Optimal. Leaf size=287 \[ -\frac {\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}-\frac {(3 b B-a C) \log (\sin (c+d x))}{a^4 d}+\frac {b^2 \left (10 a^4 b B+9 a^2 b^3 B+3 b^5 B-6 a^5 C-3 a^3 b^2 C-a b^4 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^3 d}-\frac {b \left (2 a^2 B+3 b^2 B-a b C\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4 B+6 a^2 b^2 B+3 b^4 B-3 a^3 b C-a b^3 C\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.65, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3713, 3690,
3730, 3732, 3611, 3556} \begin {gather*} -\frac {(3 b B-a C) \log (\sin (c+d x))}{a^4 d}-\frac {b \left (2 a^2 B-a b C+3 b^2 B\right )}{2 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\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 {b \left (a^4 B-3 a^3 b C+6 a^2 b^2 B-a b^3 C+3 b^4 B\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {b^2 \left (-6 a^5 C+10 a^4 b B-3 a^3 b^2 C+9 a^2 b^3 B-a b^4 C+3 b^5 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^3}-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3611
Rule 3690
Rule 3713
Rule 3730
Rule 3732
Rubi steps
\begin {align*} \int \frac {\cot ^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 {\cot ^2(c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\\ &=-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {\int \frac {\cot (c+d x) \left (3 b B-a C+a B \tan (c+d x)+3 b B \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{a}\\ &=-\frac {b \left (2 a^2 B+3 b^2 B-a b C\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {\int \frac {\cot (c+d x) \left (2 \left (a^2+b^2\right ) (3 b B-a C)+2 a^2 (a B+b C) \tan (c+d x)+2 b \left (2 a^2 B+3 b^2 B-a b C\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a^2 \left (a^2+b^2\right )}\\ &=-\frac {b \left (2 a^2 B+3 b^2 B-a b C\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4 B+6 a^2 b^2 B+3 b^4 B-3 a^3 b C-a b^3 C\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\int \frac {\cot (c+d x) \left (2 \left (a^2+b^2\right )^2 (3 b B-a C)+2 a^3 \left (a^2 B-b^2 B+2 a b C\right ) \tan (c+d x)+2 b \left (a^4 B+6 a^2 b^2 B+3 b^4 B-3 a^3 b C-a b^3 C\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^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 {b \left (2 a^2 B+3 b^2 B-a b C\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4 B+6 a^2 b^2 B+3 b^4 B-3 a^3 b C-a b^3 C\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {(3 b B-a C) \int \cot (c+d x) \, dx}{a^4}+\frac {\left (b^2 \left (10 a^4 b B+9 a^2 b^3 B+3 b^5 B-6 a^5 C-3 a^3 b^2 C-a b^4 C\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^4 \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 {(3 b B-a C) \log (\sin (c+d x))}{a^4 d}+\frac {b^2 \left (10 a^4 b B+9 a^2 b^3 B+3 b^5 B-6 a^5 C-3 a^3 b^2 C-a b^4 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^3 d}-\frac {b \left (2 a^2 B+3 b^2 B-a b C\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4 B+6 a^2 b^2 B+3 b^4 B-3 a^3 b C-a b^3 C\right )}{a^3 \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 = 6.28, size = 288, normalized size = 1.00 \begin {gather*} -\frac {B \cot (c+d x)}{a^3 d}+\frac {(B+i C) \log (i-\tan (c+d x))}{2 (i a-b)^3 d}-\frac {(3 b B-a C) \log (\tan (c+d x))}{a^4 d}-\frac {(i B+C) \log (i+\tan (c+d x))}{2 (a-i b)^3 d}+\frac {b^2 \left (10 a^4 b B+9 a^2 b^3 B+3 b^5 B-6 a^5 C-3 a^3 b^2 C-a b^4 C\right ) \log (a+b \tan (c+d x))}{a^4 \left (a^2+b^2\right )^3 d}-\frac {b^2 (b B-a C)}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {b^2 \left (4 a^2 b B+2 b^3 B-3 a^3 C-a b^2 C\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.68, size = 289, normalized size = 1.01
method | result | size |
derivativedivides | \(\frac {-\frac {b^{2} \left (4 B \,a^{2} b +2 B \,b^{3}-3 C \,a^{3}-C a \,b^{2}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{2} \left (10 B \,a^{4} b +9 B \,a^{2} b^{3}+3 B \,b^{5}-6 C \,a^{5}-3 C \,a^{3} b^{2}-C a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {\left (B b -C a \right ) b^{2}}{2 a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\frac {\left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-B \,a^{3}+3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {B}{a^{3} \tan \left (d x +c \right )}+\frac {\left (-3 B b +C a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}}{d}\) | \(289\) |
default | \(\frac {-\frac {b^{2} \left (4 B \,a^{2} b +2 B \,b^{3}-3 C \,a^{3}-C a \,b^{2}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{2} \left (10 B \,a^{4} b +9 B \,a^{2} b^{3}+3 B \,b^{5}-6 C \,a^{5}-3 C \,a^{3} b^{2}-C a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {\left (B b -C a \right ) b^{2}}{2 a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\frac {\left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-B \,a^{3}+3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {B}{a^{3} \tan \left (d x +c \right )}+\frac {\left (-3 B b +C a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}}{d}\) | \(289\) |
norman | \(\frac {\frac {b \left (3 B \,a^{4} b +11 B \,a^{2} b^{3}+6 B \,b^{5}-4 C \,a^{3} b^{2}-2 C a \,b^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{d \,a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {B \tan \left (d x +c \right )}{a d}-\frac {b^{2} \left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) x \left (\tan ^{4}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{2} \left (4 B \,a^{4} b +17 B \,a^{2} b^{3}+9 B \,b^{5}-7 C \,a^{3} b^{2}-3 C a \,b^{4}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{2 d \,a^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {2 b \left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) a x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\tan \left (d x +c \right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{2} \left (10 B \,a^{4} b +9 B \,a^{2} b^{3}+3 B \,b^{5}-6 C \,a^{5}-3 C \,a^{3} b^{2}-C a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {\left (3 B b -C a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4} d}+\frac {\left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(566\) |
risch | \(\text {Expression too large to display}\) | \(1550\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 454, normalized size = 1.58 \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 (6 \, C a^{5} b^{2} - 10 \, B a^{4} b^{3} + 3 \, C a^{3} b^{4} - 9 \, B a^{2} b^{5} + C a b^{6} - 3 \, B b^{7}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{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 \, B a^{6} + 4 \, B a^{4} b^{2} + 2 \, B a^{2} b^{4} + 2 \, {\left (B a^{4} b^{2} - 3 \, C a^{3} b^{3} + 6 \, B a^{2} b^{4} - C a b^{5} + 3 \, B b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (4 \, B a^{5} b - 7 \, C a^{4} b^{2} + 17 \, B a^{3} b^{3} - 3 \, C a^{2} b^{4} + 9 \, B a b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \tan \left (d x + c\right )} - \frac {2 \, {\left (C a - 3 \, B b\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{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. 917 vs.
\(2 (283) = 566\).
time = 3.22, size = 917, normalized size = 3.20 \begin {gather*} -\frac {2 \, B a^{9} + 6 \, B a^{7} b^{2} + 6 \, B a^{5} b^{4} + 2 \, B a^{3} b^{6} + {\left (7 \, C a^{5} b^{4} - 9 \, B a^{4} b^{5} + C a^{3} b^{6} - 3 \, B a^{2} b^{7} + 2 \, {\left (B a^{7} b^{2} + 3 \, C a^{6} b^{3} - 3 \, B a^{5} b^{4} - C a^{4} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (B a^{7} b^{2} + 4 \, C a^{6} b^{3} - 2 \, B a^{5} b^{4} - 3 \, C a^{4} b^{5} + 6 \, B a^{3} b^{6} - C a^{2} b^{7} + 3 \, B a b^{8} + 2 \, {\left (B a^{8} b + 3 \, C a^{7} b^{2} - 3 \, B a^{6} b^{3} - C a^{5} b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{2} - {\left ({\left (C a^{7} b^{2} - 3 \, B a^{6} b^{3} + 3 \, C a^{5} b^{4} - 9 \, B a^{4} b^{5} + 3 \, C a^{3} b^{6} - 9 \, B a^{2} b^{7} + C a b^{8} - 3 \, B b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (C a^{8} b - 3 \, B a^{7} b^{2} + 3 \, C a^{6} b^{3} - 9 \, B a^{5} b^{4} + 3 \, C a^{4} b^{5} - 9 \, B a^{3} b^{6} + C a^{2} b^{7} - 3 \, B a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (C a^{9} - 3 \, B a^{8} b + 3 \, C a^{7} b^{2} - 9 \, B a^{6} b^{3} + 3 \, C a^{5} b^{4} - 9 \, B a^{4} b^{5} + C a^{3} b^{6} - 3 \, B a^{2} b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left ({\left (6 \, C a^{5} b^{4} - 10 \, B a^{4} b^{5} + 3 \, C a^{3} b^{6} - 9 \, B a^{2} b^{7} + C a b^{8} - 3 \, B b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (6 \, C a^{6} b^{3} - 10 \, B a^{5} b^{4} + 3 \, C a^{4} b^{5} - 9 \, B a^{3} b^{6} + C a^{2} b^{7} - 3 \, B a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (6 \, C a^{7} b^{2} - 10 \, B a^{6} b^{3} + 3 \, C a^{5} b^{4} - 9 \, B a^{4} b^{5} + C a^{3} b^{6} - 3 \, B a^{2} b^{7}\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 (4 \, B a^{8} b + 12 \, B a^{6} b^{3} - 9 \, C a^{5} b^{4} + 23 \, B a^{4} b^{5} - 3 \, C a^{3} b^{6} + 9 \, B a^{2} b^{7} + 2 \, {\left (B a^{9} + 3 \, C a^{8} b - 3 \, B a^{7} b^{2} - C a^{6} b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{10} b^{2} + 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} + a^{4} b^{8}\right )} d \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right )} d \tan \left (d x + c\right )^{2} + {\left (a^{12} + 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} + a^{6} b^{6}\right )} d \tan \left (d x + c\right )\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 = 1.62, size = 560, normalized size = 1.95 \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 (6 \, C a^{5} b^{3} - 10 \, B a^{4} b^{4} + 3 \, C a^{3} b^{5} - 9 \, B a^{2} b^{6} + C a b^{7} - 3 \, B b^{8}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}} - \frac {18 \, C a^{5} b^{4} \tan \left (d x + c\right )^{2} - 30 \, B a^{4} b^{5} \tan \left (d x + c\right )^{2} + 9 \, C a^{3} b^{6} \tan \left (d x + c\right )^{2} - 27 \, B a^{2} b^{7} \tan \left (d x + c\right )^{2} + 3 \, C a b^{8} \tan \left (d x + c\right )^{2} - 9 \, B b^{9} \tan \left (d x + c\right )^{2} + 42 \, C a^{6} b^{3} \tan \left (d x + c\right ) - 68 \, B a^{5} b^{4} \tan \left (d x + c\right ) + 26 \, C a^{4} b^{5} \tan \left (d x + c\right ) - 66 \, B a^{3} b^{6} \tan \left (d x + c\right ) + 8 \, C a^{2} b^{7} \tan \left (d x + c\right ) - 22 \, B a b^{8} \tan \left (d x + c\right ) + 25 \, C a^{7} b^{2} - 39 \, B a^{6} b^{3} + 19 \, C a^{5} b^{4} - 41 \, B a^{4} b^{5} + 6 \, C a^{3} b^{6} - 14 \, B a^{2} b^{7}}{{\left (a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}} - \frac {2 \, {\left (C a - 3 \, B b\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} + \frac {2 \, {\left (C a \tan \left (d x + c\right ) - 3 \, B b \tan \left (d x + c\right ) + B a\right )}}{a^{4} \tan \left (d x + c\right )}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.99, size = 380, normalized size = 1.32 \begin {gather*} \frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-6\,C\,a^5+10\,B\,a^4\,b-3\,C\,a^3\,b^2+9\,B\,a^2\,b^3-C\,a\,b^4+3\,B\,b^5\right )}{a^4\,d\,{\left (a^2+b^2\right )}^3}-\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 )\right )\,\left (3\,B\,b-C\,a\right )}{a^4\,d}-\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 {B}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (B\,a^4\,b^2-3\,C\,a^3\,b^3+6\,B\,a^2\,b^4-C\,a\,b^5+3\,B\,b^6\right )}{a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (4\,B\,a^4\,b-7\,C\,a^3\,b^2+17\,B\,a^2\,b^3-3\,C\,a\,b^4+9\,B\,b^5\right )}{2\,a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,\mathrm {tan}\left (c+d\,x\right )+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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