Integrand size = 47, antiderivative size = 343 \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {(a-i b)^2 (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{3/2} f}-\frac {(a+i b)^2 (B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{3/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 b \left (6 a d \left (2 c^2 C-B c d+(A+C) d^2\right )-b \left (8 c^3 C-6 B c^2 d+c (3 A+5 C) d^2-3 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}+\frac {2 b^2 \left (4 c^2 C-3 B c d+(3 A+C) d^2\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{3 d^2 \left (c^2+d^2\right ) f} \] Output:
-(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)^(3/2)/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)^(3/2)/f-2*(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))^(1/2)+2/3*b*(6*a*d*(2*c^2*C-B*c*d+(A+C)*d^2)-b*(8*c^3*C -6*B*c^2*d+c*(3*A+5*C)*d^2-3*B*d^3))*(c+d*tan(f*x+e))^(1/2)/d^3/(c^2+d^2)/ f+2/3*b^2*(4*c^2*C-3*B*c*d+(3*A+C)*d^2)*tan(f*x+e)*(c+d*tan(f*x+e))^(1/2)/ d^2/(c^2+d^2)/f
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.36 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\frac {2 C (a+b \tan (e+f x))^2}{3 d f \sqrt {c+d \tan (e+f x)}}+\frac {2 \left (\frac {(-4 b c C+3 b B d+4 a C d) (a+b \tan (e+f x))}{d f \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {2 \left (8 b^2 c^2 C-6 b^2 B c d-16 a b c C d+3 A b^2 d^2+9 a b B d^2+8 a^2 C d^2-3 b^2 C d^2\right )}{d \sqrt {c+d \tan (e+f x)}}+\frac {2 \left (\frac {3}{2} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \left (-\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}\right )+\frac {\left (-\frac {3}{2} c \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-\frac {3}{2} \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^4\right ) \left (-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )}{(i c+d) \sqrt {c+d \tan (e+f x)}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )}{(i c-d) \sqrt {c+d \tan (e+f x)}}\right )}{d}\right )}{d}}{2 d f}\right )}{3 d} \] Input:
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])^(3/2),x]
Output:
(2*C*(a + b*Tan[e + f*x])^2)/(3*d*f*Sqrt[c + d*Tan[e + f*x]]) + (2*(((-4*b *c*C + 3*b*B*d + 4*a*C*d)*(a + b*Tan[e + f*x]))/(d*f*Sqrt[c + d*Tan[e + f* x]]) + ((-2*(8*b^2*c^2*C - 6*b^2*B*c*d - 16*a*b*c*C*d + 3*A*b^2*d^2 + 9*a* b*B*d^2 + 8*a^2*C*d^2 - 3*b^2*C*d^2))/(d*Sqrt[c + d*Tan[e + f*x]]) + (2*(( 3*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2*(((-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]]/Sq rt[c + I*d]])/Sqrt[c + I*d]))/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)*(-(Hypergeometri c2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)]/((I*c + d)*Sqrt[c + d*T an[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]])))/d))/d)/(2*d*f)))/(3*d)
Time = 3.68 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.319, Rules used = {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.
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{(c+d \tan (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {2 \int \frac {(a+b \tan (e+f x)) \left (b \left (4 C c^2-3 B d c+(3 A+C) d^2\right ) \tan ^2(e+f x)+d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+A d (a c+4 b d)+2 \left (2 b c-\frac {a d}{2}\right ) (c C-B d)\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x)) \left (b \left (4 C c^2-3 B d c+(3 A+C) d^2\right ) \tan ^2(e+f x)+d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+A d (a c+4 b d)+(4 b c-a d) (c C-B d)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x)) \left (b \left (4 C c^2-3 B d c+(3 A+C) d^2\right ) \tan (e+f x)^2+d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+A d (a c+4 b d)+(4 b c-a d) (c C-B d)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4120 |
| \(\displaystyle \frac {\frac {2 b^2 \tan (e+f x) \left (d^2 (3 A+C)-3 B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {2 \int \frac {2 c \left (4 C c^2-3 B d c+(3 A+C) d^2\right ) b^2-\left (6 a d \left (2 C c^2-B d c+(A+C) d^2\right )-b \left (8 C c^3-6 B d c^2+(3 A+5 C) d^2 c-3 B d^3\right )\right ) \tan ^2(e+f x) b-3 a d (A d (a c+4 b d)+(4 b c-a d) (c C-B d))-3 d^2 \left ((B c-(A-C) d) a^2+2 b (A c-C c+B d) a-b^2 (B c-(A-C) d)\right ) \tan (e+f x)}{2 \sqrt {c+d \tan (e+f x)}}dx}{3 d}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 b^2 \tan (e+f x) \left (d^2 (3 A+C)-3 B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {\int \frac {2 c \left (4 C c^2-3 B d c+(3 A+C) d^2\right ) b^2-\left (6 a d \left (2 C c^2-B d c+(A+C) d^2\right )-b \left (8 C c^3-6 B d c^2+(3 A+5 C) d^2 c-3 B d^3\right )\right ) \tan ^2(e+f x) b-3 a d (A d (a c+4 b d)+(4 b c-a d) (c C-B d))-3 d^2 \left ((B c-(A-C) d) a^2+2 b (A c-C c+B d) a-b^2 (B c-(A-C) d)\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{3 d}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 b^2 \tan (e+f x) \left (d^2 (3 A+C)-3 B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {\int \frac {2 c \left (4 C c^2-3 B d c+(3 A+C) d^2\right ) b^2-\left (6 a d \left (2 C c^2-B d c+(A+C) d^2\right )-b \left (8 C c^3-6 B d c^2+(3 A+5 C) d^2 c-3 B d^3\right )\right ) \tan (e+f x)^2 b-3 a d (A d (a c+4 b d)+(4 b c-a d) (c C-B d))-3 d^2 \left ((B c-(A-C) d) a^2+2 b (A c-C c+B d) a-b^2 (B c-(A-C) d)\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{3 d}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {\frac {2 b^2 \tan (e+f x) \left (d^2 (3 A+C)-3 B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {\int \frac {-3 \left ((A c-C c+B d) a^2-2 b (B c-(A-C) d) a-b^2 (A c-C c+B d)\right ) d^2-3 \left ((B c-(A-C) d) a^2+2 b (A c-C c+B d) a-b^2 (B c-(A-C) d)\right ) \tan (e+f x) d^2}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a d \left (d^2 (A+C)-B c d+2 c^2 C\right )-b \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )\right )}{d f}}{3 d}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 b^2 \tan (e+f x) \left (d^2 (3 A+C)-3 B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {\int \frac {-3 \left ((A c-C c+B d) a^2-2 b (B c-(A-C) d) a-b^2 (A c-C c+B d)\right ) d^2-3 \left ((B c-(A-C) d) a^2+2 b (A c-C c+B d) a-b^2 (B c-(A-C) d)\right ) \tan (e+f x) d^2}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a d \left (d^2 (A+C)-B c d+2 c^2 C\right )-b \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )\right )}{d f}}{3 d}}{d \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^2 \tan (e+f x) \left (d^2 (3 A+C)-3 B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {-\frac {3}{2} d^2 (a+i b)^2 (c-i d) (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {3}{2} d^2 (a-i b)^2 (c+i d) (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a d \left (d^2 (A+C)-B c d+2 c^2 C\right )-b \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )\right )}{d f}}{3 d}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^2 \tan (e+f x) \left (d^2 (3 A+C)-3 B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {-\frac {3}{2} d^2 (a+i b)^2 (c-i d) (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {3}{2} d^2 (a-i b)^2 (c+i d) (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a d \left (d^2 (A+C)-B c d+2 c^2 C\right )-b \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )\right )}{d f}}{3 d}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^2 \tan (e+f x) \left (d^2 (3 A+C)-3 B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {-\frac {3 i d^2 (a-i b)^2 (c+i d) (A-i B-C) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}+\frac {3 i d^2 (a+i b)^2 (c-i d) (A+i B-C) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a d \left (d^2 (A+C)-B c d+2 c^2 C\right )-b \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )\right )}{d f}}{3 d}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^2 \tan (e+f x) \left (d^2 (3 A+C)-3 B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {\frac {3 i d^2 (a-i b)^2 (c+i d) (A-i B-C) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {3 i d^2 (a+i b)^2 (c-i d) (A+i B-C) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a d \left (d^2 (A+C)-B c d+2 c^2 C\right )-b \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )\right )}{d f}}{3 d}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^2 \tan (e+f x) \left (d^2 (3 A+C)-3 B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {-\frac {3 d (a-i b)^2 (c+i d) (A-i B-C) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}-\frac {3 d (a+i b)^2 (c-i d) (A+i B-C) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}-\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a d \left (d^2 (A+C)-B c d+2 c^2 C\right )-b \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )\right )}{d f}}{3 d}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^2 \tan (e+f x) \left (d^2 (3 A+C)-3 B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {-\frac {3 d^2 (a-i b)^2 (c+i d) (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}-\frac {3 d^2 (a+i b)^2 (c-i d) (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}-\frac {2 b \sqrt {c+d \tan (e+f x)} \left (6 a d \left (d^2 (A+C)-B c d+2 c^2 C\right )-b \left (c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right )\right )}{d f}}{3 d}}{d \left (c^2+d^2\right )}\) |
Input:
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])^(3/2),x]
Output:
(-2*(c^2*C - B*c*d + A*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*(4*c^2*C - 3*B*c*d + (3*A + C)*d^2)*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(3*d*f) - ((-3*(a - I*b)^2*(A - I*B - C)*( c + I*d)*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)*d^2*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/ (Sqrt[c + I*d]*f) - (2*b*(6*a*d*(2*c^2*C - B*c*d + (A + C)*d^2) - b*(8*c^3 *C - 6*B*c^2*d + c*(3*A + 5*C)*d^2 - 3*B*d^3))*Sqrt[c + d*Tan[e + f*x]])/( d*f))/(3*d))/(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. \(9398\) vs. \(2(312)=624\).
Time = 0.22 (sec) , antiderivative size = 9399, normalized size of antiderivative = 27.40
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(9399\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(36710\) |
| default | \(\text {Expression too large to display}\) | \(36710\) |
Input:
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))^(3 /2),x,method=_RETURNVERBOSE)
Output:
result too large to display
Timed out. \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
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))^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
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))**(3/2),x)
Output:
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))**(3/2), x)
Timed out. \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
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))^(3/2),x, algorithm="maxima")
Output:
Timed out
\[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\int { \frac {{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
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))^(3/2),x, algorithm="giac")
Output:
integrate((C*tan(f*x + e)^2 + B*tan(f*x + e) + A)*(b*tan(f*x + e) + a)^2/( d*tan(f*x + e) + c)^(3/2), x)
Time = 62.01 (sec) , antiderivative size = 54886, normalized size of antiderivative = 160.02 \[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Too large to display} \] Input:
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))^(3/2),x)
Output:
(2*(B*b^2*c^3 + B*a^2*c*d^2 - 2*B*a*b*c^2*d))/(d^2*f*(c^2 + d^2)*(c + d*ta n(e + f*x))^(1/2)) - atan((((-(((8*B^2*a^4*c^3*f^2 + 8*B^2*b^4*c^3*f^2 - 4 8*B^2*a^2*b^2*c^3*f^2 + 32*B^2*a*b^3*d^3*f^2 - 32*B^2*a^3*b*d^3*f^2 - 24*B ^2*a^4*c*d^2*f^2 - 24*B^2*b^4*c*d^2*f^2 - 96*B^2*a*b^3*c^2*d*f^2 + 96*B^2* a^3*b*c^2*d*f^2 + 144*B^2*a^2*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^ 4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(B^4*a^8 + B^4*b^8 + 4*B^4*a^2*b^6 + 6*B^4*a^4*b^4 + 4*B^4*a^6*b^2))^(1/2) - 4*B^2*a^4*c^3*f^2 - 4*B^2*b^4*c^3* f^2 + 24*B^2*a^2*b^2*c^3*f^2 - 16*B^2*a*b^3*d^3*f^2 + 16*B^2*a^3*b*d^3*f^2 + 12*B^2*a^4*c*d^2*f^2 + 12*B^2*b^4*c*d^2*f^2 + 48*B^2*a*b^3*c^2*d*f^2 - 48*B^2*a^3*b*c^2*d*f^2 - 72*B^2*a^2*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(-((( 8*B^2*a^4*c^3*f^2 + 8*B^2*b^4*c^3*f^2 - 48*B^2*a^2*b^2*c^3*f^2 + 32*B^2*a* b^3*d^3*f^2 - 32*B^2*a^3*b*d^3*f^2 - 24*B^2*a^4*c*d^2*f^2 - 24*B^2*b^4*c*d ^2*f^2 - 96*B^2*a*b^3*c^2*d*f^2 + 96*B^2*a^3*b*c^2*d*f^2 + 144*B^2*a^2*b^2 *c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f ^4)*(B^4*a^8 + B^4*b^8 + 4*B^4*a^2*b^6 + 6*B^4*a^4*b^4 + 4*B^4*a^6*b^2))^( 1/2) - 4*B^2*a^4*c^3*f^2 - 4*B^2*b^4*c^3*f^2 + 24*B^2*a^2*b^2*c^3*f^2 - 16 *B^2*a*b^3*d^3*f^2 + 16*B^2*a^3*b*d^3*f^2 + 12*B^2*a^4*c*d^2*f^2 + 12*B^2* b^4*c*d^2*f^2 + 48*B^2*a*b^3*c^2*d*f^2 - 48*B^2*a^3*b*c^2*d*f^2 - 72*B^2*a ^2*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^...
\[ \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx =\text {Too large to display} \] Input:
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))^(3 /2),x)
Output:
( - 2*sqrt(tan(e + f*x)*d + c)*a**3 + int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**4)/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*tan(e + f *x)*b**2*c*d**2*f + int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**4)/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*b**2*c**2*d*f + 2*int((sqrt (tan(e + f*x)*d + c)*tan(e + f*x)**3)/(tan(e + f*x)**2*d**2 + 2*tan(e + f* x)*c*d + c**2),x)*tan(e + f*x)*a*b*c*d**2*f + int((sqrt(tan(e + f*x)*d + c )*tan(e + f*x)**3)/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*t an(e + f*x)*b**3*d**2*f + 2*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**3) /(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*a*b*c**2*d*f + int( (sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**3)/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*b**3*c*d*f - int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*tan(e + f*x )*a**3*d**2*f + int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f* x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*tan(e + f*x)*a**2*c*d**2*f + 3* int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**2*d**2 + 2*t an(e + f*x)*c*d + c**2),x)*tan(e + f*x)*a*b**2*d**2*f - int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*a**3*c*d*f + int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan( e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*a**2*c**2*d*f + 3*int((sq rt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**2*d**2 + 2*tan(e...