Integrand size = 47, antiderivative size = 358 \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {(a-i b)^2 (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}-\frac {(a+i b)^2 (B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f} \]
-(a-I*b)^2*(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-(a+I*b)^2*(B-I*(A-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+b*c)*(b*(4*c^4*C-B*c^3*d-2*c^2*(A-5*C)*d^2- 7*B*c*d^3+4*A*d^4)+3*a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))/d^3/(c^2+d^2)^2/f/(c +d*tan(f*x+e))^(1/2)+2/3*b^2*(4*c^2*C-B*c*d+(A+3*C)*d^2)*(c+d*tan(f*x+e))^ (1/2)/d^3/(c^2+d^2)/f-2/3*(A*d^2-B*c*d+C*c^2)*(a+b*tan(f*x+e))^2/d/(c^2+d^ 2)/f/(c+d*tan(f*x+e))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.63 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=\frac {2 C (a+b \tan (e+f x))^2}{d f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (-\frac {(-4 b c C+b B d+4 a C d) (a+b \tan (e+f x))}{d f (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {2 \left (8 b^2 c^2 C-2 b^2 B c d-16 a b c C d-A b^2 d^2+a b B d^2+8 a^2 C d^2+b^2 C d^2\right )}{3 d (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (\frac {\left (\frac {3}{2} c \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {3}{2} \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^4\right ) \left (-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )}{3 (i c+d) (c+d \tan (e+f x))^{3/2}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )}{3 (i c-d) (c+d \tan (e+f x))^{3/2}}\right )}{d}-\frac {3}{2} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \left (-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )}{(i c+d) \sqrt {c+d \tan (e+f x)}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )}{(i c-d) \sqrt {c+d \tan (e+f x)}}\right )\right )}{3 d}}{2 d f}\right )}{d} \]
Integrate[((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)) /(c + d*Tan[e + f*x])^(5/2),x]
(2*C*(a + b*Tan[e + f*x])^2)/(d*f*(c + d*Tan[e + f*x])^(3/2)) + (2*(-(((-4 *b*c*C + b*B*d + 4*a*C*d)*(a + b*Tan[e + f*x]))/(d*f*(c + d*Tan[e + f*x])^ (3/2))) - ((-2*(8*b^2*c^2*C - 2*b^2*B*c*d - 16*a*b*c*C*d - A*b^2*d^2 + a*b *B*d^2 + 8*a^2*C*d^2 + b^2*C*d^2))/(3*d*(c + d*Tan[e + f*x])^(3/2)) + (2*( (((3*c*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/2 + (3*(2*a*b*B - a^2*(A - C) + b^2*(A - C))*d^4)/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)) + Hypergeometri c2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)]/(3*(I*c - d)*(c + d*Ta n[e + f*x])^(3/2))))/d - (3*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2*(-(Hyperge ometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)]/((I*c + d)*Sqrt[c + d*Tan[e + f*x]])) + 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]])))/2))/(3*d))/(2*d*f)))/d
Time = 2.56 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.08, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.298, Rules used = {3042, 4128, 27, 3042, 4118, 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.
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{(c+d \tan (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {2 \int \frac {(a+b \tan (e+f x)) \left (b \left (4 C c^2-B d c+(A+3 C) d^2\right ) \tan ^2(e+f x)+3 d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+3 a d (A c-C c+B d)+4 b \left (C c^2-B d c+A d^2\right )\right )}{2 (c+d \tan (e+f x))^{3/2}}dx}{3 d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x)) \left (b \left (4 C c^2-B d c+(A+3 C) d^2\right ) \tan ^2(e+f x)+3 d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+3 a d (A c-C c+B d)+4 b \left (C c^2-B d c+A d^2\right )\right )}{(c+d \tan (e+f x))^{3/2}}dx}{3 d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x)) \left (b \left (4 C c^2-B d c+(A+3 C) d^2\right ) \tan (e+f x)^2+3 d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+3 a d (A c-C c+B d)+4 b \left (C c^2-B d c+A d^2\right )\right )}{(c+d \tan (e+f x))^{3/2}}dx}{3 d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4118 |
| \(\displaystyle \frac {\frac {\int \frac {\left (c^2+d^2\right ) \left (4 C c^2-B d c+(A+3 C) d^2\right ) \tan ^2(e+f x) b^2+\left (4 C c^4-B d c^3-2 (A-5 C) d^2 c^2-7 B d^3 c+4 A d^4\right ) b^2+6 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) b-3 a^2 d^2 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )-3 d^2 \left (\left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^2+2 b \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a-b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}+\frac {2 (b c-a d) \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right )\right )}{d^2 f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\left (c^2+d^2\right ) \left (4 C c^2-B d c+(A+3 C) d^2\right ) \tan (e+f x)^2 b^2+\left (4 C c^4-B d c^3-2 (A-5 C) d^2 c^2-7 B d^3 c+4 A d^4\right ) b^2+6 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) b-3 a^2 d^2 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )-3 d^2 \left (\left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^2+2 b \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a-b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}+\frac {2 (b c-a d) \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right )\right )}{d^2 f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {\frac {\int \frac {-3 \left (\left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2-2 b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) d^2-3 \left (\left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^2+2 b \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a-b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) d^2}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 b^2 \left (c^2+d^2\right ) \left (d^2 (A+3 C)-B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{d f}}{d \left (c^2+d^2\right )}+\frac {2 (b c-a d) \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right )\right )}{d^2 f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-3 \left (\left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2-2 b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) d^2-3 \left (\left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^2+2 b \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a-b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) d^2}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 b^2 \left (c^2+d^2\right ) \left (d^2 (A+3 C)-B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{d f}}{d \left (c^2+d^2\right )}+\frac {2 (b c-a d) \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right )\right )}{d^2 f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {\frac {2 (b c-a d) \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right )\right )}{d^2 f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {3}{2} d^2 (a+i b)^2 (c-i d)^2 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {3}{2} d^2 (a-i b)^2 (c+i d)^2 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 b^2 \left (c^2+d^2\right ) \left (d^2 (A+3 C)-B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{d f}}{d \left (c^2+d^2\right )}}{3 d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {\frac {2 (b c-a d) \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right )\right )}{d^2 f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {3}{2} d^2 (a+i b)^2 (c-i d)^2 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {3}{2} d^2 (a-i b)^2 (c+i d)^2 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 b^2 \left (c^2+d^2\right ) \left (d^2 (A+3 C)-B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{d f}}{d \left (c^2+d^2\right )}}{3 d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {\frac {2 (b c-a d) \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right )\right )}{d^2 f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {3 i d^2 (a-i b)^2 (c+i d)^2 (A-i B-C) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {3 i d^2 (a+i b)^2 (c-i d)^2 (A+i B-C) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}+\frac {2 b^2 \left (c^2+d^2\right ) \left (d^2 (A+3 C)-B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{d f}}{d \left (c^2+d^2\right )}}{3 d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {\frac {2 (b c-a d) \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right )\right )}{d^2 f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {3 i d^2 (a-i b)^2 (c+i d)^2 (A-i B-C) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}+\frac {3 i d^2 (a+i b)^2 (c-i d)^2 (A+i B-C) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}+\frac {2 b^2 \left (c^2+d^2\right ) \left (d^2 (A+3 C)-B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{d f}}{d \left (c^2+d^2\right )}}{3 d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {\frac {2 (b c-a d) \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right )\right )}{d^2 f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {3 d (a+i b)^2 (c-i d)^2 (A+i B-C) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}+\frac {3 d (a-i b)^2 (c+i d)^2 (A-i B-C) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b^2 \left (c^2+d^2\right ) \left (d^2 (A+3 C)-B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{d f}}{d \left (c^2+d^2\right )}}{3 d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {\frac {2 (b c-a d) \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right )\right )}{d^2 f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {3 d^2 (a-i b)^2 (c+i d)^2 (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {3 d^2 (a+i b)^2 (c-i d)^2 (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}+\frac {2 b^2 \left (c^2+d^2\right ) \left (d^2 (A+3 C)-B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{d f}}{d \left (c^2+d^2\right )}}{3 d \left (c^2+d^2\right )}\) |
Int[((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2),x]
(-2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^2)/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + ((2*(b*c - a*d)*(b*(4*c^4*C - B*c^3*d - 2*c^2*( A - 5*C)*d^2 - 7*B*c*d^3 + 4*A*d^4) + 3*a*d^2*(2*c*(A - C)*d - B*(c^2 - d^ 2))))/(d^2*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) + ((3*(a - I*b)^2*(A - I*B - C)*(c + I*d)^2*d^2*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/(Sqrt[c - I*d ]*f) + (3*(a + I*b)^2*(A + I*B - C)*(c - I*d)^2*d^2*ArcTan[Tan[e + f*x]/Sq rt[c + I*d]])/(Sqrt[c + I*d]*f) + (2*b^2*(c^2 + d^2)*(4*c^2*C - B*c*d + (A + 3*C)*d^2)*Sqrt[c + d*Tan[e + f*x]])/(d*f))/(d*(c^2 + d^2)))/(3*d*(c^2 + d^2))
3.2.23.3.1 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 - a*d))*(c^2*C - B*c*d + A*d^2)*((c + d*Tan[e + f*x])^(n + 1)/(d^2*f*(n + 1)*(c^2 + d^2))), x] + Simp[1/(d*(c^2 + d^2)) Int[(c + d*Tan[e + f*x])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b* (c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*d) *Tan[e + f*x] + b*C*(c^2 + d^2)*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[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. \(11359\) vs. \(2(325)=650\).
Time = 0.22 (sec) , antiderivative size = 11360, normalized size of antiderivative = 31.73
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(11360\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(61833\) |
| default | \(\text {Expression too large to display}\) | \(61833\) |
int((a+b*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5 /2),x,method=_RETURNVERBOSE)
Timed out. \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+ e))^(5/2),x, algorithm="fricas")
\[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Integral((a + b*tan(e + f*x))**2*(A + B*tan(e + f*x) + C*tan(e + f*x)**2)/ (c + d*tan(e + f*x))**(5/2), x)
Timed out. \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+ e))^(5/2),x, algorithm="maxima")
Timed out. \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+ e))^(5/2),x, algorithm="giac")
Time = 109.69 (sec) , antiderivative size = 88684, normalized size of antiderivative = 247.72 \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
int(((a + b*tan(e + f*x))^2*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/(c + d*tan(e + f*x))^(5/2),x)
atan((((c + d*tan(e + f*x))^(1/2)*(96*A^2*a^2*b^2*d^18*f^3 - 16*A^2*b^4*d^ 18*f^3 - 16*A^2*a^4*d^18*f^3 + 320*A^2*a^4*c^4*d^14*f^3 + 1024*A^2*a^4*c^6 *d^12*f^3 + 1440*A^2*a^4*c^8*d^10*f^3 + 1024*A^2*a^4*c^10*d^8*f^3 + 320*A^ 2*a^4*c^12*d^6*f^3 - 16*A^2*a^4*c^16*d^2*f^3 + 320*A^2*b^4*c^4*d^14*f^3 + 1024*A^2*b^4*c^6*d^12*f^3 + 1440*A^2*b^4*c^8*d^10*f^3 + 1024*A^2*b^4*c^10* d^8*f^3 + 320*A^2*b^4*c^12*d^6*f^3 - 16*A^2*b^4*c^16*d^2*f^3 - 256*A^2*a*b ^3*c*d^17*f^3 + 256*A^2*a^3*b*c*d^17*f^3 - 1280*A^2*a*b^3*c^3*d^15*f^3 - 2 304*A^2*a*b^3*c^5*d^13*f^3 - 1280*A^2*a*b^3*c^7*d^11*f^3 + 1280*A^2*a*b^3* c^9*d^9*f^3 + 2304*A^2*a*b^3*c^11*d^7*f^3 + 1280*A^2*a*b^3*c^13*d^5*f^3 + 256*A^2*a*b^3*c^15*d^3*f^3 + 1280*A^2*a^3*b*c^3*d^15*f^3 + 2304*A^2*a^3*b* c^5*d^13*f^3 + 1280*A^2*a^3*b*c^7*d^11*f^3 - 1280*A^2*a^3*b*c^9*d^9*f^3 - 2304*A^2*a^3*b*c^11*d^7*f^3 - 1280*A^2*a^3*b*c^13*d^5*f^3 - 256*A^2*a^3*b* c^15*d^3*f^3 - 1920*A^2*a^2*b^2*c^4*d^14*f^3 - 6144*A^2*a^2*b^2*c^6*d^12*f ^3 - 8640*A^2*a^2*b^2*c^8*d^10*f^3 - 6144*A^2*a^2*b^2*c^10*d^8*f^3 - 1920* A^2*a^2*b^2*c^12*d^6*f^3 + 96*A^2*a^2*b^2*c^16*d^2*f^3) + ((((8*A^2*a^4*c^ 5*f^2 + 8*A^2*b^4*c^5*f^2 - 48*A^2*a^2*b^2*c^5*f^2 - 80*A^2*a^4*c^3*d^2*f^ 2 - 80*A^2*b^4*c^3*d^2*f^2 - 32*A^2*a*b^3*d^5*f^2 + 32*A^2*a^3*b*d^5*f^2 + 40*A^2*a^4*c*d^4*f^2 + 40*A^2*b^4*c*d^4*f^2 - 160*A^2*a*b^3*c^4*d*f^2 + 1 60*A^2*a^3*b*c^4*d*f^2 + 320*A^2*a*b^3*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c*d^4 *f^2 - 320*A^2*a^3*b*c^2*d^3*f^2 + 480*A^2*a^2*b^2*c^3*d^2*f^2)^2/4 - (...