Optimal. Leaf size=682 \[ \frac{\left (6 a^2 b^2 d^2 \left (8 d^2 (A-C)+12 B c d+3 c^2 C\right )-4 a^3 b d^3 (2 B d+3 c C)+3 a^4 C d^4-12 a b^3 d \left (-24 c d^2 (A-C)-6 B c^2 d+16 B d^3+c^3 C\right )+b^4 \left (48 c^2 d^2 (A-C)-128 d^4 (A-C)-8 B c^3 d-192 B c d^3+3 c^4 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{64 b^{5/2} d^{5/2} f}+\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (64 b d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )+(b c-a d) \left (48 b d^2 (a B+A b-b C)+(b c-a d) (-3 a C d-8 b B d+3 b c C)\right )\right )}{64 b^2 d^2 f}+\frac{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)+(b c-a d) (-3 a C d-8 b B d+3 b c C)\right )}{96 b d^2 f}-\frac{(a-i b)^{3/2} (c-i d)^{3/2} (B+i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(a+i b)^{3/2} (c+i d)^{3/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(-3 a C d-8 b B d+3 b c C) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 11.8965, antiderivative size = 682, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {3647, 3655, 6725, 63, 217, 206, 93, 208} \[ \frac{\left (6 a^2 b^2 d^2 \left (8 d^2 (A-C)+12 B c d+3 c^2 C\right )-4 a^3 b d^3 (2 B d+3 c C)+3 a^4 C d^4-12 a b^3 d \left (-24 c d^2 (A-C)-6 B c^2 d+16 B d^3+c^3 C\right )+b^4 \left (48 c^2 d^2 (A-C)-128 d^4 (A-C)-8 B c^3 d-192 B c d^3+3 c^4 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{64 b^{5/2} d^{5/2} f}+\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (64 b d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )+(b c-a d) \left (48 b d^2 (a B+A b-b C)+(b c-a d) (-3 a C d-8 b B d+3 b c C)\right )\right )}{64 b^2 d^2 f}+\frac{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)+(b c-a d) (-3 a C d-8 b B d+3 b c C)\right )}{96 b d^2 f}-\frac{(a-i b)^{3/2} (c-i d)^{3/2} (B+i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(a+i b)^{3/2} (c+i d)^{3/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(-3 a C d-8 b B d+3 b c C) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3647
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\int \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (\frac{1}{2} (-3 b c C+a (8 A-5 C) d)+4 (A b+a B-b C) d \tan (e+f x)-\frac{1}{2} (3 b c C-8 b B d-3 a C d) \tan ^2(e+f x)\right ) \, dx}{4 d}\\ &=-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\int \frac{(c+d \tan (e+f x))^{3/2} \left (\frac{1}{4} \left (3 a^2 (16 A-15 C) d^2+b^2 c (3 c C-8 B d)-2 a b d (3 c C+20 B d)\right )+12 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+\frac{1}{4} \left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \tan ^2(e+f x)\right )}{\sqrt{a+b \tan (e+f x)}} \, dx}{12 d^2}\\ &=\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\int \frac{\sqrt{c+d \tan (e+f x)} \left (-\frac{3}{8} \left (3 a^3 C d^3-a^2 b d^2 (64 A c-55 c C-56 B d)-b^3 c \left (3 c^2 C-8 B c d-16 (A-C) d^2\right )+a b^2 d \left (9 c^2 C+64 B c d+48 (A-C) d^2\right )\right )+24 b d^2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \tan (e+f x)+\frac{3}{8} \left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+(b c-a d) \left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \tan ^2(e+f x)\right )}{\sqrt{a+b \tan (e+f x)}} \, dx}{24 b d^2}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+(b c-a d) \left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\int \frac{\frac{3}{16} \left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)-4 a b^3 d \left (3 c^3 C+46 B c^2 d+56 c (A-C) d^2-16 B d^3\right )+b^4 c \left (3 c^3 C-8 B c^2 d-80 c (A-C) d^2+64 B d^3\right )-2 a^2 b^2 d^2 \left (55 c^2 C+92 B c d-40 C d^2-8 A \left (8 c^2-5 d^2\right )\right )\right )-24 b^2 d^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{3}{16} \left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 B c d^3-128 (A-C) d^4\right )\right ) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{24 b^2 d^2}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+(b c-a d) \left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \frac{\frac{3}{16} \left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)-4 a b^3 d \left (3 c^3 C+46 B c^2 d+56 c (A-C) d^2-16 B d^3\right )+b^4 c \left (3 c^3 C-8 B c^2 d-80 c (A-C) d^2+64 B d^3\right )-2 a^2 b^2 d^2 \left (55 c^2 C+92 B c d-40 C d^2-8 A \left (8 c^2-5 d^2\right )\right )\right )-24 b^2 d^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 ) x+\frac{3}{16} \left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 B c d^3-128 (A-C) d^4\right )\right ) x^2}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{24 b^2 d^2 f}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+(b c-a d) \left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \left (\frac{3 \left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 B c d^3-128 (A-C) d^4\right )\right )}{16 \sqrt{a+b x} \sqrt{c+d x}}+\frac{24 \left (-b^2 d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-b^2 d^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 ) x\right )}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{24 b^2 d^2 f}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+(b c-a d) \left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \frac{-b^2 d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-b^2 d^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 ) x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b^2 d^2 f}+\frac{\left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 B c d^3-128 (A-C) d^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{128 b^2 d^2 f}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+(b c-a d) \left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \left (\frac{-i b^2 d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )+b^2 d^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 )}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{-i b^2 d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-b^2 d^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 )}{2 (i+x) \sqrt{a+b x} \sqrt{c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b^2 d^2 f}+\frac{\left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 B c d^3-128 (A-C) d^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b \tan (e+f x)}\right )}{64 b^3 d^2 f}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+(b c-a d) \left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\left ((a-i b)^2 (B+i (A-C)) (c-i d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{(i+x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac{\left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 B c d^3-128 (A-C) d^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{64 b^3 d^2 f}+\frac{\left (-i b^2 d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )+b^2 d^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 )\right ) \operatorname{Subst}\left (\int \frac{1}{(i-x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 b^2 d^2 f}\\ &=\frac{\left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 B c d^3-128 (A-C) d^4\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{64 b^{5/2} d^{5/2} f}+\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+(b c-a d) \left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\left ((a-i b)^2 (B+i (A-C)) (c-i d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a+i b-(-c+i d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{\left (-i b^2 d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )+b^2 d^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 )\right ) \operatorname{Subst}\left (\int \frac{1}{a+i b-(c+i d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{b^2 d^2 f}\\ &=-\frac{(a-i b)^{3/2} (B+i (A-C)) (c-i d)^{3/2} \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 )}{f}-\frac{(a+i b)^{3/2} (B-i (A-C)) (c+i d)^{3/2} \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 )}{f}+\frac{\left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 B c d^3-128 (A-C) d^4\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{64 b^{5/2} d^{5/2} f}+\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+(b c-a d) \left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}\\ \end{align*}
Mathematica [A] time = 8.17153, size = 1316, normalized size = 1.93 \[ \frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\frac{(-3 b c C+3 a d C+8 b B d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{6 d f}+\frac{\frac{\left (48 b (A b-C b+a B) d^2+(b c-a d) (3 b c C-3 a d C-8 b B d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{8 b f}+\frac{\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (24 b \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3-\frac{3}{8} (a d-b c) \left (48 b (A b-C b+a B) d^2+(b c-a d) (3 b c C-3 a d C-8 b B d)\right )\right )}{b f}+\frac{\frac{3 \sqrt{b} \sqrt{c-\frac{a d}{b}} \sqrt{\frac{1}{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left (c-\frac{a d}{b}\right )}}} \sqrt{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left (c-\frac{a d}{b}\right )}} \left (\left (3 C c^4-8 B d c^3+48 (A-C) d^2 c^2-192 B d^3 c-128 (A-C) d^4\right ) b^4-12 a d \left (C c^3-6 B d c^2-24 (A-C) d^2 c+16 B d^3\right ) b^3+6 a^2 d^2 \left (3 C c^2+12 B d c+8 (A-C) d^2\right ) b^2-4 a^3 d^3 (3 c C+2 B d) b+3 a^4 C d^4\right ) \sqrt{\frac{c+d \tan (e+f x)}{c-\frac{a d}{b}}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}} \sqrt{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left (c-\frac{a d}{b}\right )}}}\right )}{8 \sqrt{d} \sqrt{c+d \tan (e+f x)}}+\frac{24 \left (b^5 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 )-b^4 \sqrt{-b^2} d^2 \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 )\right ) \tan ^{-1}\left (\frac{\sqrt{-c-\frac{\sqrt{-b^2} d}{b}} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right )}{b^2 \sqrt{a+\sqrt{-b^2}} \sqrt{-c-\frac{\sqrt{-b^2} d}{b}}}+\frac{24 \left (-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 ) b^5-\sqrt{-b^2} d^2 \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 ) b^4\right ) \tan ^{-1}\left (\frac{\sqrt{c-\frac{\sqrt{-b^2} d}{b}} \sqrt{a+b \tan (e+f x)}}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+d \tan (e+f x)}}\right )}{b^2 \sqrt{\sqrt{-b^2}-a} \sqrt{c-\frac{\sqrt{-b^2} d}{b}}}}{b^2 f}}{2 b}}{3 d}}{4 d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( c+d\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( A+B\tan \left ( fx+e \right ) +C \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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 )}^{\frac{3}{2}}{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}\,{d x} \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(-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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