Optimal. Leaf size=661 \[ \frac {\log (\cos (e+f x)) \left (-\left (a^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+3 a^2 b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+3 a b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^3 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )}{f}-x \left (a^3 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+3 a^2 b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-3 a b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+\frac {(c+d \tan (e+f x))^3 \left (4 a^3 C d^3-3 a^2 b d^2 (3 c C-16 B d)+3 a b^2 d \left (20 d^2 (A-C)-5 B c d+2 c^2 C\right )-\left (b^3 \left (5 c d^2 (A-C)-2 B c^2 d+20 B d^3+c^3 C\right )\right )\right )}{60 d^4 f}+\frac {\left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {d \tan (e+f x) \left (a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}+\frac {b \tan (e+f x) (c+d \tan (e+f x))^3 \left (5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right )}{20 d^3 f}-\frac {(-a C d-2 b B d+b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac {C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f} \]
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Rubi [A] time = 2.38, antiderivative size = 661, 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 = {3647, 3637, 3630, 3528, 3525, 3475} \[ \frac {(c+d \tan (e+f x))^3 \left (-3 a^2 b d^2 (3 c C-16 B d)+4 a^3 C d^3+3 a b^2 d \left (20 d^2 (A-C)-5 B c d+2 c^2 C\right )+b^3 \left (-\left (5 c d^2 (A-C)-2 B c^2 d+20 B d^3+c^3 C\right )\right )\right )}{60 d^4 f}+\frac {\log (\cos (e+f x)) \left (3 a^2 b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+a^3 \left (-\left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+3 a b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^3 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )}{f}-x \left (3 a^2 b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+a^3 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-3 a b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+\frac {\left (3 a^2 b (A-C)+a^3 B-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {d \tan (e+f x) \left (3 a^2 b (A c-B d-c C)+a^3 (d (A-C)+B c)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}+\frac {b \tan (e+f x) (c+d \tan (e+f x))^3 \left (5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right )}{20 d^3 f}-\frac {(-a C d-2 b B d+b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac {C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3528
Rule 3630
Rule 3637
Rule 3647
Rubi steps
\begin {align*} \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^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 (c+d \tan (e+f x))^3}{6 d f}+\frac {\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2 \left (-3 (b c C-a (2 A-C) d)+6 (A b+a B-b C) d \tan (e+f x)-3 (b c C-2 b B d-a C d) \tan ^2(e+f x)\right ) \, dx}{6 d}\\ &=-\frac {(b c C-2 b B d-a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac {C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}+\frac {\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^2 \left (6 \left (a^2 (5 A-4 C) d^2+b^2 c (c C-2 B d)-a b d (2 c C+3 B d)\right )+30 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+6 \left (5 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \tan ^2(e+f x)\right ) \, dx}{30 d^2}\\ &=\frac {b \left (5 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^3 f}-\frac {(b c C-2 b B d-a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac {C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}-\frac {\int (c+d \tan (e+f x))^2 \left (-6 \left (4 a^3 (5 A-4 C) d^3+3 a b^2 c d (2 c C-5 B d)-3 a^2 b d^2 (3 c C+4 B d)-b^3 c \left (c^2 C-2 B c d+5 (A-C) d^2\right )\right )-120 \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)-6 \left (4 a^3 C d^3-3 a^2 b d^2 (3 c C-16 B d)+3 a b^2 d \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+5 c (A-C) d^2+20 B d^3\right )\right ) \tan ^2(e+f x)\right ) \, dx}{120 d^3}\\ &=\frac {\left (4 a^3 C d^3-3 a^2 b d^2 (3 c C-16 B d)+3 a b^2 d \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+5 c (A-C) d^2+20 B d^3\right )\right ) (c+d \tan (e+f x))^3}{60 d^4 f}+\frac {b \left (5 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^3 f}-\frac {(b c C-2 b B d-a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac {C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}-\frac {\int (c+d \tan (e+f x))^2 \left (120 \left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3-120 \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}{120 d^3}\\ &=\frac {\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))^2}{2 f}+\frac {\left (4 a^3 C d^3-3 a^2 b d^2 (3 c C-16 B d)+3 a b^2 d \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+5 c (A-C) d^2+20 B d^3\right )\right ) (c+d \tan (e+f x))^3}{60 d^4 f}+\frac {b \left (5 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^3 f}-\frac {(b c C-2 b B d-a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac {C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}-\frac {\int (c+d \tan (e+f x)) \left (-120 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 )-120 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}{120 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 ) x+\frac {d \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)}{f}+\frac {\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))^2}{2 f}+\frac {\left (4 a^3 C d^3-3 a^2 b d^2 (3 c C-16 B d)+3 a b^2 d \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+5 c (A-C) d^2+20 B d^3\right )\right ) (c+d \tan (e+f x))^3}{60 d^4 f}+\frac {b \left (5 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^3 f}-\frac {(b c C-2 b B d-a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac {C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}-\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 ) \int \tan (e+f x) \, dx\\ &=-\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 ) x+\frac {\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 ) \log (\cos (e+f x))}{f}+\frac {d \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)}{f}+\frac {\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))^2}{2 f}+\frac {\left (4 a^3 C d^3-3 a^2 b d^2 (3 c C-16 B d)+3 a b^2 d \left (2 c^2 C-5 B c d+20 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+5 c (A-C) d^2+20 B d^3\right )\right ) (c+d \tan (e+f x))^3}{60 d^4 f}+\frac {b \left (5 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^3}{20 d^3 f}-\frac {(b c C-2 b B d-a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac {C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}\\ \end {align*}
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Mathematica [C] time = 6.65, size = 573, normalized size = 0.87 \[ \frac {C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}+\frac {-\frac {3 (-a C d-2 b B d+b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f}+\frac {\frac {3 b \tan (e+f x) (c+d \tan (e+f x))^3 \left (5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right )}{2 d f}-\frac {\frac {(c+d \tan (e+f x))^3 \left (b \left (6 c \left (5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right )-120 d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )\right )-24 a d \left (5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right )\right )}{3 d f}-\frac {60 \left (d^2 \left (a^3 (B c-d (A-C))+3 a^2 b (A c+B d-c C)-3 a b^2 (B c-d (A-C))-b^3 (A c+B d-c C)\right ) \left (-i (c-i d)^2 \log (\tan (e+f x)+i)+i (c+i d)^2 \log (-\tan (e+f x)+i)-2 d^2 \tan (e+f x)\right )+d^2 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) \left (6 c d^2 \tan (e+f x)+(-d+i c)^3 \log (-\tan (e+f x)+i)-(d+i c)^3 \log (\tan (e+f x)+i)+d^3 \tan ^2(e+f x)\right )\right )}{f}}{4 d}}{5 d}}{6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 690, normalized size = 1.04 \[ \frac {10 \, C b^{3} d^{2} \tan \left (f x + e\right )^{6} + 12 \, {\left (2 \, C b^{3} c d + {\left (3 \, C a b^{2} + B b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{5} + 15 \, {\left (C b^{3} c^{2} + 2 \, {\left (3 \, C a b^{2} + B b^{3}\right )} c d + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{4} + 20 \, {\left ({\left (3 \, C a b^{2} + B b^{3}\right )} c^{2} + 2 \, {\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} c d + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{3} + 60 \, {\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^{2} - 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 )} c d - {\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^{2}\right )} f x + 30 \, {\left ({\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} c^{2} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} c d + {\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^{2}\right )} \tan \left (f x + e\right )^{2} - 30 \, {\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^{2} + 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 )} c d - {\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^{2}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 60 \, {\left ({\left (C a^{3} + 3 \, B a^{2} b + 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} c^{2} + 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 )} c d + {\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^{2}\right )} \tan \left (f x + e\right )}{60 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1807, normalized size = 2.73 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 691, normalized size = 1.05 \[ \frac {10 \, C b^{3} d^{2} \tan \left (f x + e\right )^{6} + 12 \, {\left (2 \, C b^{3} c d + {\left (3 \, C a b^{2} + B b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{5} + 15 \, {\left (C b^{3} c^{2} + 2 \, {\left (3 \, C a b^{2} + B b^{3}\right )} c d + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{4} + 20 \, {\left ({\left (3 \, C a b^{2} + B b^{3}\right )} c^{2} + 2 \, {\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} c d + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{3} + 30 \, {\left ({\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} c^{2} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} c d + {\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^{2}\right )} \tan \left (f x + e\right )^{2} + 60 \, {\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^{2} - 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 )} c d - {\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^{2}\right )} {\left (f x + e\right )} + 30 \, {\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^{2} + 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 )} c d - {\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^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 60 \, {\left ({\left (C a^{3} + 3 \, B a^{2} b + 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} c^{2} + 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 )} c d + {\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^{2}\right )} \tan \left (f x + e\right )}{60 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.29, size = 891, normalized size = 1.35 \[ x\,\left (A\,a^3\,c^2-A\,a^3\,d^2+B\,b^3\,c^2-C\,a^3\,c^2-B\,b^3\,d^2+C\,a^3\,d^2+2\,A\,b^3\,c\,d-2\,B\,a^3\,c\,d-2\,C\,b^3\,c\,d-3\,A\,a\,b^2\,c^2+3\,A\,a\,b^2\,d^2-3\,B\,a^2\,b\,c^2+3\,B\,a^2\,b\,d^2+3\,C\,a\,b^2\,c^2-3\,C\,a\,b^2\,d^2-6\,A\,a^2\,b\,c\,d+6\,B\,a\,b^2\,c\,d+6\,C\,a^2\,b\,c\,d\right )-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (B\,b^3\,c^2-A\,a^3\,d^2-b^2\,d\,\left (B\,b\,d+3\,C\,a\,d+2\,C\,b\,c\right )-C\,a^3\,c^2+C\,a^3\,d^2+2\,A\,b^3\,c\,d-2\,B\,a^3\,c\,d-3\,A\,a\,b^2\,c^2+3\,A\,a\,b^2\,d^2-3\,B\,a^2\,b\,c^2+3\,B\,a^2\,b\,d^2+3\,C\,a\,b^2\,c^2-6\,A\,a^2\,b\,c\,d+6\,B\,a\,b^2\,c\,d+6\,C\,a^2\,b\,c\,d\right )}{f}-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )\,\left (\frac {A\,b^3\,c^2}{2}-\frac {B\,a^3\,c^2}{2}-\frac {A\,b^3\,d^2}{2}+\frac {B\,a^3\,d^2}{2}-\frac {C\,b^3\,c^2}{2}+\frac {C\,b^3\,d^2}{2}-A\,a^3\,c\,d-B\,b^3\,c\,d+C\,a^3\,c\,d-\frac {3\,A\,a^2\,b\,c^2}{2}+\frac {3\,A\,a^2\,b\,d^2}{2}+\frac {3\,B\,a\,b^2\,c^2}{2}-\frac {3\,B\,a\,b^2\,d^2}{2}+\frac {3\,C\,a^2\,b\,c^2}{2}-\frac {3\,C\,a^2\,b\,d^2}{2}+3\,A\,a\,b^2\,c\,d+3\,B\,a^2\,b\,c\,d-3\,C\,a\,b^2\,c\,d\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (\frac {A\,b^3\,d^2}{4}+\frac {C\,b^3\,c^2}{4}-\frac {C\,b^3\,d^2}{4}+\frac {B\,b^3\,c\,d}{2}+\frac {3\,B\,a\,b^2\,d^2}{4}+\frac {3\,C\,a^2\,b\,d^2}{4}+\frac {3\,C\,a\,b^2\,c\,d}{2}\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {B\,b^3\,c^2}{3}-\frac {b^2\,d\,\left (B\,b\,d+3\,C\,a\,d+2\,C\,b\,c\right )}{3}+\frac {C\,a^3\,d^2}{3}+\frac {2\,A\,b^3\,c\,d}{3}+A\,a\,b^2\,d^2+B\,a^2\,b\,d^2+C\,a\,b^2\,c^2+2\,B\,a\,b^2\,c\,d+2\,C\,a^2\,b\,c\,d\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {A\,b^3\,c^2}{2}-\frac {A\,b^3\,d^2}{2}+\frac {B\,a^3\,d^2}{2}-\frac {C\,b^3\,c^2}{2}+\frac {C\,b^3\,d^2}{2}-B\,b^3\,c\,d+C\,a^3\,c\,d+\frac {3\,A\,a^2\,b\,d^2}{2}+\frac {3\,B\,a\,b^2\,c^2}{2}-\frac {3\,B\,a\,b^2\,d^2}{2}+\frac {3\,C\,a^2\,b\,c^2}{2}-\frac {3\,C\,a^2\,b\,d^2}{2}+3\,A\,a\,b^2\,c\,d+3\,B\,a^2\,b\,c\,d-3\,C\,a\,b^2\,c\,d\right )}{f}+\frac {b^2\,d\,{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (B\,b\,d+3\,C\,a\,d+2\,C\,b\,c\right )}{5\,f}+\frac {C\,b^3\,d^2\,{\mathrm {tan}\left (e+f\,x\right )}^6}{6\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.18, size = 1819, normalized size = 2.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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