Optimal. Leaf size=946 \[ -\frac{(i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right ) (c-i d)^{5/2}}{(a-i b)^{9/2} f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}-\frac{2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{35 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{5/2}}-\frac{(B-i (A-C)) (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{(a+i b)^{9/2} f}-\frac{2 \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^4 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^3 f (a+b \tan (e+f x))^{3/2}} \]
[Out]
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Rubi [A] time = 6.46419, antiderivative size = 946, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used = {3645, 3649, 3616, 3615, 93, 208} \[ -\frac{(i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right ) (c-i d)^{5/2}}{(a-i b)^{9/2} f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}-\frac{2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{35 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{5/2}}-\frac{(B-i (A-C)) (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{(a+i b)^{9/2} f}-\frac{2 \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^4 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^3 f (a+b \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3645
Rule 3649
Rule 3616
Rule 3615
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{9/2}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}+\frac{2 \int \frac{(c+d \tan (e+f x))^{3/2} \left (\frac{1}{2} ((b B-a C) (7 b c-5 a d)+A b (7 a c+5 b d))-\frac{7}{2} b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)-\frac{1}{2} \left (2 A b^2-2 a b B-5 a^2 C-7 b^2 C\right ) d \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx}{7 b \left (a^2+b^2\right )}\\ &=-\frac{2 \left (2 a^3 b B d+5 a^4 C d+b^4 (7 B c+5 A d)+2 a b^3 (7 A c-7 c C-6 B d)-a^2 b^2 (7 B c+9 A d-19 C d)\right ) (c+d \tan (e+f x))^{3/2}}{35 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}+\frac{4 \int \frac{\sqrt{c+d \tan (e+f x)} \left (\frac{1}{4} \left (b (5 a c+3 b d) ((b B-a C) (7 b c-5 a d)+A b (7 a c+5 b d))-(5 b c-3 a d) \left (2 a^2 b B d+5 a^3 C d+A b^2 (7 b c-9 a d)-7 b^3 (c C+B d)-7 a b^2 (B c-2 C d)\right )\right )+\frac{35}{4} b^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )-b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)+\frac{1}{4} d \left (6 a^3 b B d+15 a^4 C d-2 a b^3 (14 A c-14 c C-17 B d)-b^4 (14 B c+5 (4 A-7 C) d)+2 a^2 b^2 (7 B c+4 A d+11 C d)\right ) \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx}{35 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac{2 \left (6 a^5 b B d^2+15 a^6 C d^2+a^4 b^2 d (14 B c+8 A d+37 C d)+3 a^2 b^4 \left (35 A c^2-35 c^2 C-70 B c d-39 A d^2+54 C d^2\right )-a^3 b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right )+a b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right )+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^3 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (2 a^3 b B d+5 a^4 C d+b^4 (7 B c+5 A d)+2 a b^3 (7 A c-7 c C-6 B d)-a^2 b^2 (7 B c+9 A d-19 C d)\right ) (c+d \tan (e+f x))^{3/2}}{35 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}+\frac{8 \int \frac{\frac{1}{8} \left (6 a^5 b B d^3+15 a^6 C d^3+a^4 b^2 d^2 (14 B c+8 A d+37 C d)-a b^5 \left (315 A c^3-315 c^3 C-735 B c^2 d-497 A c d^2+497 c C d^2+71 B d^3\right )-a^3 b^3 \left (105 c^3 C+245 B c^2 d-203 c C d^2-75 B d^3-7 A \left (15 c^3-29 c d^2\right )\right )-b^6 \left (5 d \left (49 A c^2-49 c^2 C-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )+3 a^2 b^4 \left (35 B \left (3 c^3-5 c d^2\right )+d \left (245 A c^2-245 c^2 C-39 A d^2+54 C d^2\right )\right )\right )-\frac{105}{8} b^3 \left (3 a^2 b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)+\frac{1}{8} d \left (6 a^5 b B d^2+15 a^6 C d^2+a^4 b^2 d (14 B c+8 A d+37 C d)-3 a^2 b^4 \left (70 A c^2-70 c^2 C-140 B c d-66 A d^2+51 C d^2\right )+b^6 \left (70 A c^2-70 c^2 C-154 B c d-90 A d^2+105 C d^2\right )-2 a b^5 \left (224 c (A-C) d+B \left (105 c^2-122 d^2\right )\right )+2 a^3 b^3 \left (56 c (A-C) d+5 B \left (7 c^2-3 d^2\right )\right )\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}} \, dx}{105 b^3 \left (a^2+b^2\right )^3}\\ &=-\frac{2 \left (6 a^5 b B d^2+15 a^6 C d^2+a^4 b^2 d (14 B c+8 A d+37 C d)+3 a^2 b^4 \left (35 A c^2-35 c^2 C-70 B c d-39 A d^2+54 C d^2\right )-a^3 b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right )+a b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right )+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^3 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (6 a^7 b B d^3+15 a^8 C d^3+2 a^6 b^2 d^2 (7 B c+4 A d+26 C d)-2 a b^7 \left (210 A c^3-210 c^3 C-525 B c^2 d-406 A c d^2+406 c C d^2+88 B d^3\right )-a^4 b^4 \left (105 B c^3+525 A c^2 d-525 c^2 C d-749 B c d^2-311 A d^3+221 C d^3\right )+2 a^2 b^6 \left (315 B c^3+875 A c^2 d-875 c^2 C d-812 B c d^2-261 A d^3+291 C d^3\right )+2 a^5 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right )-b^8 \left (5 d \left (49 A c^2-49 c^2 C-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )-2 a^3 b^5 \left (210 c^3 C+700 B c^2 d-798 c C d^2-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^4 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (2 a^3 b B d+5 a^4 C d+b^4 (7 B c+5 A d)+2 a b^3 (7 A c-7 c C-6 B d)-a^2 b^2 (7 B c+9 A d-19 C d)\right ) (c+d \tan (e+f x))^{3/2}}{35 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}-\frac{16 \int \frac{\frac{105}{16} b^3 (b c-a d) \left (6 a^2 b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^4 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )+b^4 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-4 a^3 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+4 a b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+\frac{105}{16} b^3 (b c-a d) \left (4 a^3 b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+4 a b^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+6 a^2 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )-b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{105 b^3 \left (a^2+b^2\right )^4 (b c-a d)}\\ &=-\frac{2 \left (6 a^5 b B d^2+15 a^6 C d^2+a^4 b^2 d (14 B c+8 A d+37 C d)+3 a^2 b^4 \left (35 A c^2-35 c^2 C-70 B c d-39 A d^2+54 C d^2\right )-a^3 b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right )+a b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right )+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^3 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (6 a^7 b B d^3+15 a^8 C d^3+2 a^6 b^2 d^2 (7 B c+4 A d+26 C d)-2 a b^7 \left (210 A c^3-210 c^3 C-525 B c^2 d-406 A c d^2+406 c C d^2+88 B d^3\right )-a^4 b^4 \left (105 B c^3+525 A c^2 d-525 c^2 C d-749 B c d^2-311 A d^3+221 C d^3\right )+2 a^2 b^6 \left (315 B c^3+875 A c^2 d-875 c^2 C d-812 B c d^2-261 A d^3+291 C d^3\right )+2 a^5 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right )-b^8 \left (5 d \left (49 A c^2-49 c^2 C-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )-2 a^3 b^5 \left (210 c^3 C+700 B c^2 d-798 c C d^2-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^4 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (2 a^3 b B d+5 a^4 C d+b^4 (7 B c+5 A d)+2 a b^3 (7 A c-7 c C-6 B d)-a^2 b^2 (7 B c+9 A d-19 C d)\right ) (c+d \tan (e+f x))^{3/2}}{35 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}+\frac{\left ((A-i B-C) (c-i d)^3\right ) \int \frac{1+i \tan (e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{2 (a-i b)^4}+\frac{\left ((A+i B-C) (c+i d)^3\right ) \int \frac{1-i \tan (e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{2 (a+i b)^4}\\ &=-\frac{2 \left (6 a^5 b B d^2+15 a^6 C d^2+a^4 b^2 d (14 B c+8 A d+37 C d)+3 a^2 b^4 \left (35 A c^2-35 c^2 C-70 B c d-39 A d^2+54 C d^2\right )-a^3 b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right )+a b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right )+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^3 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (6 a^7 b B d^3+15 a^8 C d^3+2 a^6 b^2 d^2 (7 B c+4 A d+26 C d)-2 a b^7 \left (210 A c^3-210 c^3 C-525 B c^2 d-406 A c d^2+406 c C d^2+88 B d^3\right )-a^4 b^4 \left (105 B c^3+525 A c^2 d-525 c^2 C d-749 B c d^2-311 A d^3+221 C d^3\right )+2 a^2 b^6 \left (315 B c^3+875 A c^2 d-875 c^2 C d-812 B c d^2-261 A d^3+291 C d^3\right )+2 a^5 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right )-b^8 \left (5 d \left (49 A c^2-49 c^2 C-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )-2 a^3 b^5 \left (210 c^3 C+700 B c^2 d-798 c C d^2-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^4 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (2 a^3 b B d+5 a^4 C d+b^4 (7 B c+5 A d)+2 a b^3 (7 A c-7 c C-6 B d)-a^2 b^2 (7 B c+9 A d-19 C d)\right ) (c+d \tan (e+f x))^{3/2}}{35 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}+\frac{\left ((A-i B-C) (c-i d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{(1-i x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a-i b)^4 f}+\frac{\left ((A+i B-C) (c+i d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{(1+i x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a+i b)^4 f}\\ &=-\frac{2 \left (6 a^5 b B d^2+15 a^6 C d^2+a^4 b^2 d (14 B c+8 A d+37 C d)+3 a^2 b^4 \left (35 A c^2-35 c^2 C-70 B c d-39 A d^2+54 C d^2\right )-a^3 b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right )+a b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right )+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^3 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (6 a^7 b B d^3+15 a^8 C d^3+2 a^6 b^2 d^2 (7 B c+4 A d+26 C d)-2 a b^7 \left (210 A c^3-210 c^3 C-525 B c^2 d-406 A c d^2+406 c C d^2+88 B d^3\right )-a^4 b^4 \left (105 B c^3+525 A c^2 d-525 c^2 C d-749 B c d^2-311 A d^3+221 C d^3\right )+2 a^2 b^6 \left (315 B c^3+875 A c^2 d-875 c^2 C d-812 B c d^2-261 A d^3+291 C d^3\right )+2 a^5 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right )-b^8 \left (5 d \left (49 A c^2-49 c^2 C-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )-2 a^3 b^5 \left (210 c^3 C+700 B c^2 d-798 c C d^2-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^4 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (2 a^3 b B d+5 a^4 C d+b^4 (7 B c+5 A d)+2 a b^3 (7 A c-7 c C-6 B d)-a^2 b^2 (7 B c+9 A d-19 C d)\right ) (c+d \tan (e+f x))^{3/2}}{35 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}+\frac{\left ((A-i B-C) (c-i d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{i a+b-(i c+d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{(a-i b)^4 f}+\frac{\left ((A+i B-C) (c+i d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{-i a+b-(-i c+d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{(a+i b)^4 f}\\ &=-\frac{(i A+B-i C) (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{(a-i b)^{9/2} f}-\frac{(B-i (A-C)) (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{(a+i b)^{9/2} f}-\frac{2 \left (6 a^5 b B d^2+15 a^6 C d^2+a^4 b^2 d (14 B c+8 A d+37 C d)+3 a^2 b^4 \left (35 A c^2-35 c^2 C-70 B c d-39 A d^2+54 C d^2\right )-a^3 b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right )+a b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right )+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^3 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (6 a^7 b B d^3+15 a^8 C d^3+2 a^6 b^2 d^2 (7 B c+4 A d+26 C d)-2 a b^7 \left (210 A c^3-210 c^3 C-525 B c^2 d-406 A c d^2+406 c C d^2+88 B d^3\right )-a^4 b^4 \left (105 B c^3+525 A c^2 d-525 c^2 C d-749 B c d^2-311 A d^3+221 C d^3\right )+2 a^2 b^6 \left (315 B c^3+875 A c^2 d-875 c^2 C d-812 B c d^2-261 A d^3+291 C d^3\right )+2 a^5 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right )-b^8 \left (5 d \left (49 A c^2-49 c^2 C-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )-2 a^3 b^5 \left (210 c^3 C+700 B c^2 d-798 c C d^2-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^4 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (2 a^3 b B d+5 a^4 C d+b^4 (7 B c+5 A d)+2 a b^3 (7 A c-7 c C-6 B d)-a^2 b^2 (7 B c+9 A d-19 C d)\right ) (c+d \tan (e+f x))^{3/2}}{35 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\\ \end{align*}
Mathematica [C] time = 52.8871, size = 2719441, normalized size = 2874.67 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{(A+B\tan \left ( fx+e \right ) +C \left ( \tan \left ( fx+e \right ) \right ) ^{2}) \left ( c+d\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}} \left ( a+b\tan \left ( fx+e \right ) \right ) ^{-{\frac{9}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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