Integrand size = 47, antiderivative size = 585 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {(a-i b)^3 (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}-\frac {(i a-b)^3 (A+i B-C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (b \left (2 c^4 C-B c^3 d+4 c^2 C d^2-3 B c d^3+2 A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b \left (3 a b d \left (8 c^4 C-2 B c^3 d-c^2 (A-17 C) d^2-8 B c d^3+(5 A+3 C) d^4\right )-b^2 \left (16 c^5 C-8 B c^4 d+2 c^3 (A+15 C) d^2-17 B c^2 d^3+8 c (A+C) d^4-3 B d^5\right )+6 a^2 d^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^4 \left (c^2+d^2\right )^2 f}+\frac {2 b^2 \left (b \left (8 c^4 C-4 B c^3 d+c^2 (A+15 C) d^2-10 B c d^3+(7 A+C) d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right )^2 f} \] Output:
-(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)^(5/2)/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)^(5/2)/f-2/3*(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))^(3/2)-2*(b*(2*A*d^4-B*c^3*d-3*B*c*d^3+2*C*c^4+4*C*c^2*d ^2)+a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))*(a+b*tan(f*x+e))^2/d^2/(c^2+d^2)^2/f/ (c+d*tan(f*x+e))^(1/2)+2/3*b*(3*a*b*d*(8*c^4*C-2*B*c^3*d-c^2*(A-17*C)*d^2- 8*B*c*d^3+(5*A+3*C)*d^4)-b^2*(16*c^5*C-8*B*c^4*d+2*c^3*(A+15*C)*d^2-17*B*c ^2*d^3+8*c*(A+C)*d^4-3*B*d^5)+6*a^2*d^3*(2*c*(A-C)*d-B*(c^2-d^2)))*(c+d*ta n(f*x+e))^(1/2)/d^4/(c^2+d^2)^2/f+2/3*b^2*(b*(8*c^4*C-4*B*c^3*d+c^2*(A+15* C)*d^2-10*B*c*d^3+(7*A+C)*d^4)+3*a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))*tan(f*x+ e)*(c+d*tan(f*x+e))^(1/2)/d^3/(c^2+d^2)^2/f
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.59 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=\frac {2 C (a+b \tan (e+f x))^3}{3 d f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (-\frac {3 (2 b c C-b B d-2 a C d) (a+b \tan (e+f x))^2}{d f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (-\frac {3 \left (b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-b B d-2 a C d)\right ) (a+b \tan (e+f x))}{2 d f (c+d \tan (e+f x))^{3/2}}-\frac {3 \left (-\frac {2 \left (-16 b^3 c^3 C+8 b^3 B c^2 d+48 a b^2 c^2 C d-2 A b^3 c d^2-18 a b^2 B c d^2-48 a^2 b c C d^2+2 b^3 c C d^2+9 a^2 b B d^3+b^3 B d^3+16 a^3 C d^3\right )}{3 d (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (\frac {\left (\frac {3}{2} c \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^4+\frac {3}{2} \left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^5\right ) \left (-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )}{3 (i c+d) (c+d \tan (e+f x))^{3/2}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )}{3 (i c-d) (c+d \tan (e+f x))^{3/2}}\right )}{d}-\frac {3}{2} \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \left (-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )}{(i c+d) \sqrt {c+d \tan (e+f x)}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )}{(i c-d) \sqrt {c+d \tan (e+f x)}}\right )\right )}{3 d}\right )}{4 d f}\right )}{d}\right )}{3 d} \] Input:
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])^(5/2),x]
Output:
(2*C*(a + b*Tan[e + f*x])^3)/(3*d*f*(c + d*Tan[e + f*x])^(3/2)) + (2*((-3* (2*b*c*C - b*B*d - 2*a*C*d)*(a + b*Tan[e + f*x])^2)/(d*f*(c + d*Tan[e + f* x])^(3/2)) + (2*((-3*(b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - b *B*d - 2*a*C*d))*(a + b*Tan[e + f*x]))/(2*d*f*(c + d*Tan[e + f*x])^(3/2)) - (3*((-2*(-16*b^3*c^3*C + 8*b^3*B*c^2*d + 48*a*b^2*c^2*C*d - 2*A*b^3*c*d^ 2 - 18*a*b^2*B*c*d^2 - 48*a^2*b*c*C*d^2 + 2*b^3*c*C*d^2 + 9*a^2*b*B*d^3 + b^3*B*d^3 + 16*a^3*C*d^3))/(3*d*(c + d*Tan[e + f*x])^(3/2)) + (2*((((3*c*( a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^3*(A - C))*d^4)/2 + (3*(3*a^2*b*B - b^3*B - a^3*(A - C) + 3*a*b^2*(A - C))*d^5)/2)*(-1/3*Hypergeometric2F1[- 3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)]/((I*c + d)*(c + d*Tan[e + f* x])^(3/2)) + Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I* d)]/(3*(I*c - d)*(c + d*Tan[e + f*x])^(3/2))))/d - (3*(a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^3*(A - C))*d^3*(-(Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)]/((I*c + d)*Sqrt[c + d*Tan[e + f*x]])) + Hyper geometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)]/((I*c - d)*Sqrt [c + d*Tan[e + f*x]])))/2))/(3*d)))/(4*d*f)))/d))/(3*d)
Time = 7.53 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.03, 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, 4128, 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.
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{(c+d \tan (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {2 \int \frac {3 (a+b \tan (e+f x))^2 \left (b \left (2 C c^2-B d c+(A+C) d^2\right ) \tan ^2(e+f x)+d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+A d (a c+2 b d)+\frac {2}{3} \left (3 b c-\frac {3 a d}{2}\right ) (c C-B d)\right )}{2 (c+d \tan (e+f x))^{3/2}}dx}{3 d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^2 \left (b \left (2 C c^2-B d c+(A+C) d^2\right ) \tan ^2(e+f x)+d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+A d (a c+2 b d)+(2 b c-a d) (c C-B d)\right )}{(c+d \tan (e+f x))^{3/2}}dx}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^2 \left (b \left (2 C c^2-B d c+(A+C) d^2\right ) \tan (e+f x)^2+d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+A d (a c+2 b d)+(2 b c-a d) (c C-B d)\right )}{(c+d \tan (e+f x))^{3/2}}dx}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\frac {2 \int \frac {(a+b \tan (e+f x)) \left (((a c+b d) ((A-C) (b c-a d)+B (a c+b d))-(b c-a d) (b B c-b (A-C) d-a (A c-C c+B d))) \tan (e+f x) d^2+2 \left (\frac {a c}{2}+2 b d\right ) (A d (a c+2 b d)+(2 b c-a d) (c C-B d)) d+b \left (3 a \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) d^2+b \left (8 C c^4-4 B d c^3+(A+15 C) d^2 c^2-10 B d^3 c+(7 A+C) d^4\right )\right ) \tan ^2(e+f x)-(4 b c-a d) \left (a d^2 (B c-(A-C) d)-b (2 c C-B d) \left (c^2+d^2\right )\right )\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {(a+b \tan (e+f x)) \left (((a c+b d) ((A-C) (b c-a d)+B (a c+b d))-(b c-a d) (b B c-b (A-C) d-a (A c-C c+B d))) \tan (e+f x) d^2+2 \left (\frac {a c}{2}+2 b d\right ) (A d (a c+2 b d)+(2 b c-a d) (c C-B d)) d+b \left (3 a \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) d^2+b \left (8 C c^4-4 B d c^3+(A+15 C) d^2 c^2-10 B d^3 c+(7 A+C) d^4\right )\right ) \tan ^2(e+f x)-(4 b c-a d) \left (a d^2 (B c-(A-C) d)-b (2 c C-B d) \left (c^2+d^2\right )\right )\right )}{\sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {(a+b \tan (e+f x)) \left (((a c+b d) ((A-C) (b c-a d)+B (a c+b d))-(b c-a d) (b B c-b (A-C) d-a (A c-C c+B d))) \tan (e+f x) d^2+2 \left (\frac {a c}{2}+2 b d\right ) (A d (a c+2 b d)+(2 b c-a d) (c C-B d)) d+b \left (3 a \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) d^2+b \left (8 C c^4-4 B d c^3+(A+15 C) d^2 c^2-10 B d^3 c+(7 A+C) d^4\right )\right ) \tan (e+f x)^2-(4 b c-a d) \left (a d^2 (B c-(A-C) d)-b (2 c C-B d) \left (c^2+d^2\right )\right )\right )}{\sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4120 |
| \(\displaystyle \frac {\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right )\right )}{3 d f}-\frac {2 \int -\frac {-2 c \left (8 C c^4-4 B d c^3+(A+15 C) d^2 c^2-10 B d^3 c+(7 A+C) d^4\right ) b^3+6 a d \left (4 C c^4-B d c^3-2 (A-5 C) d^2 c^2-7 B d^3 c+4 A d^4\right ) b^2+\left (6 a^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) d^3+3 a b \left (8 C c^4-2 B d c^3-(A-17 C) d^2 c^2-8 B d^3 c+(5 A+3 C) d^4\right ) d-b^2 \left (16 C c^5-8 B d c^4+2 (A+15 C) d^2 c^3-17 B d^3 c^2+8 (A+C) d^4 c-3 B d^5\right )\right ) \tan ^2(e+f x) b+15 a^2 d^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) b-3 a^3 d^3 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )-3 d^3 \left (\left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^3+3 b \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a-b^3 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{2 \sqrt {c+d \tan (e+f x)}}dx}{3 d}}{d \left (c^2+d^2\right )}-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {-2 c \left (8 C c^4-4 B d c^3+(A+15 C) d^2 c^2-10 B d^3 c+(7 A+C) d^4\right ) b^3+6 a d \left (4 C c^4-B d c^3-2 (A-5 C) d^2 c^2-7 B d^3 c+4 A d^4\right ) b^2+\left (6 a^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) d^3+3 a b \left (8 C c^4-2 B d c^3-(A-17 C) d^2 c^2-8 B d^3 c+(5 A+3 C) d^4\right ) d-b^2 \left (16 C c^5-8 B d c^4+2 (A+15 C) d^2 c^3-17 B d^3 c^2+8 (A+C) d^4 c-3 B d^5\right )\right ) \tan ^2(e+f x) b+15 a^2 d^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) b-3 a^3 d^3 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )-3 d^3 \left (\left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^3+3 b \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a-b^3 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{3 d}+\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right )\right )}{3 d f}}{d \left (c^2+d^2\right )}-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {-2 c \left (8 C c^4-4 B d c^3+(A+15 C) d^2 c^2-10 B d^3 c+(7 A+C) d^4\right ) b^3+6 a d \left (4 C c^4-B d c^3-2 (A-5 C) d^2 c^2-7 B d^3 c+4 A d^4\right ) b^2+\left (6 a^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) d^3+3 a b \left (8 C c^4-2 B d c^3-(A-17 C) d^2 c^2-8 B d^3 c+(5 A+3 C) d^4\right ) d-b^2 \left (16 C c^5-8 B d c^4+2 (A+15 C) d^2 c^3-17 B d^3 c^2+8 (A+C) d^4 c-3 B d^5\right )\right ) \tan (e+f x)^2 b+15 a^2 d^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) b-3 a^3 d^3 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )-3 d^3 \left (\left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^3+3 b \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a-b^3 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{3 d}+\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right )\right )}{3 d f}}{d \left (c^2+d^2\right )}-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {-3 \left (\left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^3-3 b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) d^3-3 \left (\left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^3+3 b \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a-b^3 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) \tan (e+f x) d^3}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a b d \left (-c^2 d^2 (A-17 C)+d^4 (5 A+3 C)-2 B c^3 d-8 B c d^3+8 c^4 C\right )-b^2 \left (2 c^3 d^2 (A+15 C)+8 c d^4 (A+C)-8 B c^4 d-17 B c^2 d^3-3 B d^5+16 c^5 C\right )\right )}{d f}}{3 d}+\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right )\right )}{3 d f}}{d \left (c^2+d^2\right )}-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {-3 \left (\left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^3-3 b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) d^3-3 \left (\left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^3+3 b \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a-b^3 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) \tan (e+f x) d^3}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a b d \left (-c^2 d^2 (A-17 C)+d^4 (5 A+3 C)-2 B c^3 d-8 B c d^3+8 c^4 C\right )-b^2 \left (2 c^3 d^2 (A+15 C)+8 c d^4 (A+C)-8 B c^4 d-17 B c^2 d^3-3 B d^5+16 c^5 C\right )\right )}{d f}}{3 d}+\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right )\right )}{3 d f}}{d \left (c^2+d^2\right )}-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right )\right )}{3 d f}+\frac {\frac {3}{2} d^3 (a+i b)^3 (c-i d)^2 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {3}{2} d^3 (a-i b)^3 (c+i d)^2 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a b d \left (-c^2 d^2 (A-17 C)+d^4 (5 A+3 C)-2 B c^3 d-8 B c d^3+8 c^4 C\right )-b^2 \left (2 c^3 d^2 (A+15 C)+8 c d^4 (A+C)-8 B c^4 d-17 B c^2 d^3-3 B d^5+16 c^5 C\right )\right )}{d f}}{3 d}}{d \left (c^2+d^2\right )}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right )\right )}{3 d f}+\frac {\frac {3}{2} d^3 (a+i b)^3 (c-i d)^2 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {3}{2} d^3 (a-i b)^3 (c+i d)^2 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a b d \left (-c^2 d^2 (A-17 C)+d^4 (5 A+3 C)-2 B c^3 d-8 B c d^3+8 c^4 C\right )-b^2 \left (2 c^3 d^2 (A+15 C)+8 c d^4 (A+C)-8 B c^4 d-17 B c^2 d^3-3 B d^5+16 c^5 C\right )\right )}{d f}}{3 d}}{d \left (c^2+d^2\right )}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right )\right )}{3 d f}+\frac {\frac {3 i d^3 (a-i b)^3 (c+i d)^2 (A-i B-C) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {3 i d^3 (a+i b)^3 (c-i d)^2 (A+i B-C) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a b d \left (-c^2 d^2 (A-17 C)+d^4 (5 A+3 C)-2 B c^3 d-8 B c d^3+8 c^4 C\right )-b^2 \left (2 c^3 d^2 (A+15 C)+8 c d^4 (A+C)-8 B c^4 d-17 B c^2 d^3-3 B d^5+16 c^5 C\right )\right )}{d f}}{3 d}}{d \left (c^2+d^2\right )}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right )\right )}{3 d f}+\frac {-\frac {3 i d^3 (a-i b)^3 (c+i d)^2 (A-i B-C) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}+\frac {3 i d^3 (a+i b)^3 (c-i d)^2 (A+i B-C) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a b d \left (-c^2 d^2 (A-17 C)+d^4 (5 A+3 C)-2 B c^3 d-8 B c d^3+8 c^4 C\right )-b^2 \left (2 c^3 d^2 (A+15 C)+8 c d^4 (A+C)-8 B c^4 d-17 B c^2 d^3-3 B d^5+16 c^5 C\right )\right )}{d f}}{3 d}}{d \left (c^2+d^2\right )}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right )\right )}{3 d f}+\frac {\frac {3 d^2 (a-i b)^3 (c+i d)^2 (A-i B-C) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}+\frac {3 d^2 (a+i b)^3 (c-i d)^2 (A+i B-C) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a b d \left (-c^2 d^2 (A-17 C)+d^4 (5 A+3 C)-2 B c^3 d-8 B c d^3+8 c^4 C\right )-b^2 \left (2 c^3 d^2 (A+15 C)+8 c d^4 (A+C)-8 B c^4 d-17 B c^2 d^3-3 B d^5+16 c^5 C\right )\right )}{d f}}{3 d}}{d \left (c^2+d^2\right )}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^3}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {-\frac {2 (a+b \tan (e+f x))^2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^2 \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right )\right )}{3 d f}+\frac {\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a^2 d^3 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+3 a b d \left (-c^2 d^2 (A-17 C)+d^4 (5 A+3 C)-2 B c^3 d-8 B c d^3+8 c^4 C\right )-b^2 \left (2 c^3 d^2 (A+15 C)+8 c d^4 (A+C)-8 B c^4 d-17 B c^2 d^3-3 B d^5+16 c^5 C\right )\right )}{d f}+\frac {3 d^3 (a-i b)^3 (c+i d)^2 (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {3 d^3 (a+i b)^3 (c-i d)^2 (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}}{3 d}}{d \left (c^2+d^2\right )}}{d \left (c^2+d^2\right )}\) |
Input:
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])^(5/2),x]
Output:
(-2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^3)/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + ((-2*(b*(2*c^4*C - B*c^3*d + 4*c^2*C*d^2 - 3*B* c*d^3 + 2*A*d^4) + a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*(a + b*Tan[e + f *x])^2)/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) + ((2*b^2*(b*(8*c^4*C - 4*B*c^3*d + c^2*(A + 15*C)*d^2 - 10*B*c*d^3 + (7*A + C)*d^4) + 3*a*d^2*(2 *c*(A - C)*d - B*(c^2 - d^2)))*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(3*d *f) + ((3*(a - I*b)^3*(A - I*B - C)*(c + I*d)^2*d^3*ArcTan[Tan[e + f*x]/Sq rt[c - I*d]])/(Sqrt[c - I*d]*f) + (3*(a + I*b)^3*(A + I*B - C)*(c - I*d)^2 *d^3*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) + (2*b*(3*a*b*d *(8*c^4*C - 2*B*c^3*d - c^2*(A - 17*C)*d^2 - 8*B*c*d^3 + (5*A + 3*C)*d^4) - b^2*(16*c^5*C - 8*B*c^4*d + 2*c^3*(A + 15*C)*d^2 - 17*B*c^2*d^3 + 8*c*(A + C)*d^4 - 3*B*d^5) + 6*a^2*d^3*(2*c*(A - C)*d - B*(c^2 - d^2)))*Sqrt[c + d*Tan[e + f*x]])/(d*f))/(3*d))/(d*(c^2 + d^2)))/(d*(c^2 + d^2))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(13618\) vs. \(2(550)=1100\).
Time = 0.53 (sec) , antiderivative size = 13619, normalized size of antiderivative = 23.28
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(13619\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(85156\) |
| default | \(\text {Expression too large to display}\) | \(85156\) |
Input:
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))^(5 /2),x,method=_RETURNVERBOSE)
Output:
result too large to display
Timed out. \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
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))^(5/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:
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))**(5/2),x)
Output:
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))**(5/2), x)
Timed out. \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
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))^(5/2),x, algorithm="maxima")
Output:
Timed out
Timed out. \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
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))^(5/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Hanged} \] Input:
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))^(5/2),x)
Output:
\text{Hanged}
\[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {too large to display} \] Input:
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))^(5 /2),x)
Output:
( - 2*sqrt(tan(e + f*x)*d + c)*a**4 + 3*int((sqrt(tan(e + f*x)*d + c)*tan( e + f*x)**5)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e + f*x)**2*b**3*c*d**3*f + 6*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**5)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c* d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e + f*x)*b**3*c**2*d**2*f + 3* int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**5)/(tan(e + f*x)**3*d**3 + 3*t an(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*b**3*c**3*d*f + 9 *int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**4)/(tan(e + f*x)**3*d**3 + 3* tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e + f*x)**2* a*b**2*c*d**3*f + 3*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**4)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3), x)*tan(e + f*x)**2*b**4*d**3*f + 18*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**4)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x) *c**2*d + c**3),x)*tan(e + f*x)*a*b**2*c**2*d**2*f + 6*int((sqrt(tan(e + f *x)*d + c)*tan(e + f*x)**4)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d* *2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e + f*x)*b**4*c*d**2*f + 9*int(( sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**4)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*a*b**2*c**3*d*f + 3*in t((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**4)/(tan(e + f*x)**3*d**3 + 3*tan (e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*b**4*c**2*d*f + ...