Optimal. Leaf size=352 \[ -\frac {\left (a \left (c^3 C-3 B c^2 d-3 c C d^2+B d^3-A \left (c^3-3 c d^2\right )\right )-b \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (c^2+d^2\right )^3}+\frac {\left (b \left (c^3 C-3 B c^2 d-3 c C d^2+B d^3\right )-a \left (B c^3+3 c^2 C d-3 B c d^2-C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{\left (c^2+d^2\right )^3 f}+\frac {(b c-a d) \left (c^2 C-B c d+A d^2\right )}{2 d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {b \left (c^4 C-c^2 (A-3 C) d^2-2 B c d^3+A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )}{d^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \]
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Rubi [A]
time = 0.51, antiderivative size = 349, normalized size of antiderivative = 0.99, number of steps
used = 4, number of rules used = 4, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {3716, 3709,
3612, 3611} \begin {gather*} \frac {(b c-a d) \left (A d^2-B c d+c^2 C\right )}{2 d^2 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}-\frac {a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-c^2 d^2 (A-3 C)+A d^4-2 B c d^3+c^4 C\right )}{d^2 f \left (c^2+d^2\right )^2 (c+d \tan (e+f x))}+\frac {\left (a A d \left (3 c^2-d^2\right )-a \left (B c^3-3 B c d^2+3 c^2 C d-C d^3\right )-A b \left (c^3-3 c d^2\right )+b \left (-3 B c^2 d+B d^3+c^3 C-3 c C d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^3}+\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 d (A-C) \left (3 c^2-d^2\right )-b B \left (c^3-3 c d^2\right )\right )}{\left (c^2+d^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3611
Rule 3612
Rule 3709
Rule 3716
Rubi steps
\begin {align*} \int \frac {(a+b \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^3} \, dx &=\frac {(b c-a d) \left (c^2 C-B c d+A d^2\right )}{2 d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {\int \frac {a d (A c-c C+B d)+b \left (c^2 C-B c d+A d^2\right )+d (A b c+a B c-b c C-a A d+b B d+a C d) \tan (e+f x)+b C \left (c^2+d^2\right ) \tan ^2(e+f x)}{(c+d \tan (e+f x))^2} \, dx}{d \left (c^2+d^2\right )}\\ &=\frac {(b c-a d) \left (c^2 C-B c d+A d^2\right )}{2 d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {b \left (c^4 C-c^2 (A-3 C) d^2-2 B c d^3+A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )}{d^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\int \frac {-d \left (a \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )-d \left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )-a B \left (c^2-d^2\right )+b \left (c^2 C-2 B c d-C d^2\right )\right ) \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{d \left (c^2+d^2\right )^2}\\ &=\frac {\left (b (A-C) d \left (3 c^2-d^2\right )-b B \left (c^3-3 c d^2\right )-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 )\right ) x}{\left (c^2+d^2\right )^3}+\frac {(b c-a d) \left (c^2 C-B c d+A d^2\right )}{2 d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {b \left (c^4 C-c^2 (A-3 C) d^2-2 B c d^3+A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )}{d^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\left (a A d \left (3 c^2-d^2\right )-A b \left (c^3-3 c d^2\right )+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 )\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{\left (c^2+d^2\right )^3}\\ &=\frac {\left (b (A-C) d \left (3 c^2-d^2\right )-b B \left (c^3-3 c d^2\right )-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 )\right ) x}{\left (c^2+d^2\right )^3}+\frac {\left (a A d \left (3 c^2-d^2\right )-A b \left (c^3-3 c d^2\right )+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 )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{\left (c^2+d^2\right )^3 f}+\frac {(b c-a d) \left (c^2 C-B c d+A d^2\right )}{2 d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {b \left (c^4 C-c^2 (A-3 C) d^2-2 B c d^3+A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )}{d^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.62, size = 331, normalized size = 0.94 \begin {gather*} \frac {\frac {a C d-b (c C+B d)}{(c+d \tan (e+f x))^2}-\frac {2 C d (a+b \tan (e+f x))}{(c+d \tan (e+f x))^2}+2 (A b+a B-b C) d \left (-\frac {i \log (i-\tan (e+f x))}{2 (c+i d)^2}+\frac {i \log (i+\tan (e+f x))}{2 (c-i d)^2}+\frac {d \left (2 c \log (c+d \tan (e+f x))-\frac {c^2+d^2}{c+d \tan (e+f x)}\right )}{\left (c^2+d^2\right )^2}\right )-d (A b c+a B c-b c C-a A d+b B d+a C d) \left (\frac {\log (i-\tan (e+f x))}{(-i c+d)^3}+\frac {\log (i+\tan (e+f x))}{(i c+d)^3}+\frac {d \left (\left (6 c^2-2 d^2\right ) \log (c+d \tan (e+f x))-\frac {\left (c^2+d^2\right ) \left (5 c^2+d^2+4 c d \tan (e+f x)\right )}{(c+d \tan (e+f x))^2}\right )}{\left (c^2+d^2\right )^3}\right )}{2 d^2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 493, normalized size = 1.40
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-3 A a \,c^{2} d +A a \,d^{3}+A b \,c^{3}-3 A b c \,d^{2}+B a \,c^{3}-3 B a c \,d^{2}+3 B b \,c^{2} d -B b \,d^{3}+3 C a \,c^{2} d -C a \,d^{3}-C b \,c^{3}+3 C b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A a \,c^{3}-3 A a c \,d^{2}+3 A b \,c^{2} d -A b \,d^{3}+3 B a \,c^{2} d -B a \,d^{3}-B b \,c^{3}+3 B b c \,d^{2}-C a \,c^{3}+3 C a c \,d^{2}-3 C b \,c^{2} d +C b \,d^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}+\frac {\left (3 A a \,c^{2} d -A a \,d^{3}-A b \,c^{3}+3 A b c \,d^{2}-B a \,c^{3}+3 B a c \,d^{2}-3 B b \,c^{2} d +B b \,d^{3}-3 C a \,c^{2} d +C a \,d^{3}+C b \,c^{3}-3 C b c \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}-\frac {A a \,d^{3}-A b c \,d^{2}-B a c \,d^{2}+B b \,c^{2} d +C a \,c^{2} d -C b \,c^{3}}{2 d^{2} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {2 A a c \,d^{3}-A b \,c^{2} d^{2}+A b \,d^{4}-B a \,c^{2} d^{2}+B a \,d^{4}-2 B b c \,d^{3}-2 C a c \,d^{3}+C b \,c^{4}+3 C b \,c^{2} d^{2}}{\left (c^{2}+d^{2}\right )^{2} d^{2} \left (c +d \tan \left (f x +e \right )\right )}}{f}\) | \(493\) |
default | \(\frac {\frac {\frac {\left (-3 A a \,c^{2} d +A a \,d^{3}+A b \,c^{3}-3 A b c \,d^{2}+B a \,c^{3}-3 B a c \,d^{2}+3 B b \,c^{2} d -B b \,d^{3}+3 C a \,c^{2} d -C a \,d^{3}-C b \,c^{3}+3 C b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A a \,c^{3}-3 A a c \,d^{2}+3 A b \,c^{2} d -A b \,d^{3}+3 B a \,c^{2} d -B a \,d^{3}-B b \,c^{3}+3 B b c \,d^{2}-C a \,c^{3}+3 C a c \,d^{2}-3 C b \,c^{2} d +C b \,d^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}+\frac {\left (3 A a \,c^{2} d -A a \,d^{3}-A b \,c^{3}+3 A b c \,d^{2}-B a \,c^{3}+3 B a c \,d^{2}-3 B b \,c^{2} d +B b \,d^{3}-3 C a \,c^{2} d +C a \,d^{3}+C b \,c^{3}-3 C b c \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}-\frac {A a \,d^{3}-A b c \,d^{2}-B a c \,d^{2}+B b \,c^{2} d +C a \,c^{2} d -C b \,c^{3}}{2 d^{2} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {2 A a c \,d^{3}-A b \,c^{2} d^{2}+A b \,d^{4}-B a \,c^{2} d^{2}+B a \,d^{4}-2 B b c \,d^{3}-2 C a c \,d^{3}+C b \,c^{4}+3 C b \,c^{2} d^{2}}{\left (c^{2}+d^{2}\right )^{2} d^{2} \left (c +d \tan \left (f x +e \right )\right )}}{f}\) | \(493\) |
norman | \(\frac {\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 ) c^{2} x}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (c^{2}+d^{2}\right )}+\frac {d^{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 ) x \left (\tan ^{2}\left (f x +e \right )\right )}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (c^{2}+d^{2}\right )}-\frac {5 A a \,c^{2} d^{3}+A a \,d^{5}-3 A b \,c^{3} d^{2}+A b c \,d^{4}-3 B a \,c^{3} d^{2}+B a c \,d^{4}+B b \,c^{4} d -3 B b \,c^{2} d^{3}+C a \,c^{4} d -3 C a \,c^{2} d^{3}+C b \,c^{5}+5 C b \,c^{3} d^{2}}{2 f \,d^{2} \left (c^{4}+2 c^{2} d^{2}+d^{4}\right )}-\frac {\left (2 A a c \,d^{3}-A b \,c^{2} d^{2}+A b \,d^{4}-B a \,c^{2} d^{2}+B a \,d^{4}-2 B b c \,d^{3}-2 C a c \,d^{3}+C b \,c^{4}+3 C b \,c^{2} d^{2}\right ) \tan \left (f x +e \right )}{f d \left (c^{4}+2 c^{2} d^{2}+d^{4}\right )}+\frac {2 d \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 ) c x \tan \left (f x +e \right )}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (c^{2}+d^{2}\right )}}{\left (c +d \tan \left (f x +e \right )\right )^{2}}+\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 (c +d \tan \left (f x +e \right )\right )}{f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}-\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 (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}\) | \(877\) |
risch | \(\text {Expression too large to display}\) | \(2383\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 549, normalized size = 1.56 \begin {gather*} \frac {\frac {2 \, {\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 )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {2 \, {\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 (d \tan \left (f x + e\right ) + c\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} + \frac {{\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 )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {C b c^{5} + A a d^{5} + {\left (C a + B b\right )} c^{4} d - {\left (3 \, B a + {\left (3 \, A - 5 \, C\right )} b\right )} c^{3} d^{2} + {\left ({\left (5 \, A - 3 \, C\right )} a - 3 \, B b\right )} c^{2} d^{3} + {\left (B a + A b\right )} c d^{4} + 2 \, {\left (C b c^{4} d - {\left (B a + {\left (A - 3 \, C\right )} b\right )} c^{2} d^{3} + 2 \, {\left ({\left (A - C\right )} a - B b\right )} c d^{4} + {\left (B a + A b\right )} d^{5}\right )} \tan \left (f x + e\right )}{c^{6} d^{2} + 2 \, c^{4} d^{4} + c^{2} d^{6} + {\left (c^{4} d^{4} + 2 \, c^{2} d^{6} + d^{8}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (c^{5} d^{3} + 2 \, c^{3} d^{5} + c d^{7}\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. 906 vs.
\(2 (354) = 708\).
time = 4.61, size = 906, normalized size = 2.57 \begin {gather*} \frac {C b c^{5} - A a d^{5} - 3 \, {\left (C a + B b\right )} c^{4} d + 5 \, {\left (B a + {\left (A - C\right )} b\right )} c^{3} d^{2} - {\left ({\left (7 \, A - 3 \, C\right )} a - 3 \, B b\right )} c^{2} d^{3} - {\left (B a + A b\right )} c d^{4} + 2 \, {\left ({\left ({\left (A - C\right )} a - B b\right )} c^{5} + 3 \, {\left (B a + {\left (A - C\right )} b\right )} c^{4} d - 3 \, {\left ({\left (A - C\right )} a - B b\right )} c^{3} d^{2} - {\left (B a + {\left (A - C\right )} b\right )} c^{2} d^{3}\right )} f x + {\left (C b c^{5} - A a d^{5} + {\left (C a + B b\right )} c^{4} d - {\left (3 \, B a + {\left (3 \, A - 7 \, C\right )} b\right )} c^{3} d^{2} + 5 \, {\left ({\left (A - C\right )} a - B b\right )} c^{2} d^{3} + 3 \, {\left (B a + A b\right )} c d^{4} + 2 \, {\left ({\left ({\left (A - C\right )} a - B b\right )} c^{3} d^{2} + 3 \, {\left (B a + {\left (A - C\right )} b\right )} c^{2} d^{3} - 3 \, {\left ({\left (A - C\right )} a - B b\right )} c d^{4} - {\left (B a + {\left (A - C\right )} b\right )} d^{5}\right )} f x\right )} \tan \left (f x + e\right )^{2} - {\left ({\left (B a + {\left (A - C\right )} b\right )} c^{5} - 3 \, {\left ({\left (A - C\right )} a - B b\right )} c^{4} d - 3 \, {\left (B a + {\left (A - C\right )} b\right )} c^{3} d^{2} + {\left ({\left (A - C\right )} a - B b\right )} c^{2} d^{3} + {\left ({\left (B a + {\left (A - C\right )} b\right )} c^{3} d^{2} - 3 \, {\left ({\left (A - C\right )} a - B b\right )} c^{2} d^{3} - 3 \, {\left (B a + {\left (A - C\right )} b\right )} c d^{4} + {\left ({\left (A - C\right )} a - B b\right )} d^{5}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left ({\left (B a + {\left (A - C\right )} b\right )} c^{4} d - 3 \, {\left ({\left (A - C\right )} a - B b\right )} c^{3} d^{2} - 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^{4}\right )} \tan \left (f x + e\right )\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 ) + 2 \, {\left ({\left (C a + B b\right )} c^{5} - {\left (2 \, B a + {\left (2 \, A - 3 \, C\right )} b\right )} c^{4} d + 3 \, {\left ({\left (A - C\right )} a - B b\right )} c^{3} d^{2} + 3 \, {\left (B a + {\left (A - C\right )} b\right )} c^{2} d^{3} - {\left ({\left (3 \, A - 2 \, C\right )} a - 2 \, B b\right )} c d^{4} - {\left (B a + A b\right )} d^{5} + 2 \, {\left ({\left ({\left (A - C\right )} a - B b\right )} c^{4} d + 3 \, {\left (B a + {\left (A - C\right )} b\right )} c^{3} d^{2} - 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^{4}\right )} f x\right )} \tan \left (f x + e\right )}{2 \, {\left ({\left (c^{6} d^{2} + 3 \, c^{4} d^{4} + 3 \, c^{2} d^{6} + d^{8}\right )} f \tan \left (f x + e\right )^{2} + 2 \, {\left (c^{7} d + 3 \, c^{5} d^{3} + 3 \, c^{3} d^{5} + c d^{7}\right )} f \tan \left (f x + e\right ) + {\left (c^{8} + 3 \, c^{6} d^{2} + 3 \, c^{4} d^{4} + c^{2} d^{6}\right )} f\right )}} \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. 1037 vs.
\(2 (354) = 708\).
time = 0.95, size = 1037, normalized size = 2.95 \begin {gather*} \frac {\frac {2 \, {\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 )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} + \frac {{\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 )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {2 \, {\left (B a c^{3} d + A b c^{3} d - C b c^{3} d - 3 \, A a c^{2} d^{2} + 3 \, C a c^{2} d^{2} + 3 \, B b c^{2} d^{2} - 3 \, B a c d^{3} - 3 \, A b c d^{3} + 3 \, C b c d^{3} + A a d^{4} - C a d^{4} - B b d^{4}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{c^{6} d + 3 \, c^{4} d^{3} + 3 \, c^{2} d^{5} + d^{7}} + \frac {3 \, B a c^{3} d^{4} \tan \left (f x + e\right )^{2} + 3 \, A b c^{3} d^{4} \tan \left (f x + e\right )^{2} - 3 \, C b c^{3} d^{4} \tan \left (f x + e\right )^{2} - 9 \, A a c^{2} d^{5} \tan \left (f x + e\right )^{2} + 9 \, C a c^{2} d^{5} \tan \left (f x + e\right )^{2} + 9 \, B b c^{2} d^{5} \tan \left (f x + e\right )^{2} - 9 \, B a c d^{6} \tan \left (f x + e\right )^{2} - 9 \, A b c d^{6} \tan \left (f x + e\right )^{2} + 9 \, C b c d^{6} \tan \left (f x + e\right )^{2} + 3 \, A a d^{7} \tan \left (f x + e\right )^{2} - 3 \, C a d^{7} \tan \left (f x + e\right )^{2} - 3 \, B b d^{7} \tan \left (f x + e\right )^{2} - 2 \, C b c^{6} d \tan \left (f x + e\right ) + 8 \, B a c^{4} d^{3} \tan \left (f x + e\right ) + 8 \, A b c^{4} d^{3} \tan \left (f x + e\right ) - 14 \, C b c^{4} d^{3} \tan \left (f x + e\right ) - 22 \, A a c^{3} d^{4} \tan \left (f x + e\right ) + 22 \, C a c^{3} d^{4} \tan \left (f x + e\right ) + 22 \, B b c^{3} d^{4} \tan \left (f x + e\right ) - 18 \, B a c^{2} d^{5} \tan \left (f x + e\right ) - 18 \, A b c^{2} d^{5} \tan \left (f x + e\right ) + 12 \, C b c^{2} d^{5} \tan \left (f x + e\right ) + 2 \, A a c d^{6} \tan \left (f x + e\right ) - 2 \, C a c d^{6} \tan \left (f x + e\right ) - 2 \, B b c d^{6} \tan \left (f x + e\right ) - 2 \, B a d^{7} \tan \left (f x + e\right ) - 2 \, A b d^{7} \tan \left (f x + e\right ) - C b c^{7} - C a c^{6} d - B b c^{6} d + 6 \, B a c^{5} d^{2} + 6 \, A b c^{5} d^{2} - 9 \, C b c^{5} d^{2} - 14 \, A a c^{4} d^{3} + 11 \, C a c^{4} d^{3} + 11 \, B b c^{4} d^{3} - 7 \, B a c^{3} d^{4} - 7 \, A b c^{3} d^{4} + 4 \, C b c^{3} d^{4} - 3 \, A a c^{2} d^{5} - B a c d^{6} - A b c d^{6} - A a d^{7}}{{\left (c^{6} d^{2} + 3 \, c^{4} d^{4} + 3 \, c^{2} d^{6} + d^{8}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{2}}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 16.53, size = 502, normalized size = 1.43 \begin {gather*} -\frac {\frac {A\,a\,d^5+C\,b\,c^5+A\,b\,c\,d^4+B\,a\,c\,d^4+B\,b\,c^4\,d+C\,a\,c^4\,d+5\,A\,a\,c^2\,d^3-3\,A\,b\,c^3\,d^2-3\,B\,a\,c^3\,d^2-3\,B\,b\,c^2\,d^3-3\,C\,a\,c^2\,d^3+5\,C\,b\,c^3\,d^2}{2\,d^2\,\left (c^4+2\,c^2\,d^2+d^4\right )}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (A\,b\,d^4+B\,a\,d^4+C\,b\,c^4+2\,A\,a\,c\,d^3-2\,B\,b\,c\,d^3-2\,C\,a\,c\,d^3-A\,b\,c^2\,d^2-B\,a\,c^2\,d^2+3\,C\,b\,c^2\,d^2\right )}{d\,\left (c^4+2\,c^2\,d^2+d^4\right )}}{f\,\left (c^2+2\,c\,d\,\mathrm {tan}\left (e+f\,x\right )+d^2\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (B\,b+A\,b\,1{}\mathrm {i}+B\,a\,1{}\mathrm {i}-A\,a+C\,a-C\,b\,1{}\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 {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (A\,b+B\,a-C\,b-A\,a\,1{}\mathrm {i}+B\,b\,1{}\mathrm {i}+C\,a\,1{}\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 {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (\left (A\,b+B\,a-C\,b\right )\,c^3+\left (3\,B\,b-3\,A\,a+3\,C\,a\right )\,c^2\,d+\left (3\,C\,b-3\,B\,a-3\,A\,b\right )\,c\,d^2+\left (A\,a-B\,b-C\,a\right )\,d^3\right )}{f\,\left (c^6+3\,c^4\,d^2+3\,c^2\,d^4+d^6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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