Optimal. Leaf size=331 \[ \frac {\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a^2 \left (a^4 b B+3 a^2 b^3 B+6 b^5 B-3 a^5 C-9 a^3 b^2 C-10 a b^4 C\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.61, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3713, 3686,
3726, 3728, 3707, 3698, 31, 3556} \begin {gather*} \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {a \left (-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {\left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {x \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right )}{\left (a^2+b^2\right )^3}-\frac {\left (-3 a^4 C+a^3 b B-6 a^2 b^2 C+3 a b^3 B-b^4 C\right ) \tan (c+d x)}{b^3 d \left (a^2+b^2\right )^2}+\frac {a^2 \left (-3 a^5 C+a^4 b B-9 a^3 b^2 C+3 a^2 b^3 B-10 a b^4 C+6 b^5 B\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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[Out]
Rule 31
Rule 3556
Rule 3686
Rule 3698
Rule 3707
Rule 3713
Rule 3726
Rule 3728
Rubi steps
\begin {align*} \int \frac {\tan ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx &=\int \frac {\tan ^4(c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\\ &=\frac {a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\tan ^2(c+d x) \left (-3 a (b B-a C)+2 b (b B-a C) \tan (c+d x)-\left (a b B-3 a^2 C-2 b^2 C\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=\frac {a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\tan (c+d x) \left (-2 a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right )-2 b^2 \left (a^2 B-b^2 B+2 a b C\right ) \tan (c+d x)-2 \left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac {\left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {2 a \left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right )-2 b^3 \left (2 a b B-a^2 C+b^2 C\right ) \tan (c+d x)+2 \left (a^2+b^2\right )^2 (b B-3 a C) \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{2 b^3 \left (a^2+b^2\right )^2}\\ &=\frac {\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}-\frac {\left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^3}+\frac {\left (a^2 \left (a^4 b B+3 a^2 b^3 B+6 b^5 B-3 a^5 C-9 a^3 b^2 C-10 a b^4 C\right )\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^3 \left (a^2+b^2\right )^3}\\ &=\frac {\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac {\left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left (a^2 \left (a^4 b B+3 a^2 b^3 B+6 b^5 B-3 a^5 C-9 a^3 b^2 C-10 a b^4 C\right )\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^4 \left (a^2+b^2\right )^3 d}\\ &=\frac {\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a^2 \left (a^4 b B+3 a^2 b^3 B+6 b^5 B-3 a^5 C-9 a^3 b^2 C-10 a b^4 C\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 6.61, size = 1146, normalized size = 3.46 \begin {gather*} \frac {a^4 (-b B+a C) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) (B+C \tan (c+d x))}{2 (a-i b)^2 (a+i b)^2 b^2 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{(a-i b)^3 (a+i b)^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {\left (i a^{11} b^4 B+a^{10} b^5 B+5 i a^9 b^6 B+5 a^8 b^7 B+13 i a^7 b^8 B+13 a^6 b^9 B+15 i a^5 b^{10} B+15 a^4 b^{11} B+6 i a^3 b^{12} B+6 a^2 b^{13} B-3 i a^{12} b^3 C-3 a^{11} b^4 C-15 i a^{10} b^5 C-15 a^9 b^6 C-31 i a^8 b^7 C-31 a^7 b^8 C-29 i a^6 b^9 C-29 a^5 b^{10} C-10 i a^4 b^{11} C-10 a^3 b^{12} C\right ) (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{(a-i b)^6 (a+i b)^5 b^7 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}-\frac {i \left (a^6 b B+3 a^4 b^3 B+6 a^2 b^5 B-3 a^7 C-9 a^5 b^2 C-10 a^3 b^4 C\right ) \text {ArcTan}(\tan (c+d x)) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {(-b B+3 a C) \log (\cos (c+d x)) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{b^4 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {\left (a^6 b B+3 a^4 b^3 B+6 a^2 b^5 B-3 a^7 C-9 a^5 b^2 C-10 a^3 b^4 C\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{2 b^4 \left (a^2+b^2\right )^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \left (-a^4 b B \sin (c+d x)-4 a^2 b^3 B \sin (c+d x)+2 a^5 C \sin (c+d x)+5 a^3 b^2 C \sin (c+d x)\right ) (B+C \tan (c+d x))}{(a-i b)^2 (a+i b)^2 b^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {C \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \tan (c+d x) (B+C \tan (c+d x))}{b^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 263, normalized size = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {C \tan \left (d x +c \right )}{b^{3}}+\frac {a^{2} \left (B \,a^{4} b +3 B \,a^{2} b^{3}+6 B \,b^{5}-3 C \,a^{5}-9 C \,a^{3} b^{2}-10 C a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {a^{4} \left (B b -C a \right )}{2 b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{3} \left (2 B \,a^{2} b +4 B \,b^{3}-3 C \,a^{3}-5 C a \,b^{2}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (-3 B \,a^{2} b +B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(263\) |
default | \(\frac {\frac {C \tan \left (d x +c \right )}{b^{3}}+\frac {a^{2} \left (B \,a^{4} b +3 B \,a^{2} b^{3}+6 B \,b^{5}-3 C \,a^{5}-9 C \,a^{3} b^{2}-10 C a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {a^{4} \left (B b -C a \right )}{2 b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{3} \left (2 B \,a^{2} b +4 B \,b^{3}-3 C \,a^{3}-5 C a \,b^{2}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (-3 B \,a^{2} b +B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(263\) |
norman | \(\frac {\frac {C \left (\tan ^{3}\left (d x +c \right )\right )}{b d}+\frac {\left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) a^{2} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{2} \left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {a \left (2 B \,a^{4} b +4 B \,a^{2} b^{3}-6 C \,a^{5}-11 C \,a^{3} b^{2}-3 C a \,b^{4}\right ) \tan \left (d x +c \right )}{d \,b^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {a^{2} \left (3 B \,a^{4} b +7 B \,a^{2} b^{3}-9 C \,a^{5}-17 C \,a^{3} b^{2}-4 C a \,b^{4}\right )}{2 d \,b^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b \left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) a x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{2} \left (B \,a^{4} b +3 B \,a^{2} b^{3}+6 B \,b^{5}-3 C \,a^{5}-9 C \,a^{3} b^{2}-10 C a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b^{4} d}-\frac {\left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(512\) |
risch | \(\text {Expression too large to display}\) | \(1551\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 389, normalized size = 1.18 \begin {gather*} \frac {\frac {2 \, {\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, C a^{7} - B a^{6} b + 9 \, C a^{5} b^{2} - 3 \, B a^{4} b^{3} + 10 \, C a^{3} b^{4} - 6 \, B a^{2} b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {5 \, C a^{7} - 3 \, B a^{6} b + 9 \, C a^{5} b^{2} - 7 \, B a^{4} b^{3} + 2 \, {\left (3 \, C a^{6} b - 2 \, B a^{5} b^{2} + 5 \, C a^{4} b^{3} - 4 \, B a^{3} b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b^{4} + 2 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )} + \frac {2 \, C \tan \left (d x + c\right )}{b^{3}}}{2 \, d} \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. 890 vs.
\(2 (328) = 656\).
time = 8.00, size = 890, normalized size = 2.69 \begin {gather*} -\frac {3 \, C a^{7} b^{2} - B a^{6} b^{3} + 9 \, C a^{5} b^{4} - 7 \, B a^{4} b^{5} - 2 \, {\left (C a^{6} b^{3} + 3 \, C a^{4} b^{5} + 3 \, C a^{2} b^{7} + C b^{9}\right )} \tan \left (d x + c\right )^{3} - 2 \, {\left (B a^{5} b^{4} + 3 \, C a^{4} b^{5} - 3 \, B a^{3} b^{6} - C a^{2} b^{7}\right )} d x - {\left (9 \, C a^{7} b^{2} - 3 \, B a^{6} b^{3} + 23 \, C a^{5} b^{4} - 9 \, B a^{4} b^{5} + 12 \, C a^{3} b^{6} + 4 \, C a b^{8} + 2 \, {\left (B a^{3} b^{6} + 3 \, C a^{2} b^{7} - 3 \, B a b^{8} - C b^{9}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (3 \, C a^{9} - B a^{8} b + 9 \, C a^{7} b^{2} - 3 \, B a^{6} b^{3} + 10 \, C a^{5} b^{4} - 6 \, B a^{4} b^{5} + {\left (3 \, C a^{7} b^{2} - B a^{6} b^{3} + 9 \, C a^{5} b^{4} - 3 \, B a^{4} b^{5} + 10 \, C a^{3} b^{6} - 6 \, B a^{2} b^{7}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, C a^{8} b - B a^{7} b^{2} + 9 \, C a^{6} b^{3} - 3 \, B a^{5} b^{4} + 10 \, C a^{4} b^{5} - 6 \, B a^{3} b^{6}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (3 \, C a^{9} - B a^{8} b + 9 \, C a^{7} b^{2} - 3 \, B a^{6} b^{3} + 9 \, C a^{5} b^{4} - 3 \, B a^{4} b^{5} + 3 \, C a^{3} b^{6} - B a^{2} b^{7} + {\left (3 \, C a^{7} b^{2} - B a^{6} b^{3} + 9 \, C a^{5} b^{4} - 3 \, B a^{4} b^{5} + 9 \, C a^{3} b^{6} - 3 \, B a^{2} b^{7} + 3 \, C a b^{8} - B b^{9}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, C a^{8} b - B a^{7} b^{2} + 9 \, C a^{6} b^{3} - 3 \, B a^{5} b^{4} + 9 \, C a^{4} b^{5} - 3 \, B a^{3} b^{6} + 3 \, C a^{2} b^{7} - B a b^{8}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (3 \, C a^{8} b - B a^{7} b^{2} + 6 \, C a^{6} b^{3} - 3 \, B a^{5} b^{4} - 2 \, C a^{4} b^{5} + 4 \, B a^{3} b^{6} + C a^{2} b^{7} + 2 \, {\left (B a^{4} b^{5} + 3 \, C a^{3} b^{6} - 3 \, B a^{2} b^{7} - C a b^{8}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{6} + 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} + b^{12}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{5} + 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} + a b^{11}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} b^{4} + 3 \, a^{6} b^{6} + 3 \, a^{4} b^{8} + a^{2} b^{10}\right )} d\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 [A]
time = 1.10, size = 505, normalized size = 1.53 \begin {gather*} \frac {\frac {2 \, {\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, C a^{7} - B a^{6} b + 9 \, C a^{5} b^{2} - 3 \, B a^{4} b^{3} + 10 \, C a^{3} b^{4} - 6 \, B a^{2} b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {2 \, C \tan \left (d x + c\right )}{b^{3}} + \frac {9 \, C a^{7} b^{2} \tan \left (d x + c\right )^{2} - 3 \, B a^{6} b^{3} \tan \left (d x + c\right )^{2} + 27 \, C a^{5} b^{4} \tan \left (d x + c\right )^{2} - 9 \, B a^{4} b^{5} \tan \left (d x + c\right )^{2} + 30 \, C a^{3} b^{6} \tan \left (d x + c\right )^{2} - 18 \, B a^{2} b^{7} \tan \left (d x + c\right )^{2} + 12 \, C a^{8} b \tan \left (d x + c\right ) - 2 \, B a^{7} b^{2} \tan \left (d x + c\right ) + 38 \, C a^{6} b^{3} \tan \left (d x + c\right ) - 6 \, B a^{5} b^{4} \tan \left (d x + c\right ) + 50 \, C a^{4} b^{5} \tan \left (d x + c\right ) - 28 \, B a^{3} b^{6} \tan \left (d x + c\right ) + 4 \, C a^{9} + 13 \, C a^{7} b^{2} + B a^{6} b^{3} + 21 \, C a^{5} b^{4} - 11 \, B a^{4} b^{5}}{{\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.43, size = 335, normalized size = 1.01 \begin {gather*} \frac {C\,\mathrm {tan}\left (c+d\,x\right )}{b^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\frac {5\,C\,a^7-3\,B\,a^6\,b+9\,C\,a^5\,b^2-7\,B\,a^4\,b^3}{2\,b\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (3\,C\,a^6-2\,B\,a^5\,b+5\,C\,a^4\,b^2-4\,B\,a^3\,b^3\right )}{a^4+2\,a^2\,b^2+b^4}}{d\,\left (a^2\,b^3+2\,a\,b^4\,\mathrm {tan}\left (c+d\,x\right )+b^5\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {a^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-3\,C\,a^5+B\,a^4\,b-9\,C\,a^3\,b^2+3\,B\,a^2\,b^3-10\,C\,a\,b^4+6\,B\,b^5\right )}{b^4\,d\,{\left (a^2+b^2\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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