Optimal. Leaf size=545 \[ -\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac{2 (c+d \tan (e+f x))^{3/2} \left (a^2 b^2 (d (A-11 C)+3 B c)+2 a^3 b B d-5 a^4 C d-2 a b^3 (3 A c-4 B d-3 c C)-b^4 (5 A d+3 B c)\right )}{3 b^2 f \left (a^2+b^2\right )^2 \sqrt{a+b \tan (e+f x)}}-\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (2 a^2 b^2 (B c-5 C d)+2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-3 B d-2 c C)-b^4 (d (4 A+C)+2 B c)\right )}{b^3 f \left (a^2+b^2\right )^2}-\frac{(c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{5/2}}-\frac{(c+i d)^{5/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{5/2}}+\frac{d^{3/2} (-5 a C d+2 b B d+5 b c C) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{7/2} f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 11.0662, antiderivative size = 545, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184, Rules used = {3645, 3647, 3655, 6725, 63, 217, 206, 93, 208} \[ -\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac{2 (c+d \tan (e+f x))^{3/2} \left (a^2 b^2 (d (A-11 C)+3 B c)+2 a^3 b B d-5 a^4 C d-2 a b^3 (3 A c-4 B d-3 c C)-b^4 (5 A d+3 B c)\right )}{3 b^2 f \left (a^2+b^2\right )^2 \sqrt{a+b \tan (e+f x)}}-\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (2 a^2 b^2 (B c-5 C d)+2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-3 B d-2 c C)-b^4 (d (4 A+C)+2 B c)\right )}{b^3 f \left (a^2+b^2\right )^2}-\frac{(c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{5/2}}-\frac{(c+i d)^{5/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{5/2}}+\frac{d^{3/2} (-5 a C d+2 b B d+5 b c C) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{7/2} f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3645
Rule 3647
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
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))^{5/2}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac{2 \int \frac{(c+d \tan (e+f x))^{3/2} \left (\frac{1}{2} ((b B-a C) (3 b c-5 a d)+A b (3 a c+5 b d))-\frac{3}{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+3 b^2 C\right ) d \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx}{3 b \left (a^2+b^2\right )}\\ &=\frac{2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac{4 \int \frac{\sqrt{c+d \tan (e+f x)} \left (\frac{1}{4} \left (b (a c+3 b d) ((b B-a C) (3 b c-5 a d)+A b (3 a c+5 b d))+(b c-3 a d) \left (2 a^2 b B d-5 a^3 C d-A b^2 (3 b c-a d)+3 b^3 (c C+B d)+3 a b^2 (B c-2 C d)\right )\right )+\frac{3}{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{3}{4} d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \tan ^2(e+f x)\right )}{\sqrt{a+b \tan (e+f x)}} \, dx}{3 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac{d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac{2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac{4 \int \frac{-\frac{3}{8} \left (5 a^5 C d^3+10 a^3 b^2 C d^3-a^4 b d^2 (5 c C+2 B d)-2 a^2 b^3 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+8 c C d^2+3 B d^3\right )-b^5 c \left (2 c^2 C+6 B c d-C d^2-2 A \left (c^2-3 d^2\right )\right )-a b^4 \left (4 A d \left (3 c^2-d^2\right )-C d \left (12 c^2+d^2\right )+4 B \left (c^3-3 c d^2\right )\right )\right )+\frac{3}{4} b^3 \left (2 a b \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^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )-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{3}{8} \left (a^2+b^2\right )^2 d^2 (5 b c C+2 b B d-5 a C d) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{3 b^3 \left (a^2+b^2\right )^2}\\ &=-\frac{d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac{2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac{4 \operatorname{Subst}\left (\int \frac{-\frac{3}{8} \left (5 a^5 C d^3+10 a^3 b^2 C d^3-a^4 b d^2 (5 c C+2 B d)-2 a^2 b^3 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+8 c C d^2+3 B d^3\right )-b^5 c \left (2 c^2 C+6 B c d-C d^2-2 A \left (c^2-3 d^2\right )\right )-a b^4 \left (4 A d \left (3 c^2-d^2\right )-C d \left (12 c^2+d^2\right )+4 B \left (c^3-3 c d^2\right )\right )\right )+\frac{3}{4} b^3 \left (2 a b \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^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x+\frac{3}{8} \left (a^2+b^2\right )^2 d^2 (5 b c C+2 b B d-5 a C d) x^2}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{3 b^3 \left (a^2+b^2\right )^2 f}\\ &=-\frac{d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac{2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac{4 \operatorname{Subst}\left (\int \left (\frac{3 \left (a^2+b^2\right )^2 d^2 (5 b c C+2 b B d-5 a C d)}{8 \sqrt{a+b x} \sqrt{c+d x}}+\frac{3 \left (-b^3 \left (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^2 \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 )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-b^3 \left (2 a 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 )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x\right )}{4 \sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{3 b^3 \left (a^2+b^2\right )^2 f}\\ &=-\frac{d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac{2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{-b^3 \left (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^2 \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 )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-b^3 \left (2 a 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 )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c 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^3 \left (a^2+b^2\right )^2 f}+\frac{\left (d^2 (5 b c C+2 b B d-5 a C d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 b^3 f}\\ &=-\frac{d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac{2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac{\operatorname{Subst}\left (\int \left (\frac{-i b^3 \left (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^2 \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 )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (2 a 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 )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{-i b^3 \left (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^2 \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 )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-b^3 \left (2 a 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 )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c 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^3 \left (a^2+b^2\right )^2 f}+\frac{\left (d^2 (5 b c C+2 b B d-5 a C d)\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 )}{b^4 f}\\ &=-\frac{d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac{2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac{\left ((i A+B-i C) (c-i d)^3\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 (a-i b)^2 f}+\frac{\left (d^2 (5 b c C+2 b B d-5 a C d)\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 )}{b^4 f}+\frac{\left (-i b^3 \left (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^2 \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 )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (2 a 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 )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c 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^3 \left (a^2+b^2\right )^2 f}\\ &=\frac{d^{3/2} (5 b c C+2 b B d-5 a C d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{7/2} f}-\frac{d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac{2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac{\left ((i A+B-i C) (c-i d)^3\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 )}{(a-i b)^2 f}+\frac{\left (-i b^3 \left (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^2 \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 )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (2 a 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 )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c 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^3 \left (a^2+b^2\right )^2 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)^{5/2} 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)^{5/2} f}+\frac{d^{3/2} (5 b c C+2 b B d-5 a C d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{7/2} f}-\frac{d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac{2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 46.4635, size = 2018669, normalized size = 3703.98 \[ \text{Result too large to show} \]
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{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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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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