 A+C) d^2\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d^2 \left (c^2+d^2\right ) f} \]
-(a-I*b)^3*(I*A+B-I*C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I* d)^(3/2)/f-(I*a-b)^3*(A+I*B-C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2 ))/(c+I*d)^(3/2)/f+2/15*b*(6*a^2*d^2*(12*c^2*C-5*B*c*d+(5*A+7*C)*d^2)-15*a *b*d*(8*c^3*C-6*B*c^2*d+c*(3*A+5*C)*d^2-3*B*d^3)+b^2*(48*c^4*C-40*B*c^3*d+ 6*c^2*(5*A+3*C)*d^2-25*B*c*d^3+15*(A-C)*d^4))*(c+d*tan(f*x+e))^(1/2)/d^4/( c^2+d^2)/f-2/15*b^2*(4*(-a*d+b*c)*(6*c^2*C-5*B*c*d+(5*A+C)*d^2)-5*d^2*((A- C)*(-a*d+b*c)+B*(a*c+b*d)))*(c+d*tan(f*x+e))^(1/2)*tan(f*x+e)/d^3/(c^2+d^2 )/f+2/5*b*(6*c^2*C-5*B*c*d+(5*A+C)*d^2)*(c+d*tan(f*x+e))^(1/2)*(a+b*tan(f* x+e))^2/d^2/(c^2+d^2)/f-2*(A*d^2-B*c*d+C*c^2)*(a+b*tan(f*x+e))^3/d/(c^2+d^ 2)/f/(c+d*tan(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.88 (sec) , antiderivative size = 920, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\frac {2 C (a+b \tan (e+f x))^3}{5 d f \sqrt {c+d \tan (e+f x)}}+\frac {2 \left (\frac {(-6 b c C+5 b B d+6 a C d) (a+b \tan (e+f x))^2}{3 d f \sqrt {c+d \tan (e+f x)}}+\frac {2 \left (\frac {\left (15 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-5 b B d-6 a C d)\right ) (a+b \tan (e+f x))}{2 d f \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {2 \left (-48 b^3 c^3 C+40 b^3 B c^2 d+144 a b^2 c^2 C d-30 A b^3 c d^2-110 a b^2 B c d^2-144 a^2 b c C d^2+30 b^3 c C d^2+60 a A b^2 d^3+85 a^2 b B d^3-15 b^3 B d^3+48 a^3 C d^3-60 a b^2 C d^3\right )}{d \sqrt {c+d \tan (e+f x)}}+\frac {2 \left (\frac {1}{2} \left (45 a^2 A b d^3-15 A b^3 d^3+15 a^3 B d^3-45 a b^2 B d^3-45 a^2 b C d^3+15 b^3 C d^3\right ) \left (-\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}\right )+\frac {\left (-\frac {1}{2} c d \left (45 a^2 A b d^3-15 A b^3 d^3+15 a^3 B d^3-45 a b^2 B d^3-45 a^2 b C d^3+15 b^3 C d^3\right )+d^2 \left (\frac {1}{2} \left (-48 b^3 c^3 C+40 b^3 B c^2 d+144 a b^2 c^2 C d-30 A b^3 c d^2-110 a b^2 B c d^2-144 a^2 b c C d^2+30 b^3 c C d^2+15 a^3 A d^3+15 a A b^2 d^3+40 a^2 b B d^3+33 a^3 C d^3-15 a b^2 C d^3\right )+\frac {1}{2} \left (48 b^3 c^3 C-40 b^3 B c^2 d-144 a b^2 c^2 C d+30 A b^3 c d^2+110 a b^2 B c d^2+144 a^2 b c C d^2-30 b^3 c C d^2-60 a A b^2 d^3-85 a^2 b B d^3+15 b^3 B d^3-48 a^3 C d^3+60 a b^2 C d^3\right )\right )\right ) \left (-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )}{(i c+d) \sqrt {c+d \tan (e+f x)}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )}{(i c-d) \sqrt {c+d \tan (e+f x)}}\right )}{d}\right )}{d}}{4 d f}\right )}{3 d}\right )}{5 d} \]
Integrate[((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)) /(c + d*Tan[e + f*x])^(3/2),x]
(2*C*(a + b*Tan[e + f*x])^3)/(5*d*f*Sqrt[c + d*Tan[e + f*x]]) + (2*(((-6*b *c*C + 5*b*B*d + 6*a*C*d)*(a + b*Tan[e + f*x])^2)/(3*d*f*Sqrt[c + d*Tan[e + f*x]]) + (2*(((15*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 5*b *B*d - 6*a*C*d))*(a + b*Tan[e + f*x]))/(2*d*f*Sqrt[c + d*Tan[e + f*x]]) + ((-2*(-48*b^3*c^3*C + 40*b^3*B*c^2*d + 144*a*b^2*c^2*C*d - 30*A*b^3*c*d^2 - 110*a*b^2*B*c*d^2 - 144*a^2*b*c*C*d^2 + 30*b^3*c*C*d^2 + 60*a*A*b^2*d^3 + 85*a^2*b*B*d^3 - 15*b^3*B*d^3 + 48*a^3*C*d^3 - 60*a*b^2*C*d^3))/(d*Sqrt[ c + d*Tan[e + f*x]]) + (2*(((45*a^2*A*b*d^3 - 15*A*b^3*d^3 + 15*a^3*B*d^3 - 45*a*b^2*B*d^3 - 45*a^2*b*C*d^3 + 15*b^3*C*d^3)*(((-I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + (I*ArcTanh[Sqrt[c + d*Tan[ e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d]))/2 + ((-1/2*(c*d*(45*a^2*A*b*d^3 - 15*A*b^3*d^3 + 15*a^3*B*d^3 - 45*a*b^2*B*d^3 - 45*a^2*b*C*d^3 + 15*b^3*C *d^3)) + d^2*((-48*b^3*c^3*C + 40*b^3*B*c^2*d + 144*a*b^2*c^2*C*d - 30*A*b ^3*c*d^2 - 110*a*b^2*B*c*d^2 - 144*a^2*b*c*C*d^2 + 30*b^3*c*C*d^2 + 15*a^3 *A*d^3 + 15*a*A*b^2*d^3 + 40*a^2*b*B*d^3 + 33*a^3*C*d^3 - 15*a*b^2*C*d^3)/ 2 + (48*b^3*c^3*C - 40*b^3*B*c^2*d - 144*a*b^2*c^2*C*d + 30*A*b^3*c*d^2 + 110*a*b^2*B*c*d^2 + 144*a^2*b*c*C*d^2 - 30*b^3*c*C*d^2 - 60*a*A*b^2*d^3 - 85*a^2*b*B*d^3 + 15*b^3*B*d^3 - 48*a^3*C*d^3 + 60*a*b^2*C*d^3)/2))*(-(Hype rgeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)]/((I*c + d)*Sqr t[c + d*Tan[e + f*x]])) + Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e ...
Time = 3.84 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.383, Rules used = {3042, 4128, 27, 3042, 4130, 27, 3042, 4120, 27, 3042, 4113, 3042, 4022, 3042, 4020, 25, 73, 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.
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{(c+d \tan (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {2 \int \frac {(a+b \tan (e+f x))^2 \left (b \left (6 C c^2-5 B d c+(5 A+C) d^2\right ) \tan ^2(e+f x)+d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+A d (a c+6 b d)+2 \left (3 b c-\frac {a d}{2}\right ) (c C-B d)\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^2 \left (b \left (6 C c^2-5 B d c+(5 A+C) d^2\right ) \tan ^2(e+f x)+d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+A d (a c+6 b d)+(6 b c-a d) (c C-B d)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^2 \left (b \left (6 C c^2-5 B d c+(5 A+C) d^2\right ) \tan (e+f x)^2+d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+A d (a c+6 b d)+(6 b c-a d) (c C-B d)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {2 \int -\frac {(a+b \tan (e+f x)) \left (-5 \left ((B c-(A-C) d) a^2+2 b (A c-C c+B d) a-b^2 (B c-(A-C) d)\right ) \tan (e+f x) d^2-5 a (A d (a c+6 b d)+(6 b c-a d) (c C-B d)) d+b \left (4 (b c-a d) \left (6 C c^2-5 B d c+(5 A+C) d^2\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \tan ^2(e+f x)+b (4 b c+a d) \left (6 C c^2-5 B d c+(5 A+C) d^2\right )\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{5 d}+\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\int \frac {(a+b \tan (e+f x)) \left (-5 \left ((B c-(A-C) d) a^2+2 b (A c-C c+B d) a-b^2 (B c-(A-C) d)\right ) \tan (e+f x) d^2-5 a (A d (a c+6 b d)+(6 b c-a d) (c C-B d)) d+b \left (4 (b c-a d) \left (6 C c^2-5 B d c+(5 A+C) d^2\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \tan ^2(e+f x)+b (4 b c+a d) \left (6 C c^2-5 B d c+(5 A+C) d^2\right )\right )}{\sqrt {c+d \tan (e+f x)}}dx}{5 d}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\int \frac {(a+b \tan (e+f x)) \left (-5 \left ((B c-(A-C) d) a^2+2 b (A c-C c+B d) a-b^2 (B c-(A-C) d)\right ) \tan (e+f x) d^2-5 a (A d (a c+6 b d)+(6 b c-a d) (c C-B d)) d+b \left (4 (b c-a d) \left (6 C c^2-5 B d c+(5 A+C) d^2\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \tan (e+f x)^2+b (4 b c+a d) \left (6 C c^2-5 B d c+(5 A+C) d^2\right )\right )}{\sqrt {c+d \tan (e+f x)}}dx}{5 d}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4120 |
| \(\displaystyle \frac {\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (4 (b c-a d) \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{3 d f}-\frac {2 \int \frac {2 c \left (24 C c^3-20 B d c^2+3 (5 A+3 C) d^2 c-5 B d^3\right ) b^3-30 a c d \left (4 C c^2-3 B d c+(3 A+C) d^2\right ) b^2+\left (\left (48 C c^4-40 B d c^3+6 (5 A+3 C) d^2 c^2-25 B d^3 c+15 (A-C) d^4\right ) b^2-15 a d \left (8 C c^3-6 B d c^2+(3 A+5 C) d^2 c-3 B d^3\right ) b+6 a^2 d^2 \left (12 C c^2-5 B d c+(5 A+7 C) d^2\right )\right ) \tan ^2(e+f x) b+3 a^2 d^2 \left (24 C c^2-25 B d c+(25 A-C) d^2\right ) b+15 a^3 d^3 (A c-C c+B d)+15 d^3 \left ((B c-(A-C) d) a^3+3 b (A c-C c+B d) a^2-3 b^2 (B c-(A-C) d) a-b^3 (A c-C c+B d)\right ) \tan (e+f x)}{2 \sqrt {c+d \tan (e+f x)}}dx}{3 d}}{5 d}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (4 (b c-a d) \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{3 d f}-\frac {\int \frac {2 c \left (24 C c^3-20 B d c^2+3 (5 A+3 C) d^2 c-5 B d^3\right ) b^3-30 a c d \left (4 C c^2-3 B d c+(3 A+C) d^2\right ) b^2+\left (\left (48 C c^4-40 B d c^3+6 (5 A+3 C) d^2 c^2-25 B d^3 c+15 (A-C) d^4\right ) b^2-15 a d \left (8 C c^3-6 B d c^2+(3 A+5 C) d^2 c-3 B d^3\right ) b+6 a^2 d^2 \left (12 C c^2-5 B d c+(5 A+7 C) d^2\right )\right ) \tan ^2(e+f x) b+3 a^2 d^2 \left (24 C c^2-25 B d c+(25 A-C) d^2\right ) b+15 a^3 d^3 (A c-C c+B d)+15 d^3 \left ((B c-(A-C) d) a^3+3 b (A c-C c+B d) a^2-3 b^2 (B c-(A-C) d) a-b^3 (A c-C c+B d)\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{3 d}}{5 d}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (4 (b c-a d) \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{3 d f}-\frac {\int \frac {2 c \left (24 C c^3-20 B d c^2+3 (5 A+3 C) d^2 c-5 B d^3\right ) b^3-30 a c d \left (4 C c^2-3 B d c+(3 A+C) d^2\right ) b^2+\left (\left (48 C c^4-40 B d c^3+6 (5 A+3 C) d^2 c^2-25 B d^3 c+15 (A-C) d^4\right ) b^2-15 a d \left (8 C c^3-6 B d c^2+(3 A+5 C) d^2 c-3 B d^3\right ) b+6 a^2 d^2 \left (12 C c^2-5 B d c+(5 A+7 C) d^2\right )\right ) \tan (e+f x)^2 b+3 a^2 d^2 \left (24 C c^2-25 B d c+(25 A-C) d^2\right ) b+15 a^3 d^3 (A c-C c+B d)+15 d^3 \left ((B c-(A-C) d) a^3+3 b (A c-C c+B d) a^2-3 b^2 (B c-(A-C) d) a-b^3 (A c-C c+B d)\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{3 d}}{5 d}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (4 (b c-a d) \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{3 d f}-\frac {\int \frac {15 \left ((A c-C c+B d) a^3-3 b (B c-(A-C) d) a^2-3 b^2 (A c-C c+B d) a+b^3 (B c-(A-C) d)\right ) d^3+15 \left ((B c-(A-C) d) a^3+3 b (A c-C c+B d) a^2-3 b^2 (B c-(A-C) d) a-b^3 (A c-C c+B d)\right ) \tan (e+f x) d^3}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^2 \left (d^2 (5 A+7 C)-5 B c d+12 c^2 C\right )-15 a b d \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )+b^2 \left (6 c^2 d^2 (5 A+3 C)+15 d^4 (A-C)-40 B c^3 d-25 B c d^3+48 c^4 C\right )\right )}{d f}}{3 d}}{5 d}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (4 (b c-a d) \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{3 d f}-\frac {\int \frac {15 \left ((A c-C c+B d) a^3-3 b (B c-(A-C) d) a^2-3 b^2 (A c-C c+B d) a+b^3 (B c-(A-C) d)\right ) d^3+15 \left ((B c-(A-C) d) a^3+3 b (A c-C c+B d) a^2-3 b^2 (B c-(A-C) d) a-b^3 (A c-C c+B d)\right ) \tan (e+f x) d^3}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^2 \left (d^2 (5 A+7 C)-5 B c d+12 c^2 C\right )-15 a b d \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )+b^2 \left (6 c^2 d^2 (5 A+3 C)+15 d^4 (A-C)-40 B c^3 d-25 B c d^3+48 c^4 C\right )\right )}{d f}}{3 d}}{5 d}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (4 (b c-a d) \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{3 d f}-\frac {\frac {15}{2} d^3 (a+i b)^3 (c-i d) (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {15}{2} d^3 (a-i b)^3 (c+i d) (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^2 \left (d^2 (5 A+7 C)-5 B c d+12 c^2 C\right )-15 a b d \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )+b^2 \left (6 c^2 d^2 (5 A+3 C)+15 d^4 (A-C)-40 B c^3 d-25 B c d^3+48 c^4 C\right )\right )}{d f}}{3 d}}{5 d}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (4 (b c-a d) \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{3 d f}-\frac {\frac {15}{2} d^3 (a+i b)^3 (c-i d) (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {15}{2} d^3 (a-i b)^3 (c+i d) (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^2 \left (d^2 (5 A+7 C)-5 B c d+12 c^2 C\right )-15 a b d \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )+b^2 \left (6 c^2 d^2 (5 A+3 C)+15 d^4 (A-C)-40 B c^3 d-25 B c d^3+48 c^4 C\right )\right )}{d f}}{3 d}}{5 d}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (4 (b c-a d) \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{3 d f}-\frac {\frac {15 i d^3 (a-i b)^3 (c+i d) (A-i B-C) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {15 i d^3 (a+i b)^3 (c-i d) (A+i B-C) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^2 \left (d^2 (5 A+7 C)-5 B c d+12 c^2 C\right )-15 a b d \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )+b^2 \left (6 c^2 d^2 (5 A+3 C)+15 d^4 (A-C)-40 B c^3 d-25 B c d^3+48 c^4 C\right )\right )}{d f}}{3 d}}{5 d}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (4 (b c-a d) \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{3 d f}-\frac {-\frac {15 i d^3 (a-i b)^3 (c+i d) (A-i B-C) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}+\frac {15 i d^3 (a+i b)^3 (c-i d) (A+i B-C) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^2 \left (d^2 (5 A+7 C)-5 B c d+12 c^2 C\right )-15 a b d \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )+b^2 \left (6 c^2 d^2 (5 A+3 C)+15 d^4 (A-C)-40 B c^3 d-25 B c d^3+48 c^4 C\right )\right )}{d f}}{3 d}}{5 d}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (4 (b c-a d) \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{3 d f}-\frac {\frac {15 d^2 (a-i b)^3 (c+i d) (A-i B-C) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}+\frac {15 d^2 (a+i b)^3 (c-i d) (A+i B-C) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^2 \left (d^2 (5 A+7 C)-5 B c d+12 c^2 C\right )-15 a b d \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )+b^2 \left (6 c^2 d^2 (5 A+3 C)+15 d^4 (A-C)-40 B c^3 d-25 B c d^3+48 c^4 C\right )\right )}{d f}}{3 d}}{5 d}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right ) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (4 (b c-a d) \left (d^2 (5 A+C)-5 B c d+6 c^2 C\right )-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{3 d f}-\frac {\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^2 \left (d^2 (5 A+7 C)-5 B c d+12 c^2 C\right )-15 a b d \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )+b^2 \left (6 c^2 d^2 (5 A+3 C)+15 d^4 (A-C)-40 B c^3 d-25 B c d^3+48 c^4 C\right )\right )}{d f}+\frac {15 d^3 (a-i b)^3 (c+i d) (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {15 d^3 (a+i b)^3 (c-i d) (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}}{3 d}}{5 d}}{d \left (c^2+d^2\right )}\) |
Int[((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2),x]
(-2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^3)/(d*(c^2 + d^2)*f*Sqrt[ c + d*Tan[e + f*x]]) + ((2*b*(6*c^2*C - 5*B*c*d + (5*A + C)*d^2)*(a + b*Ta n[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/(5*d*f) - ((2*b^2*(4*(b*c - a*d)*( 6*c^2*C - 5*B*c*d + (5*A + C)*d^2) - 5*d^2*((A - C)*(b*c - a*d) + B*(a*c + b*d)))*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(3*d*f) - ((15*(a - I*b)^3* (A - I*B - C)*(c + I*d)*d^3*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) + (15*(a + I*b)^3*(A + I*B - C)*(c - I*d)*d^3*ArcTan[Tan[e + f*x]/ Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) + (2*b*(6*a^2*d^2*(12*c^2*C - 5*B*c*d + (5*A + 7*C)*d^2) - 15*a*b*d*(8*c^3*C - 6*B*c^2*d + c*(3*A + 5*C)*d^2 - 3*B *d^3) + b^2*(48*c^4*C - 40*B*c^3*d + 6*c^2*(5*A + 3*C)*d^2 - 25*B*c*d^3 + 15*(A - C)*d^4))*Sqrt[c + d*Tan[e + f*x]])/(d*f))/(3*d))/(5*d))/(d*(c^2 + d^2))
3.2.16.3.1 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_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)])^(n_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 2))), x] - Simp[1/(d*(n + 2)) Int[(c + d*Tan[e + f*x])^n*Si mp[b*c*C - a*A*d*(n + 2) - (A*b + a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C* d*(n + 2) - b*(c*C - B*d*(n + 2)))*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[c^2 + d^2, 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*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])))
Leaf count of result is larger than twice the leaf count of optimal. \(11254\) vs. \(2(476)=952\).
Time = 0.44 (sec) , antiderivative size = 11255, normalized size of antiderivative = 22.03
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(11255\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(49725\) |
| default | \(\text {Expression too large to display}\) | \(49725\) |
int((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(3 /2),x,method=_RETURNVERBOSE)
Timed out. \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+ e))^(3/2),x, algorithm="fricas")
\[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Integral((a + b*tan(e + f*x))**3*(A + B*tan(e + f*x) + C*tan(e + f*x)**2)/ (c + d*tan(e + f*x))**(3/2), x)
Timed out. \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+ e))^(3/2),x, algorithm="maxima")
Timed out. \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+ e))^(3/2),x, algorithm="giac")
Timed out. \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Hanged} \]