Optimal. Leaf size=804 \[ -\frac {\left (3 a b^2 \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )+a^3 \left (c^3 C-3 B c^2 d-3 c C d^2+B d^3-A \left (c^3-3 c d^2\right )\right )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (c^2+d^2\right )^3}-\frac {\left (3 a^2 b \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )-b^3 \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )-a^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) \log (\cos (e+f x))}{\left (c^2+d^2\right )^3 f}-\frac {(b c-a d) \left (b^2 \left (3 c^6 C-B c^5 d+9 c^4 C d^2-3 B c^3 d^3-c^2 (A-10 C) d^4-6 B c d^5+3 A d^6\right )+a^2 d^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+a b d^2 \left (8 c (A-C) d^3-B \left (c^4+6 c^2 d^2-3 d^4\right )\right )\right ) \log (c+d \tan (e+f x))}{d^4 \left (c^2+d^2\right )^3 f}+\frac {b^2 \left (b \left (3 c^4 C-B c^3 d+6 c^2 C d^2-3 B c d^3+(2 A+C) d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{d^3 \left (c^2+d^2\right )^2 f}-\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {\left (b \left (3 c^4 C-B c^3 d-c^2 (A-7 C) d^2-5 B c d^3+3 A d^4\right )+2 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{2 d^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \]
[Out]
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Rubi [A]
time = 1.79, antiderivative size = 804, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3726, 3718,
3707, 3698, 31, 3556} \begin {gather*} -\frac {\left (C c^2-B d c+A d^2\right ) (a+b \tan (e+f x))^3}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {\left (2 a \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) d^2+b \left (3 C c^4-B d c^3-(A-7 C) d^2 c^2-5 B d^3 c+3 A d^4\right )\right ) (a+b \tan (e+f x))^2}{2 d^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}-\frac {\left (\left (C c^3-3 B d c^2-3 C d^2 c+B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-3 b \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right ) a^2+3 b^2 \left (A c^3-C c^3+3 B d c^2-3 A d^2 c+3 C d^2 c-B d^3\right ) 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 ) x}{\left (c^2+d^2\right )^3}-\frac {\left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right ) a^3\right )+3 b \left (A c^3-C c^3+3 B d c^2-3 A d^2 c+3 C d^2 c-B d^3\right ) a^2+3 b^2 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right ) a-b^3 \left (A c^3-C c^3+3 B d c^2-3 A d^2 c+3 C d^2 c-B d^3\right )\right ) \log (\cos (e+f x))}{\left (c^2+d^2\right )^3 f}-\frac {(b c-a d) \left (a^2 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right ) d^3+a b \left (8 c (A-C) d^3-B \left (c^4+6 d^2 c^2-3 d^4\right )\right ) d^2+b^2 \left (3 C c^6-B d c^5+9 C d^2 c^4-3 B d^3 c^3-(A-10 C) d^4 c^2-6 B d^5 c+3 A d^6\right )\right ) \log (c+d \tan (e+f x))}{d^4 \left (c^2+d^2\right )^3 f}+\frac {b^2 \left (a \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) d^2+b \left (3 C c^4-B d c^3+6 C d^2 c^2-3 B d^3 c+(2 A+C) d^4\right )\right ) \tan (e+f x)}{d^3 \left (c^2+d^2\right )^2 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 3556
Rule 3698
Rule 3707
Rule 3718
Rule 3726
Rubi steps
\begin {align*} \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx &=-\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {\int \frac {(a+b \tan (e+f x))^2 \left (A d (2 a c+3 b d)+(3 b c-2 a d) (c C-B d)+2 d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+b \left (3 c^2 C-B c d+(A+2 C) d^2\right ) \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx}{2 d \left (c^2+d^2\right )}\\ &=-\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {\left (b \left (3 c^4 C-B c^3 d-c^2 (A-7 C) d^2-5 B c d^3+3 A d^4\right )+2 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{2 d^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\int \frac {(a+b \tan (e+f x)) \left (d (a c+2 b d) (A d (2 a c+3 b d)+(3 b c-2 a d) (c C-B d))-(2 b c-a d) \left (2 a d^2 (B c-(A-C) d)-b \left (3 c^3 C-B c^2 d-c (A-4 C) d^2-2 B d^3\right )\right )+2 d^2 ((a c+b d) ((A-C) (b c-a d)+B (a c+b d))-(b c-a d) (b B c-b (A-C) d-a (A c-c C+B d))) \tan (e+f x)+2 b \left (b \left (3 c^4 C-B c^3 d+6 c^2 C d^2-3 B c d^3+(2 A+C) d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx}{2 d^2 \left (c^2+d^2\right )^2}\\ &=\frac {b^2 \left (b \left (3 c^4 C-B c^3 d+6 c^2 C d^2-3 B c d^3+(2 A+C) d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{d^3 \left (c^2+d^2\right )^2 f}-\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {\left (b \left (3 c^4 C-B c^3 d-c^2 (A-7 C) d^2-5 B c d^3+3 A d^4\right )+2 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{2 d^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}-\frac {\int \frac {-2 \left (3 a b^2 d \left (c^4 C-c^2 (A-3 C) d^2-2 B c d^3+A d^4\right )-b^3 c \left (3 c^4 C-B c^3 d+6 c^2 C d^2-3 B c d^3+(2 A+C) d^4\right )-a^3 d^3 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+3 a^2 b d^3 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )+2 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)+2 b^2 (3 b c C-b B d-3 a C d) \left (c^2+d^2\right )^2 \tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx}{2 d^3 \left (c^2+d^2\right )^2}\\ &=-\frac {\left (3 a 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^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 )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (c^2+d^2\right )^3}+\frac {b^2 \left (b \left (3 c^4 C-B c^3 d+6 c^2 C d^2-3 B c d^3+(2 A+C) d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{d^3 \left (c^2+d^2\right )^2 f}-\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {\left (b \left (3 c^4 C-B c^3 d-c^2 (A-7 C) d^2-5 B c d^3+3 A d^4\right )+2 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{2 d^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\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 (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^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 ) \int \tan (e+f x) \, dx}{\left (c^2+d^2\right )^3}-\frac {\left ((b c-a d) \left (b^2 \left (3 c^6 C-B c^5 d+9 c^4 C d^2-3 B c^3 d^3-c^2 (A-10 C) d^4-6 B c d^5+3 A d^6\right )+a^2 d^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+a b d^2 \left (8 c (A-C) d^3-B \left (c^4+6 c^2 d^2-3 d^4\right )\right )\right )\right ) \int \frac {1+\tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx}{d^3 \left (c^2+d^2\right )^3}\\ &=-\frac {\left (3 a 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^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 )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (c^2+d^2\right )^3}-\frac {\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 (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^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 ) \log (\cos (e+f x))}{\left (c^2+d^2\right )^3 f}+\frac {b^2 \left (b \left (3 c^4 C-B c^3 d+6 c^2 C d^2-3 B c d^3+(2 A+C) d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{d^3 \left (c^2+d^2\right )^2 f}-\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {\left (b \left (3 c^4 C-B c^3 d-c^2 (A-7 C) d^2-5 B c d^3+3 A d^4\right )+2 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{2 d^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}-\frac {\left ((b c-a d) \left (b^2 \left (3 c^6 C-B c^5 d+9 c^4 C d^2-3 B c^3 d^3-c^2 (A-10 C) d^4-6 B c d^5+3 A d^6\right )+a^2 d^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+a b d^2 \left (8 c (A-C) d^3-B \left (c^4+6 c^2 d^2-3 d^4\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{c+x} \, dx,x,d \tan (e+f x)\right )}{d^4 \left (c^2+d^2\right )^3 f}\\ &=-\frac {\left (3 a 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^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 )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (c^2+d^2\right )^3}-\frac {\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 (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^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 ) \log (\cos (e+f x))}{\left (c^2+d^2\right )^3 f}-\frac {(b c-a d) \left (b^2 \left (3 c^6 C-B c^5 d+9 c^4 C d^2-3 B c^3 d^3-c^2 (A-10 C) d^4-6 B c d^5+3 A d^6\right )+a^2 d^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )+a b d^2 \left (8 c (A-C) d^3-B \left (c^4+6 c^2 d^2-3 d^4\right )\right )\right ) \log (c+d \tan (e+f x))}{d^4 \left (c^2+d^2\right )^3 f}+\frac {b^2 \left (b \left (3 c^4 C-B c^3 d+6 c^2 C d^2-3 B c d^3+(2 A+C) d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{d^3 \left (c^2+d^2\right )^2 f}-\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^3}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {\left (b \left (3 c^4 C-B c^3 d-c^2 (A-7 C) d^2-5 B c d^3+3 A d^4\right )+2 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^2}{2 d^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 13.71, size = 1445, normalized size = 1.80 \begin {gather*} \frac {\left (3 a 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^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 )-3 a^2 b \left ((A-C) d \left (-3 c^2+d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (-3 c^2+d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) (e+f x) (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}{\left (c^2+d^2\right )^3 f (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}-\frac {b^2 (-3 b c C+b B d+3 a C d) \log \left (1-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}{d^4 f (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}+\frac {\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 (-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^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 ) \log \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}{\left (c^2+d^2\right )^3 f (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}+\frac {(-b c+a d) \left (b^2 \left (3 c^6 C-B c^5 d+9 c^4 C d^2-3 B c^3 d^3-c^2 (A-10 C) d^4-6 B c d^5+3 A d^6\right )+a^2 d^3 \left (-\left ((A-C) d \left (-3 c^2+d^2\right )\right )-B \left (c^3-3 c d^2\right )\right )-a b d^2 \left (8 c (-A+C) d^3+B \left (c^4+6 c^2 d^2-3 d^4\right )\right )\right ) \log \left (-2 d \tan \left (\frac {1}{2} (e+f x)\right )+c \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}{d^4 \left (c^2+d^2\right )^3 f (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}-\frac {2 b^3 C (c \cos (e+f x)+d \sin (e+f x))^3 \tan \left (\frac {1}{2} (e+f x)\right ) (a+b \tan (e+f x))^3}{d^3 f (a \cos (e+f x)+b \sin (e+f x))^3 \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (c+d \tan (e+f x))^3}+\frac {2 (b c-a d)^3 \left (c^2 C-B c d+A d^2\right ) (c \cos (e+f x)+d \sin (e+f x))^3 \left (c+2 d \tan \left (\frac {1}{2} (e+f x)\right )\right ) (a+b \tan (e+f x))^3}{c^3 d^2 \left (c^2+d^2\right ) f (a \cos (e+f x)+b \sin (e+f x))^3 \left (c+2 d \tan \left (\frac {1}{2} (e+f x)\right )-c \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2 (c+d \tan (e+f x))^3}-\frac {2 (b c-a d)^2 (c \cos (e+f x)+d \sin (e+f x))^3 \left (a d \left (c^2 (A+C) d^3+A d^5+c^5 C \tan \left (\frac {1}{2} (e+f x)\right )+c d^4 \left (-B+A \tan \left (\frac {1}{2} (e+f x)\right )\right )+c^4 d \left (C-2 B \tan \left (\frac {1}{2} (e+f x)\right )\right )-c^3 d^2 \left (B-3 A \tan \left (\frac {1}{2} (e+f x)\right )+C \tan \left (\frac {1}{2} (e+f x)\right )\right )\right )+b c \left (-A d^5+2 c^5 C \tan \left (\frac {1}{2} (e+f x)\right )+c d^4 \left (B+2 A \tan \left (\frac {1}{2} (e+f x)\right )\right )-c^4 d \left (C+B \tan \left (\frac {1}{2} (e+f x)\right )\right )-c^2 d^3 \left (A+C+3 B \tan \left (\frac {1}{2} (e+f x)\right )\right )+c^3 d^2 \left (B+4 C \tan \left (\frac {1}{2} (e+f x)\right )\right )\right )\right ) (a+b \tan (e+f x))^3}{c^3 d^3 \left (c^2+d^2\right )^2 f (a \cos (e+f x)+b \sin (e+f x))^3 \left (-2 d \tan \left (\frac {1}{2} (e+f x)\right )+c \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) (c+d \tan (e+f x))^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 1.01, size = 1271, normalized size = 1.58 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.60, size = 1117, normalized size = 1.39 \begin {gather*} \frac {\frac {2 \, C b^{3} \tan \left (f x + e\right )}{d^{3}} + \frac {2 \, {\left ({\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} + B b^{3}\right )} c^{3} + 3 \, {\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - C\right )} b^{3}\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} + B b^{3}\right )} c d^{2} - {\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - C\right )} b^{3}\right )} d^{3}\right )} {\left (f x + e\right )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {2 \, {\left (3 \, C b^{3} c^{7} + 9 \, C b^{3} c^{5} d^{2} - {\left (3 \, C a b^{2} + B b^{3}\right )} c^{6} d - 3 \, {\left (3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{3} + {\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - 10 \, C\right )} b^{3}\right )} c^{3} d^{4} - 3 \, {\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, {\left (A - 2 \, C\right )} a b^{2} + 2 \, B b^{3}\right )} c^{2} d^{5} - 3 \, {\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} c d^{6} + {\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )} d^{7}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{6} d^{4} + 3 \, c^{4} d^{6} + 3 \, c^{2} d^{8} + d^{10}} + \frac {{\left ({\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - C\right )} b^{3}\right )} c^{3} - 3 \, {\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} + B b^{3}\right )} c^{2} d - 3 \, {\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - C\right )} b^{3}\right )} c d^{2} + {\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} + B b^{3}\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {5 \, C b^{3} c^{7} + A a^{3} d^{7} - 3 \, {\left (3 \, C a b^{2} + B b^{3}\right )} c^{6} d + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A + 9 \, C\right )} b^{3}\right )} c^{5} d^{2} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, {\left (A - 7 \, C\right )} a b^{2} - 7 \, B b^{3}\right )} c^{4} d^{3} - {\left (3 \, B a^{3} + 3 \, {\left (3 \, A - 5 \, C\right )} a^{2} b - 15 \, B a b^{2} - 5 \, A b^{3}\right )} c^{3} d^{4} + {\left ({\left (5 \, A - 3 \, C\right )} a^{3} - 9 \, B a^{2} b - 9 \, A a b^{2}\right )} c^{2} d^{5} + {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{6} + 2 \, {\left (3 \, C b^{3} c^{6} d - 2 \, {\left (3 \, C a b^{2} + B b^{3}\right )} c^{5} d^{2} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A + 5 \, C\right )} b^{3}\right )} c^{4} d^{3} - 4 \, {\left (3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{4} - {\left (B a^{3} + 3 \, {\left (A - 3 \, C\right )} a^{2} b - 9 \, B a b^{2} - 3 \, A b^{3}\right )} c^{2} d^{5} + 2 \, {\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )} c d^{6} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{7}\right )} \tan \left (f x + e\right )}{c^{6} d^{4} + 2 \, c^{4} d^{6} + c^{2} d^{8} + {\left (c^{4} d^{6} + 2 \, c^{2} d^{8} + d^{10}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (c^{5} d^{5} + 2 \, c^{3} d^{7} + c d^{9}\right )} \tan \left (f x + e\right )}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2503 vs.
\(2 (804) = 1608\).
time = 4.64, size = 2503, normalized size = 3.11 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2505 vs.
\(2 (804) = 1608\).
time = 1.49, size = 2505, normalized size = 3.12 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 20.60, size = 1172, normalized size = 1.46 \begin {gather*} \frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (d^3\,\left (3\,B\,b^3\,c^4+9\,C\,a\,b^2\,c^4\right )-d^6\,\left (3\,A\,b^3\,c-3\,B\,a^3\,c-9\,A\,a^2\,b\,c+9\,B\,a\,b^2\,c+9\,C\,a^2\,b\,c\right )+d^5\,\left (3\,A\,a^3\,c^2+6\,B\,b^3\,c^2-3\,C\,a^3\,c^2-9\,A\,a\,b^2\,c^2-9\,B\,a^2\,b\,c^2+18\,C\,a\,b^2\,c^2\right )+d^4\,\left (A\,b^3\,c^3-B\,a^3\,c^3-10\,C\,b^3\,c^3-3\,A\,a^2\,b\,c^3+3\,B\,a\,b^2\,c^3+3\,C\,a^2\,b\,c^3\right )+d^7\,\left (C\,a^3-A\,a^3+3\,A\,a\,b^2+3\,B\,a^2\,b\right )+d\,\left (B\,b^3\,c^6+3\,C\,a\,b^2\,c^6\right )-3\,C\,b^3\,c^7-9\,C\,b^3\,c^5\,d^2\right )}{f\,\left (c^6\,d^4+3\,c^4\,d^6+3\,c^2\,d^8+d^{10}\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (A\,a^3+A\,b^3\,1{}\mathrm {i}-B\,a^3\,1{}\mathrm {i}+B\,b^3-C\,a^3-C\,b^3\,1{}\mathrm {i}-3\,A\,a\,b^2-A\,a^2\,b\,3{}\mathrm {i}+B\,a\,b^2\,3{}\mathrm {i}-3\,B\,a^2\,b+3\,C\,a\,b^2+C\,a^2\,b\,3{}\mathrm {i}\right )}{2\,f\,\left (-c^3\,1{}\mathrm {i}-3\,c^2\,d+c\,d^2\,3{}\mathrm {i}+d^3\right )}-\frac {\frac {A\,a^3\,d^7+5\,C\,b^3\,c^7+B\,a^3\,c\,d^6-3\,B\,b^3\,c^6\,d+5\,A\,a^3\,c^2\,d^5+5\,A\,b^3\,c^3\,d^4+A\,b^3\,c^5\,d^2-3\,B\,a^3\,c^3\,d^4-7\,B\,b^3\,c^4\,d^3-3\,C\,a^3\,c^2\,d^5+C\,a^3\,c^4\,d^3+9\,C\,b^3\,c^5\,d^2-9\,A\,a\,b^2\,c^2\,d^5+3\,A\,a\,b^2\,c^4\,d^3-9\,A\,a^2\,b\,c^3\,d^4+15\,B\,a\,b^2\,c^3\,d^4+3\,B\,a\,b^2\,c^5\,d^2-9\,B\,a^2\,b\,c^2\,d^5+3\,B\,a^2\,b\,c^4\,d^3-21\,C\,a\,b^2\,c^4\,d^3+15\,C\,a^2\,b\,c^3\,d^4+3\,C\,a^2\,b\,c^5\,d^2+3\,A\,a^2\,b\,c\,d^6-9\,C\,a\,b^2\,c^6\,d}{2\,d\,\left (c^4+2\,c^2\,d^2+d^4\right )}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (B\,a^3\,d^6+3\,C\,b^3\,c^6+3\,A\,a^2\,b\,d^6+2\,A\,a^3\,c\,d^5-2\,B\,b^3\,c^5\,d-2\,C\,a^3\,c\,d^5+3\,A\,b^3\,c^2\,d^4+A\,b^3\,c^4\,d^2-B\,a^3\,c^2\,d^4-4\,B\,b^3\,c^3\,d^3+5\,C\,b^3\,c^4\,d^2-3\,A\,a^2\,b\,c^2\,d^4+9\,B\,a\,b^2\,c^2\,d^4+3\,B\,a\,b^2\,c^4\,d^2-12\,C\,a\,b^2\,c^3\,d^3+9\,C\,a^2\,b\,c^2\,d^4+3\,C\,a^2\,b\,c^4\,d^2-6\,A\,a\,b^2\,c\,d^5-6\,B\,a^2\,b\,c\,d^5-6\,C\,a\,b^2\,c^5\,d\right )}{c^4+2\,c^2\,d^2+d^4}}{f\,\left (c^2\,d^3+2\,c\,d^4\,\mathrm {tan}\left (e+f\,x\right )+d^5\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (A\,b^3-B\,a^3-C\,b^3-3\,A\,a^2\,b+3\,B\,a\,b^2+3\,C\,a^2\,b+A\,a^3\,1{}\mathrm {i}+B\,b^3\,1{}\mathrm {i}-C\,a^3\,1{}\mathrm {i}-A\,a\,b^2\,3{}\mathrm {i}-B\,a^2\,b\,3{}\mathrm {i}+C\,a\,b^2\,3{}\mathrm {i}\right )}{2\,f\,\left (-c^3-c^2\,d\,3{}\mathrm {i}+3\,c\,d^2+d^3\,1{}\mathrm {i}\right )}+\frac {C\,b^3\,\mathrm {tan}\left (e+f\,x\right )}{d^3\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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