Integrand size = 47, antiderivative size = 550 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {(i a+b)^3 (A-i B-C) (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(a+i b)^3 (i A-B-i C) (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac {2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac {2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f} \]
[Out]
Time = 2.85 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {3728, 3718, 3711, 3609, 3620, 3618, 65, 214} \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{3465 d^4 f}+\frac {2 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}+\frac {(a+i b)^3 (c+i d)^{3/2} (i A-B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {(b+i a)^3 (c-i d)^{3/2} (A-i B-C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{693 d^3 f}-\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f} \]
[In]
[Out]
Rule 65
Rule 214
Rule 3609
Rule 3618
Rule 3620
Rule 3711
Rule 3718
Rule 3728
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \text {arctanh}\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} \text {arctanh}\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*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1290\) vs. \(2(550)=1100\).
Time = 6.57 (sec) , antiderivative size = 1290, normalized size of antiderivative = 2.35 \[ \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 \left (\frac {(-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}}{9 d f}+\frac {2 \left (\frac {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}}{14 d f}-\frac {2 \left (\frac {2 \left (-\frac {7}{8} a d \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 )+b \left (-\frac {693}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {1}{4} c \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 )\right )\right ) (c+d \tan (e+f x))^{5/2}}{5 d f}+\frac {i \left (-\frac {7}{8} a d \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 {1}{4} b c \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 )+\frac {7}{8} a d \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 )+\frac {7}{2} i d \left (\frac {99}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {1}{4} b \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 {1}{4} 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 )\right )-b \left (-\frac {693}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {1}{4} c \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 )\right )\right ) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c-i d) \left (\frac {2 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{-c+i d}+2 \sqrt {c+d \tan (e+f x)}\right )\right )}{2 f}-\frac {i \left (-\frac {7}{8} a d \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 {1}{4} b c \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 )+\frac {7}{8} a d \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 )-\frac {7}{2} i d \left (\frac {99}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {1}{4} b \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 {1}{4} 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 )\right )-b \left (-\frac {693}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {1}{4} c \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 )\right )\right ) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c+i d) \left (\frac {2 (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{-c-i d}+2 \sqrt {c+d \tan (e+f x)}\right )\right )}{2 f}\right )}{7 d}\right )}{9 d}\right )}{11 d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(10951\) vs. \(2(507)=1014\).
Time = 0.34 (sec) , antiderivative size = 10952, normalized size of antiderivative = 19.91
method | result | size |
parts | \(\text {Expression too large to display}\) | \(10952\) |
derivativedivides | \(\text {Expression too large to display}\) | \(11056\) |
default | \(\text {Expression too large to display}\) | \(11056\) |
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Leaf count of result is larger than twice the leaf count of optimal. 84950 vs. \(2 (498) = 996\).
Time = 194.27 (sec) , antiderivative size = 84950, normalized size of antiderivative = 154.45 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Too large to display} \]
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\[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]
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Timed out. \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Hanged} \]
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