Optimal. Leaf size=550 \[ \frac{2 (c+d \tan (e+f x))^{5/2} \left (-2 a^2 b d^2 (192 c C-847 B d)+168 a^3 C d^3+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )+b^3 \left (-\left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{3465 d^4 f}+\frac{2 \left (3 a^2 b (A-C)+a^3 B-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \sqrt{c+d \tan (e+f x)} \left (3 a^2 b (A c-B d-c C)+a^3 (d (A-C)+B c)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}+\frac{2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{693 d^3 f}+\frac{(a+i b)^3 (c+i d)^{3/2} (i A-B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f}+\frac{(b+i a)^3 (c-i d)^{3/2} (A-i B-C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f}-\frac{2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f} \]
[Out]
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Rubi [A] time = 2.73385, antiderivative size = 550, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.17, Rules used = {3647, 3637, 3630, 3528, 3539, 3537, 63, 208} \[ \frac{2 (c+d \tan (e+f x))^{5/2} \left (-2 a^2 b d^2 (192 c C-847 B d)+168 a^3 C d^3+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )+b^3 \left (-\left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{3465 d^4 f}+\frac{2 \left (3 a^2 b (A-C)+a^3 B-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \sqrt{c+d \tan (e+f x)} \left (3 a^2 b (A c-B d-c C)+a^3 (d (A-C)+B c)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}+\frac{2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{693 d^3 f}+\frac{(a+i b)^3 (c+i d)^{3/2} (i A-B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f}+\frac{(b+i a)^3 (c-i d)^{3/2} (A-i B-C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f}-\frac{2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3647
Rule 3637
Rule 3630
Rule 3528
Rule 3539
Rule 3537
Rule 63
Rule 208
Rubi steps
\begin{align*} \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}+\frac{2 \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left (\frac{1}{2} (-6 b c C+a (11 A-5 C) d)+\frac{11}{2} (A b+a B-b C) d \tan (e+f x)-\frac{1}{2} (6 b c C-11 b B d-6 a C d) \tan ^2(e+f x)\right ) \, dx}{11 d}\\ &=-\frac{2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}+\frac{4 \int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \left (\frac{1}{4} \left (3 a^2 (33 A-25 C) d^2+4 b^2 c (6 c C-11 B d)-a b d (48 c C+55 B d)\right )+\frac{99}{4} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+\frac{1}{4} \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan ^2(e+f x)\right ) \, dx}{99 d^2}\\ &=\frac{2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac{2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac{8 \int (c+d \tan (e+f x))^{3/2} \left (\frac{1}{8} \left (-21 a^3 (33 A-25 C) d^3-66 a b^2 c d (4 c C-9 B d)+a^2 b d^2 (384 c C+385 B d)+2 b^3 c \left (24 c^2 C-44 B c d+99 (A-C) d^2\right )\right )-\frac{693}{8} \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)-\frac{1}{8} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) \tan ^2(e+f x)\right ) \, dx}{693 d^3}\\ &=\frac{2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac{2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac{2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac{8 \int (c+d \tan (e+f x))^{3/2} \left (\frac{693}{8} \left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3-\frac{693}{8} \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)\right ) \, dx}{693 d^3}\\ &=\frac{2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac{2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac{2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac{8 \int \sqrt{c+d \tan (e+f x)} \left (-\frac{693}{8} d^3 \left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)-3 a^2 b (B c+(A-C) d)+b^3 (B c+(A-C) d)\right )-\frac{693}{8} d^3 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \tan (e+f x)\right ) \, dx}{693 d^3}\\ &=\frac{2 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac{2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac{2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac{8 \int \frac{\frac{693}{8} d^3 \left (a^3 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-3 a b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+3 a^2 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )+\frac{693}{8} d^3 \left (3 a^2 b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^3 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 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}{693 d^3}\\ &=\frac{2 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac{2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac{2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}+\frac{1}{2} \left ((a-i b)^3 (A-i B-C) (c-i d)^2\right ) \int \frac{1+i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx+\frac{1}{2} \left ((a+i b)^3 (A+i B-C) (c+i d)^2\right ) \int \frac{1-i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx\\ &=\frac{2 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac{2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac{2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}+\frac{\left ((a-i b)^3 (i A+B-i C) (c-i d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f}-\frac{\left (i (a+i b)^3 (A+i B-C) (c+i d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}\\ &=\frac{2 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac{2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac{2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac{\left ((a-i b)^3 (A-i B-C) (c-i d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-\frac{i c}{d}+\frac{i x^2}{d}} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{d f}-\frac{\left ((a+i b)^3 (A+i B-C) (c+i d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+\frac{i c}{d}-\frac{i x^2}{d}} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{d f}\\ &=-\frac{(a-i b)^3 (i A+B-i C) (c-i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f}-\frac{(i a-b)^3 (A+i B-C) (c+i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f}+\frac{2 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac{2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac{2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}\\ \end{align*}
Mathematica [B] time = 6.4178, size = 1290, normalized size = 2.35 \[ \frac{2 C (c+d \tan (e+f x))^{5/2} (a+b \tan (e+f x))^3}{11 d f}+\frac{2 \left (\frac{(-6 b c C+6 a d C+11 b B d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}+\frac{2 \left (\frac{b \left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{14 d f}-\frac{2 \left (\frac{2 \left (b \left (\frac{1}{4} c \left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right )-\frac{693}{8} \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3\right )-\frac{7}{8} a d \left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right )\right ) (c+d \tan (e+f x))^{5/2}}{5 d f}+\frac{i \left (-\frac{7}{8} a d \left (4 c (6 c C-11 B d) b^2-a d (48 c C+55 B d) b+3 a^2 (33 A-25 C) d^2\right )+\frac{1}{4} b c \left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right )+\frac{7}{8} a d \left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right )+\frac{7}{2} i d \left (\frac{99}{4} a \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2+\frac{1}{4} b \left (4 c (6 c C-11 B d) b^2-a d (48 c C+55 B d) b+3 a^2 (33 A-25 C) d^2\right )-\frac{1}{4} b \left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right )\right )-b \left (\frac{1}{4} c \left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right )-\frac{693}{8} \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3\right )\right ) \left (\frac{2}{3} (c+d \tan (e+f x))^{3/2}+(c-i d) \left (\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right ) (c-i d)^{3/2}}{i d-c}+2 \sqrt{c+d \tan (e+f x)}\right )\right )}{2 f}-\frac{i \left (-\frac{7}{8} a d \left (4 c (6 c C-11 B d) b^2-a d (48 c C+55 B d) b+3 a^2 (33 A-25 C) d^2\right )+\frac{1}{4} b c \left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right )+\frac{7}{8} a d \left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right )-\frac{7}{2} i d \left (\frac{99}{4} a \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2+\frac{1}{4} b \left (4 c (6 c C-11 B d) b^2-a d (48 c C+55 B d) b+3 a^2 (33 A-25 C) d^2\right )-\frac{1}{4} b \left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right )\right )-b \left (\frac{1}{4} c \left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right )-\frac{693}{8} \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3\right )\right ) \left (\frac{2}{3} (c+d \tan (e+f x))^{3/2}+(c+i d) \left (\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right ) (c+i d)^{3/2}}{-c-i d}+2 \sqrt{c+d \tan (e+f x)}\right )\right )}{2 f}\right )}{7 d}\right )}{9 d}\right )}{11 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.198, size = 11056, normalized size = 20.1 \begin{align*} \text{output too large to display} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\left (e + f x \right )}\right )^{3} \left (c + d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}} \left (A + B \tan{\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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 )}^{3}{\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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