Optimal. Leaf size=363 \[ -\frac {\left (a \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{a^2+b^2}-\frac {\left (b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right ) f}+\frac {\left (A b^2-a (b B-a C)\right ) (b c-a d)^3 \log (a+b \tan (e+f x))}{b^4 \left (a^2+b^2\right ) f}+\frac {d \left (b^2 d (B c+(A-C) d)+(b c-a d) (b c C+b B d-a C d)\right ) \tan (e+f x)}{b^3 f}+\frac {(b c C+b B d-a C d) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac {C (c+d \tan (e+f x))^3}{3 b f} \]
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Rubi [A]
time = 1.04, antiderivative size = 363, 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 = {3728, 3718,
3707, 3698, 31, 3556} \begin {gather*} -\frac {\log (\cos (e+f x)) \left (A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )+a \left (B c^3-3 B c d^2-3 c^2 C d+C d^3\right )+b \left (3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )\right )}{f \left (a^2+b^2\right )}-\frac {x \left (a \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )-b \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{a^2+b^2}+\frac {(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \log (a+b \tan (e+f x))}{b^4 f \left (a^2+b^2\right )}+\frac {d \tan (e+f x) \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b^3 f}+\frac {(-a C d+b B d+b c C) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac {C (c+d \tan (e+f x))^3}{3 b f} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 3556
Rule 3698
Rule 3707
Rule 3718
Rule 3728
Rubi steps
\begin {align*} \int \frac {(c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx &=\frac {C (c+d \tan (e+f x))^3}{3 b f}+\frac {\int \frac {(c+d \tan (e+f x))^2 \left (3 (A b c-a C d)+3 b (B c+(A-C) d) \tan (e+f x)+3 (b c C+b B d-a C d) \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx}{3 b}\\ &=\frac {(b c C+b B d-a C d) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac {C (c+d \tan (e+f x))^3}{3 b f}+\frac {\int \frac {(c+d \tan (e+f x)) \left (6 \left (A b^2 c^2+a d (a C d-b (2 c C+B d))\right )+6 b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) \tan (e+f x)+6 \left (b^2 d (B c+(A-C) d)+(b c-a d) (b c C+b B d-a C d)\right ) \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx}{6 b^2}\\ &=\frac {d \left (b^2 d (B c+(A-C) d)+(b c-a d) (b c C+b B d-a C d)\right ) \tan (e+f x)}{b^3 f}+\frac {(b c C+b B d-a C d) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac {C (c+d \tan (e+f x))^3}{3 b f}-\frac {\int \frac {-6 \left (A b^2 \left (b c^3-a d^3\right )-a d \left (a^2 C d^2-a b d (3 c C+B d)+b^2 \left (3 c^2 C+3 B c d-C d^2\right )\right )\right )-6 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) \tan (e+f x)+6 \left (a^3 C d^3-a^2 b d^2 (3 c C+B d)+a b^2 d \left (3 c^2 C+3 B c d+(A-C) d^2\right )-b^3 \left (c^3 C+3 B c^2 d+3 c (A-C) d^2-B d^3\right )\right ) \tan ^2(e+f x)}{a+b \tan (e+f x)} \, dx}{6 b^3}\\ &=-\frac {\left (a \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 )-b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{a^2+b^2}+\frac {d \left (b^2 d (B c+(A-C) d)+(b c-a d) (b c C+b B d-a C d)\right ) \tan (e+f x)}{b^3 f}+\frac {(b c C+b B d-a C d) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac {C (c+d \tan (e+f x))^3}{3 b f}+\frac {\left (\left (A b^2-a (b B-a C)\right ) (b c-a d)^3\right ) \int \frac {1+\tan ^2(e+f x)}{a+b \tan (e+f x)} \, dx}{b^3 \left (a^2+b^2\right )}+\frac {\left (b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )+A \left (a 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}{a^2+b^2}\\ &=-\frac {\left (a \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 )-b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{a^2+b^2}-\frac {\left (b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )+A \left (a 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 (a^2+b^2\right ) f}+\frac {d \left (b^2 d (B c+(A-C) d)+(b c-a d) (b c C+b B d-a C d)\right ) \tan (e+f x)}{b^3 f}+\frac {(b c C+b B d-a C d) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac {C (c+d \tan (e+f x))^3}{3 b f}+\frac {\left (\left (A b^2-a (b B-a C)\right ) (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (e+f x)\right )}{b^4 \left (a^2+b^2\right ) f}\\ &=-\frac {\left (a \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 )-b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{a^2+b^2}-\frac {\left (b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )+A \left (a 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 (a^2+b^2\right ) f}+\frac {\left (A b^2-a (b B-a C)\right ) (b c-a d)^3 \log (a+b \tan (e+f x))}{b^4 \left (a^2+b^2\right ) f}+\frac {d \left (b^2 d (B c+(A-C) d)+(b c-a d) (b c C+b B d-a C d)\right ) \tan (e+f x)}{b^3 f}+\frac {(b c C+b B d-a C d) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac {C (c+d \tan (e+f x))^3}{3 b f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.03, size = 255, normalized size = 0.70 \begin {gather*} \frac {\frac {3 b^2 (-i A+B+i C) (c+i d)^3 \log (i-\tan (e+f x))}{a+i b}-\frac {3 b^2 (A-i B-C) (i c+d)^3 \log (i+\tan (e+f x))}{a-i b}+\frac {6 \left (A b^2+a (-b B+a C)\right ) (b c-a d)^3 \log (a+b \tan (e+f x))}{b^2 \left (a^2+b^2\right )}+6 b d^2 (B c+(A-C) d) \tan (e+f x)+\frac {6 d (b c-a d) (b c C+b B d-a C d) \tan (e+f x)}{b}+3 (b c C+b B d-a C d) (c+d \tan (e+f x))^2+2 b C (c+d \tan (e+f x))^3}{6 b^2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 542, normalized size = 1.49
method | result | size |
norman | \(\frac {\left (A a \,c^{3}-3 A a c \,d^{2}+3 A b \,c^{2} d -A b \,d^{3}-3 B a \,c^{2} d +B a \,d^{3}+B b \,c^{3}-3 B b c \,d^{2}-C a \,c^{3}+3 C a c \,d^{2}-3 C b \,c^{2} d +C b \,d^{3}\right ) x}{a^{2}+b^{2}}+\frac {\left (A \,b^{2} d^{2}-B a b \,d^{2}+3 B \,b^{2} c d +a^{2} C \,d^{2}-3 C a b c d +3 C \,b^{2} c^{2}-C \,b^{2} d^{2}\right ) d \tan \left (f x +e \right )}{f \,b^{3}}+\frac {C \,d^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{3 b f}+\frac {d^{2} \left (B b d -a C d +3 C b c \right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 b^{2} f}+\frac {\left (3 A a \,c^{2} d -A a \,d^{3}-A b \,c^{3}+3 A b c \,d^{2}+B a \,c^{3}-3 B a c \,d^{2}+3 B b \,c^{2} d -B b \,d^{3}-3 C a \,c^{2} d +C a \,d^{3}+C b \,c^{3}-3 C b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \left (a^{2}+b^{2}\right )}-\frac {\left (A \,a^{3} d^{3} b^{2}-3 A \,b^{3} c \,d^{2} a^{2}+3 A \,b^{4} c^{2} d a -A \,b^{5} c^{3}-B \,a^{4} d^{3} b +3 B \,a^{3} c \,d^{2} b^{2}-3 B \,b^{3} c^{2} d \,a^{2}+B \,b^{4} c^{3} a +a^{5} C \,d^{3}-3 C \,a^{4} c \,d^{2} b +3 C \,a^{3} c^{2} d \,b^{2}-C \,b^{3} c^{3} a^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right ) b^{4} f}\) | \(501\) |
derivativedivides | \(\frac {\frac {\frac {\left (3 A a \,c^{2} d -A a \,d^{3}-A b \,c^{3}+3 A b c \,d^{2}+B a \,c^{3}-3 B a c \,d^{2}+3 B b \,c^{2} d -B b \,d^{3}-3 C a \,c^{2} d +C a \,d^{3}+C b \,c^{3}-3 C b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A a \,c^{3}-3 A a c \,d^{2}+3 A b \,c^{2} d -A b \,d^{3}-3 B a \,c^{2} d +B a \,d^{3}+B b \,c^{3}-3 B b c \,d^{2}-C a \,c^{3}+3 C a c \,d^{2}-3 C b \,c^{2} d +C b \,d^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{a^{2}+b^{2}}+\frac {d \left (\frac {C \,b^{2} d^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {B \,b^{2} d^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {C a b \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}+\frac {3 C \,b^{2} c d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \,b^{2} d^{2} \tan \left (f x +e \right )-B a b \,d^{2} \tan \left (f x +e \right )+3 B \,b^{2} c d \tan \left (f x +e \right )+a^{2} C \,d^{2} \tan \left (f x +e \right )-3 C a b c d \tan \left (f x +e \right )+3 C \,b^{2} c^{2} \tan \left (f x +e \right )-C \,b^{2} d^{2} \tan \left (f x +e \right )\right )}{b^{3}}+\frac {\left (-A \,a^{3} d^{3} b^{2}+3 A \,b^{3} c \,d^{2} a^{2}-3 A \,b^{4} c^{2} d a +A \,b^{5} c^{3}+B \,a^{4} d^{3} b -3 B \,a^{3} c \,d^{2} b^{2}+3 B \,b^{3} c^{2} d \,a^{2}-B \,b^{4} c^{3} a -a^{5} C \,d^{3}+3 C \,a^{4} c \,d^{2} b -3 C \,a^{3} c^{2} d \,b^{2}+C \,b^{3} c^{3} a^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )}}{f}\) | \(542\) |
default | \(\frac {\frac {\frac {\left (3 A a \,c^{2} d -A a \,d^{3}-A b \,c^{3}+3 A b c \,d^{2}+B a \,c^{3}-3 B a c \,d^{2}+3 B b \,c^{2} d -B b \,d^{3}-3 C a \,c^{2} d +C a \,d^{3}+C b \,c^{3}-3 C b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A a \,c^{3}-3 A a c \,d^{2}+3 A b \,c^{2} d -A b \,d^{3}-3 B a \,c^{2} d +B a \,d^{3}+B b \,c^{3}-3 B b c \,d^{2}-C a \,c^{3}+3 C a c \,d^{2}-3 C b \,c^{2} d +C b \,d^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{a^{2}+b^{2}}+\frac {d \left (\frac {C \,b^{2} d^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {B \,b^{2} d^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {C a b \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}+\frac {3 C \,b^{2} c d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \,b^{2} d^{2} \tan \left (f x +e \right )-B a b \,d^{2} \tan \left (f x +e \right )+3 B \,b^{2} c d \tan \left (f x +e \right )+a^{2} C \,d^{2} \tan \left (f x +e \right )-3 C a b c d \tan \left (f x +e \right )+3 C \,b^{2} c^{2} \tan \left (f x +e \right )-C \,b^{2} d^{2} \tan \left (f x +e \right )\right )}{b^{3}}+\frac {\left (-A \,a^{3} d^{3} b^{2}+3 A \,b^{3} c \,d^{2} a^{2}-3 A \,b^{4} c^{2} d a +A \,b^{5} c^{3}+B \,a^{4} d^{3} b -3 B \,a^{3} c \,d^{2} b^{2}+3 B \,b^{3} c^{2} d \,a^{2}-B \,b^{4} c^{3} a -a^{5} C \,d^{3}+3 C \,a^{4} c \,d^{2} b -3 C \,a^{3} c^{2} d \,b^{2}+C \,b^{3} c^{3} a^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )}}{f}\) | \(542\) |
risch | \(\text {Expression too large to display}\) | \(2490\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 442, normalized size = 1.22 \begin {gather*} \frac {\frac {6 \, {\left ({\left ({\left (A - C\right )} a + B b\right )} c^{3} - 3 \, {\left (B a - {\left (A - C\right )} b\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a + B b\right )} c d^{2} + {\left (B a - {\left (A - C\right )} b\right )} d^{3}\right )} {\left (f x + e\right )}}{a^{2} + b^{2}} + \frac {6 \, {\left ({\left (C a^{2} b^{3} - B a b^{4} + A b^{5}\right )} c^{3} - 3 \, {\left (C a^{3} b^{2} - B a^{2} b^{3} + A a b^{4}\right )} c^{2} d + 3 \, {\left (C a^{4} b - B a^{3} b^{2} + A a^{2} b^{3}\right )} c d^{2} - {\left (C a^{5} - B a^{4} b + A a^{3} b^{2}\right )} d^{3}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{2} b^{4} + b^{6}} + \frac {3 \, {\left ({\left (B a - {\left (A - C\right )} b\right )} c^{3} + 3 \, {\left ({\left (A - C\right )} a + B b\right )} c^{2} d - 3 \, {\left (B a - {\left (A - C\right )} b\right )} c d^{2} - {\left ({\left (A - C\right )} a + B b\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {2 \, C b^{2} d^{3} \tan \left (f x + e\right )^{3} + 3 \, {\left (3 \, C b^{2} c d^{2} - {\left (C a b - B b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} + 6 \, {\left (3 \, C b^{2} c^{2} d - 3 \, {\left (C a b - B b^{2}\right )} c d^{2} + {\left (C a^{2} - B a b + {\left (A - C\right )} b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )}{b^{3}}}{6 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.58, size = 630, normalized size = 1.74 \begin {gather*} \frac {2 \, {\left (C a^{2} b^{3} + C b^{5}\right )} d^{3} \tan \left (f x + e\right )^{3} + 6 \, {\left ({\left ({\left (A - C\right )} a b^{4} + B b^{5}\right )} c^{3} - 3 \, {\left (B a b^{4} - {\left (A - C\right )} b^{5}\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a b^{4} + B b^{5}\right )} c d^{2} + {\left (B a b^{4} - {\left (A - C\right )} b^{5}\right )} d^{3}\right )} f x + 3 \, {\left (3 \, {\left (C a^{2} b^{3} + C b^{5}\right )} c d^{2} - {\left (C a^{3} b^{2} - B a^{2} b^{3} + C a b^{4} - B b^{5}\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} + 3 \, {\left ({\left (C a^{2} b^{3} - B a b^{4} + A b^{5}\right )} c^{3} - 3 \, {\left (C a^{3} b^{2} - B a^{2} b^{3} + A a b^{4}\right )} c^{2} d + 3 \, {\left (C a^{4} b - B a^{3} b^{2} + A a^{2} b^{3}\right )} c d^{2} - {\left (C a^{5} - B a^{4} b + A a^{3} b^{2}\right )} d^{3}\right )} \log \left (\frac {b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 \, {\left ({\left (C a^{2} b^{3} + C b^{5}\right )} c^{3} - 3 \, {\left (C a^{3} b^{2} - B a^{2} b^{3} + C a b^{4} - B b^{5}\right )} c^{2} d + 3 \, {\left (C a^{4} b - B a^{3} b^{2} + A a^{2} b^{3} - B a b^{4} + {\left (A - C\right )} b^{5}\right )} c d^{2} - {\left (C a^{5} - B a^{4} b + A a^{3} b^{2} + {\left (A - C\right )} a b^{4} + B b^{5}\right )} d^{3}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 6 \, {\left (3 \, {\left (C a^{2} b^{3} + C b^{5}\right )} c^{2} d - 3 \, {\left (C a^{3} b^{2} - B a^{2} b^{3} + C a b^{4} - B b^{5}\right )} c d^{2} + {\left (C a^{4} b - B a^{3} b^{2} + A a^{2} b^{3} - B a b^{4} + {\left (A - C\right )} b^{5}\right )} d^{3}\right )} \tan \left (f x + e\right )}{6 \, {\left (a^{2} b^{4} + b^{6}\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 25.44, size = 7096, normalized size = 19.55 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.10, size = 573, normalized size = 1.58 \begin {gather*} \frac {\frac {6 \, {\left (A a c^{3} - C a c^{3} + B b c^{3} - 3 \, B a c^{2} d + 3 \, A b c^{2} d - 3 \, C b c^{2} d - 3 \, A a c d^{2} + 3 \, C a c d^{2} - 3 \, B b c d^{2} + B a d^{3} - A b d^{3} + C b d^{3}\right )} {\left (f x + e\right )}}{a^{2} + b^{2}} + \frac {3 \, {\left (B a c^{3} - A b c^{3} + C b c^{3} + 3 \, A a c^{2} d - 3 \, C a c^{2} d + 3 \, B b c^{2} d - 3 \, B a c d^{2} + 3 \, A b c d^{2} - 3 \, C b c d^{2} - A a d^{3} + C a d^{3} - B b d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {6 \, {\left (C a^{2} b^{3} c^{3} - B a b^{4} c^{3} + A b^{5} c^{3} - 3 \, C a^{3} b^{2} c^{2} d + 3 \, B a^{2} b^{3} c^{2} d - 3 \, A a b^{4} c^{2} d + 3 \, C a^{4} b c d^{2} - 3 \, B a^{3} b^{2} c d^{2} + 3 \, A a^{2} b^{3} c d^{2} - C a^{5} d^{3} + B a^{4} b d^{3} - A a^{3} b^{2} d^{3}\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{2} b^{4} + b^{6}} + \frac {2 \, C b^{2} d^{3} \tan \left (f x + e\right )^{3} + 9 \, C b^{2} c d^{2} \tan \left (f x + e\right )^{2} - 3 \, C a b d^{3} \tan \left (f x + e\right )^{2} + 3 \, B b^{2} d^{3} \tan \left (f x + e\right )^{2} + 18 \, C b^{2} c^{2} d \tan \left (f x + e\right ) - 18 \, C a b c d^{2} \tan \left (f x + e\right ) + 18 \, B b^{2} c d^{2} \tan \left (f x + e\right ) + 6 \, C a^{2} d^{3} \tan \left (f x + e\right ) - 6 \, B a b d^{3} \tan \left (f x + e\right ) + 6 \, A b^{2} d^{3} \tan \left (f x + e\right ) - 6 \, C b^{2} d^{3} \tan \left (f x + e\right )}{b^{3}}}{6 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.00, size = 508, normalized size = 1.40 \begin {gather*} \frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {B\,d^3+3\,C\,c\,d^2}{2\,b}-\frac {C\,a\,d^3}{2\,b^2}\right )}{f}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {a\,\left (\frac {B\,d^3+3\,C\,c\,d^2}{b}-\frac {C\,a\,d^3}{b^2}\right )}{b}-\frac {3\,C\,c^2\,d+3\,B\,c\,d^2+A\,d^3}{b}+\frac {C\,d^3}{b}\right )}{f}-\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (b^4\,\left (B\,a\,c^3+3\,A\,a\,d\,c^2\right )-b^3\,\left (C\,a^2\,c^3+3\,B\,a^2\,c^2\,d+3\,A\,a^2\,c\,d^2\right )+b^2\,\left (3\,C\,a^3\,c^2\,d+3\,B\,a^3\,c\,d^2+A\,a^3\,d^3\right )-b\,\left (B\,a^4\,d^3+3\,C\,c\,a^4\,d^2\right )-A\,b^5\,c^3+C\,a^5\,d^3\right )}{f\,\left (a^2\,b^4+b^6\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (A\,c^3+A\,d^3\,1{}\mathrm {i}-B\,c^3\,1{}\mathrm {i}+B\,d^3-C\,c^3-C\,d^3\,1{}\mathrm {i}-3\,A\,c\,d^2-A\,c^2\,d\,3{}\mathrm {i}+B\,c\,d^2\,3{}\mathrm {i}-3\,B\,c^2\,d+3\,C\,c\,d^2+C\,c^2\,d\,3{}\mathrm {i}\right )}{2\,f\,\left (b+a\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (A\,d^3-B\,c^3-C\,d^3-3\,A\,c^2\,d+3\,B\,c\,d^2+3\,C\,c^2\,d+A\,c^3\,1{}\mathrm {i}+B\,d^3\,1{}\mathrm {i}-C\,c^3\,1{}\mathrm {i}-A\,c\,d^2\,3{}\mathrm {i}-B\,c^2\,d\,3{}\mathrm {i}+C\,c\,d^2\,3{}\mathrm {i}\right )}{2\,f\,\left (a+b\,1{}\mathrm {i}\right )}+\frac {C\,d^3\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3\,b\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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