Optimal. Leaf size=337 \[ \frac {\left (a^3 (A c-c C+B d)-3 a b^2 (A c-c C+B d)-3 a^2 b (B c-(A-C) d)+b^3 (B c-(A-C) d)\right ) x}{c^2+d^2}-\frac {\left (3 a^2 b (A c-c C+B d)-b^3 (A c-c C+B d)+a^3 (B c-(A-C) d)-3 a b^2 (B c-(A-C) d)\right ) \log (\cos (e+f x))}{\left (c^2+d^2\right ) f}-\frac {(b c-a d)^3 \left (c^2 C-B c d+A d^2\right ) \log (c+d \tan (e+f x))}{d^4 \left (c^2+d^2\right ) f}+\frac {b \left (b (A b+a B-b C) d^2+(b c-a d) (b c C-b B d-a C d)\right ) \tan (e+f x)}{d^3 f}-\frac {(b c C-b B d-a C d) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d f} \]
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Rubi [A]
time = 1.01, antiderivative size = 337, 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^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 )}{f \left (c^2+d^2\right )}+\frac {x \left (a^3 (A c+B d-c C)-3 a^2 b (B c-d (A-C))-3 a b^2 (A c+B d-c C)+b^3 (B c-d (A-C))\right )}{c^2+d^2}-\frac {(b c-a d)^3 \left (A d^2-B c d+c^2 C\right ) \log (c+d \tan (e+f x))}{d^4 f \left (c^2+d^2\right )}+\frac {b \tan (e+f x) \left (b d^2 (a B+A b-b C)+(b c-a d) (-a C d-b B d+b c C)\right )}{d^3 f}-\frac {(-a C d-b B d+b c C) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d 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 {(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)} \, dx &=\frac {C (a+b \tan (e+f x))^3}{3 d f}+\frac {\int \frac {(a+b \tan (e+f x))^2 \left (-3 (b c C-a A d)+3 (A b+a B-b C) d \tan (e+f x)-3 (b c C-b B d-a C d) \tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx}{3 d}\\ &=-\frac {(b c C-b B d-a C d) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d f}+\frac {\int \frac {(a+b \tan (e+f x)) \left (-6 \left (2 a b c C d-a^2 A d^2-b^2 c (c C-B d)\right )+6 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+6 \left (b (A b+a B-b C) d^2+(b c-a d) (b c C-b B d-a C d)\right ) \tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx}{6 d^2}\\ &=\frac {b \left (b (A b+a B-b C) d^2+(b c-a d) (b c C-b B d-a C d)\right ) \tan (e+f x)}{d^3 f}-\frac {(b c C-b B d-a C d) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d f}-\frac {\int \frac {6 \left (3 a^2 b c C d^2-a^3 A d^3-3 a b^2 c d (c C-B d)+b^3 c \left (c^2 C-B c d+(A-C) d^2\right )\right )-6 \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 (a^3 C d^3-3 a^2 b d^2 (c C-B d)+3 a b^2 d \left (c^2 C-B c d+(A-C) d^2\right )-b^3 \left (c^3 C-B c^2 d+c (A-C) d^2+B d^3\right )\right ) \tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx}{6 d^3}\\ &=\frac {\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 ) x}{c^2+d^2}+\frac {b \left (b (A b+a B-b C) d^2+(b c-a d) (b c C-b B d-a C d)\right ) \tan (e+f x)}{d^3 f}-\frac {(b c C-b B d-a C d) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d f}-\frac {\left ((b c-a d)^3 \left (c^2 C-B c d+A d^2\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 )}+\frac {\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 ) \int \tan (e+f x) \, dx}{c^2+d^2}\\ &=\frac {\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 ) x}{c^2+d^2}-\frac {\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 ) \log (\cos (e+f x))}{\left (c^2+d^2\right ) f}+\frac {b \left (b (A b+a B-b C) d^2+(b c-a d) (b c C-b B d-a C d)\right ) \tan (e+f x)}{d^3 f}-\frac {(b c C-b B d-a C d) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d f}-\frac {\left ((b c-a d)^3 \left (c^2 C-B c d+A d^2\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 ) f}\\ &=\frac {\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 ) x}{c^2+d^2}-\frac {\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 ) \log (\cos (e+f x))}{\left (c^2+d^2\right ) f}-\frac {(b c-a d)^3 \left (c^2 C-B c d+A d^2\right ) \log (c+d \tan (e+f x))}{d^4 \left (c^2+d^2\right ) f}+\frac {b \left (b (A b+a B-b C) d^2+(b c-a d) (b c C-b B d-a C d)\right ) \tan (e+f x)}{d^3 f}-\frac {(b c C-b B d-a C d) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.86, size = 258, normalized size = 0.77 \begin {gather*} \frac {\frac {3 (a+i b)^3 (-i A+B+i C) d^2 \log (i-\tan (e+f x))}{c+i d}+\frac {3 (a-i b)^3 (i A+B-i C) d^2 \log (i+\tan (e+f x))}{c-i d}+\frac {6 (-b c+a d)^3 \left (c^2 C-B c d+A d^2\right ) \log (c+d \tan (e+f x))}{d^2 \left (c^2+d^2\right )}+6 b^2 (A b+a B-b C) d \tan (e+f x)-\frac {6 b (b c-a d) (-b c C+b B d+a C d) \tan (e+f x)}{d}-3 (b c C-b B d-a C d) (a+b \tan (e+f x))^2+2 C d (a+b \tan (e+f x))^3}{6 d^2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 542, normalized size = 1.61
method | result | size |
norman | \(\frac {\left (A \,a^{3} c +3 A \,a^{2} b d -3 A a \,b^{2} c -A \,b^{3} d +B \,a^{3} d -3 B \,a^{2} b c -3 B a \,b^{2} d +B \,b^{3} c -C \,a^{3} c -3 C \,a^{2} b d +3 C a \,b^{2} c +C \,b^{3} d \right ) x}{c^{2}+d^{2}}+\frac {\left (A \,b^{2} d^{2}+3 B a b \,d^{2}-B \,b^{2} c d +3 a^{2} C \,d^{2}-3 C a b c d +C \,b^{2} c^{2}-C \,b^{2} d^{2}\right ) b \tan \left (f x +e \right )}{f \,d^{3}}+\frac {C \,b^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{3 d f}+\frac {b^{2} \left (B b d +3 a C d -C b c \right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 d^{2} f}+\frac {\left (A \,a^{3} d^{5}-3 A \,a^{2} b c \,d^{4}+3 A a \,b^{2} c^{2} d^{3}-A \,b^{3} c^{3} d^{2}-B \,a^{3} c \,d^{4}+3 B \,a^{2} b \,c^{2} d^{3}-3 B a \,b^{2} c^{3} d^{2}+B \,b^{3} c^{4} d +C \,a^{3} c^{2} d^{3}-3 C \,a^{2} b \,c^{3} d^{2}+3 C a \,b^{2} c^{4} d -C \,b^{3} c^{5}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right ) d^{4} f}-\frac {\left (A \,a^{3} d -3 A \,a^{2} b c -3 A a \,b^{2} d +A \,b^{3} c -B \,a^{3} c -3 B \,a^{2} b d +3 B a \,b^{2} c +B \,b^{3} d -a^{3} C d +3 C \,a^{2} b c +3 C a \,b^{2} d -C \,b^{3} c \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \left (c^{2}+d^{2}\right )}\) | \(500\) |
derivativedivides | \(\frac {\frac {b \left (\frac {C \,b^{2} d^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {B \,b^{2} d^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}+\frac {3 C a b \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {C \,b^{2} c d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \,b^{2} d^{2} \tan \left (f x +e \right )+3 B a b \,d^{2} \tan \left (f x +e \right )-B \,b^{2} c d \tan \left (f x +e \right )+3 a^{2} C \,d^{2} \tan \left (f x +e \right )-3 C a b c d \tan \left (f x +e \right )+C \,b^{2} c^{2} \tan \left (f x +e \right )-C \,b^{2} d^{2} \tan \left (f x +e \right )\right )}{d^{3}}+\frac {\left (A \,a^{3} d^{5}-3 A \,a^{2} b c \,d^{4}+3 A a \,b^{2} c^{2} d^{3}-A \,b^{3} c^{3} d^{2}-B \,a^{3} c \,d^{4}+3 B \,a^{2} b \,c^{2} d^{3}-3 B a \,b^{2} c^{3} d^{2}+B \,b^{3} c^{4} d +C \,a^{3} c^{2} d^{3}-3 C \,a^{2} b \,c^{3} d^{2}+3 C a \,b^{2} c^{4} d -C \,b^{3} c^{5}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{d^{4} \left (c^{2}+d^{2}\right )}+\frac {\frac {\left (-A \,a^{3} d +3 A \,a^{2} b c +3 A a \,b^{2} d -A \,b^{3} c +B \,a^{3} c +3 B \,a^{2} b d -3 B a \,b^{2} c -B \,b^{3} d +a^{3} C d -3 C \,a^{2} b c -3 C a \,b^{2} d +C \,b^{3} c \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A \,a^{3} c +3 A \,a^{2} b d -3 A a \,b^{2} c -A \,b^{3} d +B \,a^{3} d -3 B \,a^{2} b c -3 B a \,b^{2} d +B \,b^{3} c -C \,a^{3} c -3 C \,a^{2} b d +3 C a \,b^{2} c +C \,b^{3} d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{c^{2}+d^{2}}}{f}\) | \(542\) |
default | \(\frac {\frac {b \left (\frac {C \,b^{2} d^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {B \,b^{2} d^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}+\frac {3 C a b \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {C \,b^{2} c d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \,b^{2} d^{2} \tan \left (f x +e \right )+3 B a b \,d^{2} \tan \left (f x +e \right )-B \,b^{2} c d \tan \left (f x +e \right )+3 a^{2} C \,d^{2} \tan \left (f x +e \right )-3 C a b c d \tan \left (f x +e \right )+C \,b^{2} c^{2} \tan \left (f x +e \right )-C \,b^{2} d^{2} \tan \left (f x +e \right )\right )}{d^{3}}+\frac {\left (A \,a^{3} d^{5}-3 A \,a^{2} b c \,d^{4}+3 A a \,b^{2} c^{2} d^{3}-A \,b^{3} c^{3} d^{2}-B \,a^{3} c \,d^{4}+3 B \,a^{2} b \,c^{2} d^{3}-3 B a \,b^{2} c^{3} d^{2}+B \,b^{3} c^{4} d +C \,a^{3} c^{2} d^{3}-3 C \,a^{2} b \,c^{3} d^{2}+3 C a \,b^{2} c^{4} d -C \,b^{3} c^{5}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{d^{4} \left (c^{2}+d^{2}\right )}+\frac {\frac {\left (-A \,a^{3} d +3 A \,a^{2} b c +3 A a \,b^{2} d -A \,b^{3} c +B \,a^{3} c +3 B \,a^{2} b d -3 B a \,b^{2} c -B \,b^{3} d +a^{3} C d -3 C \,a^{2} b c -3 C a \,b^{2} d +C \,b^{3} c \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A \,a^{3} c +3 A \,a^{2} b d -3 A a \,b^{2} c -A \,b^{3} d +B \,a^{3} d -3 B \,a^{2} b c -3 B a \,b^{2} d +B \,b^{3} c -C \,a^{3} c -3 C \,a^{2} b d +3 C a \,b^{2} c +C \,b^{3} d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{c^{2}+d^{2}}}{f}\) | \(542\) |
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 = 0.56, size = 451, normalized size = 1.34 \begin {gather*} \frac {\frac {6 \, {\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 + {\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\right )} {\left (f x + e\right )}}{c^{2} + d^{2}} - \frac {6 \, {\left (C b^{3} c^{5} - A a^{3} d^{5} - {\left (3 \, C a b^{2} + B b^{3}\right )} c^{4} d + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{2} - {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{4}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{2} d^{4} + d^{6}} + \frac {3 \, {\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 - {\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\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{2} + d^{2}} + \frac {2 \, C b^{3} d^{2} \tan \left (f x + e\right )^{3} - 3 \, {\left (C b^{3} c d - {\left (3 \, C a b^{2} + B b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{2} + 6 \, {\left (C b^{3} c^{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 )}{d^{3}}}{6 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.20, size = 634, normalized size = 1.88 \begin {gather*} \frac {2 \, {\left (C b^{3} c^{2} d^{3} + C b^{3} d^{5}\right )} \tan \left (f x + e\right )^{3} + 6 \, {\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 d^{4} + {\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^{5}\right )} f x - 3 \, {\left (C b^{3} c^{3} d^{2} + C b^{3} c d^{4} - {\left (3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{3} - {\left (3 \, C a b^{2} + B b^{3}\right )} d^{5}\right )} \tan \left (f x + e\right )^{2} - 3 \, {\left (C b^{3} c^{5} - A a^{3} d^{5} - {\left (3 \, C a b^{2} + B b^{3}\right )} c^{4} d + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{2} - {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{4}\right )} \log \left (\frac {d^{2} \tan \left (f x + e\right )^{2} + 2 \, c d \tan \left (f x + e\right ) + c^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 3 \, {\left (C b^{3} c^{5} - {\left (3 \, C a b^{2} + B b^{3}\right )} c^{4} d + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{2} - {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{3} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} c d^{4} - {\left (C a^{3} + 3 \, B a^{2} b + 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} d^{5}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 6 \, {\left (C b^{3} c^{4} d - {\left (3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{2} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{3} - {\left (3 \, C a b^{2} + B b^{3}\right )} c d^{4} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} d^{5}\right )} \tan \left (f x + e\right )}{6 \, {\left (c^{2} d^{4} + d^{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 = 25.85, size = 7096, normalized size = 21.06 \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 = 1.12, size = 573, normalized size = 1.70 \begin {gather*} \frac {\frac {6 \, {\left (A a^{3} c - C a^{3} c - 3 \, B a^{2} b c - 3 \, A a b^{2} c + 3 \, C a b^{2} c + B b^{3} c + B a^{3} d + 3 \, A a^{2} b d - 3 \, C a^{2} b d - 3 \, B a b^{2} d - A b^{3} d + C b^{3} d\right )} {\left (f x + e\right )}}{c^{2} + d^{2}} + \frac {3 \, {\left (B a^{3} c + 3 \, A a^{2} b c - 3 \, C a^{2} b c - 3 \, B a b^{2} c - A b^{3} c + C b^{3} c - A a^{3} d + C a^{3} d + 3 \, B a^{2} b d + 3 \, A a b^{2} d - 3 \, C a b^{2} d - B b^{3} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{2} + d^{2}} - \frac {6 \, {\left (C b^{3} c^{5} - 3 \, C a b^{2} c^{4} d - B b^{3} c^{4} d + 3 \, C a^{2} b c^{3} d^{2} + 3 \, B a b^{2} c^{3} d^{2} + A b^{3} c^{3} d^{2} - C a^{3} c^{2} d^{3} - 3 \, B a^{2} b c^{2} d^{3} - 3 \, A a b^{2} c^{2} d^{3} + B a^{3} c d^{4} + 3 \, A a^{2} b c d^{4} - A a^{3} d^{5}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{c^{2} d^{4} + d^{6}} + \frac {2 \, C b^{3} d^{2} \tan \left (f x + e\right )^{3} - 3 \, C b^{3} c d \tan \left (f x + e\right )^{2} + 9 \, C a b^{2} d^{2} \tan \left (f x + e\right )^{2} + 3 \, B b^{3} d^{2} \tan \left (f x + e\right )^{2} + 6 \, C b^{3} c^{2} \tan \left (f x + e\right ) - 18 \, C a b^{2} c d \tan \left (f x + e\right ) - 6 \, B b^{3} c d \tan \left (f x + e\right ) + 18 \, C a^{2} b d^{2} \tan \left (f x + e\right ) + 18 \, B a b^{2} d^{2} \tan \left (f x + e\right ) + 6 \, A b^{3} d^{2} \tan \left (f x + e\right ) - 6 \, C b^{3} d^{2} \tan \left (f x + e\right )}{d^{3}}}{6 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.39, size = 508, normalized size = 1.51 \begin {gather*} \frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {B\,b^3+3\,C\,a\,b^2}{2\,d}-\frac {C\,b^3\,c}{2\,d^2}\right )}{f}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {c\,\left (\frac {B\,b^3+3\,C\,a\,b^2}{d}-\frac {C\,b^3\,c}{d^2}\right )}{d}-\frac {3\,C\,a^2\,b+3\,B\,a\,b^2+A\,b^3}{d}+\frac {C\,b^3}{d}\right )}{f}-\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (d^4\,\left (B\,c\,a^3+3\,A\,b\,c\,a^2\right )-d^3\,\left (C\,a^3\,c^2+3\,B\,a^2\,b\,c^2+3\,A\,a\,b^2\,c^2\right )+d^2\,\left (3\,C\,a^2\,b\,c^3+3\,B\,a\,b^2\,c^3+A\,b^3\,c^3\right )-d\,\left (B\,b^3\,c^4+3\,C\,a\,b^2\,c^4\right )-A\,a^3\,d^5+C\,b^3\,c^5\right )}{f\,\left (c^2\,d^4+d^6\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 (d+c\,1{}\mathrm {i}\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+d\,1{}\mathrm {i}\right )}+\frac {C\,b^3\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3\,d\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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