Optimal. Leaf size=278 \[ -\frac {4 b \log (\sin (c+d x))}{a^5 d}-\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {x \left (a^4-6 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4}-\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left (a^2+b^2\right )^4}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3} \]
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Rubi [A] time = 0.85, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3569, 3649, 3651, 3530, 3475} \[ -\frac {b \left (13 a^4 b^2+12 a^2 b^4+a^6+4 b^6\right )}{a^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {b \left (4 a^2 b^2+a^4+2 b^4\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {4 b^3 \left (6 a^4 b^2+4 a^2 b^4+5 a^6+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left (a^2+b^2\right )^4}-\frac {x \left (-6 a^2 b^2+a^4+b^4\right )}{\left (a^2+b^2\right )^4}-\frac {4 b \log (\sin (c+d x))}{a^5 d}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3530
Rule 3569
Rule 3649
Rule 3651
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx &=-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {\int \frac {\cot (c+d x) \left (4 b+a \tan (c+d x)+4 b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^4} \, dx}{a}\\ &=-\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {\int \frac {\cot (c+d x) \left (12 b \left (a^2+b^2\right )+3 a^3 \tan (c+d x)+3 b \left (3 a^2+4 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 a^2 \left (a^2+b^2\right )}\\ &=-\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {\int \frac {\cot (c+d x) \left (24 b \left (a^2+b^2\right )^2+6 a^3 \left (a^2-b^2\right ) \tan (c+d x)+12 b \left (a^4+4 a^2 b^2+2 b^4\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{6 a^3 \left (a^2+b^2\right )^2}\\ &=-\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {\int \frac {\cot (c+d x) \left (24 b \left (a^2+b^2\right )^3+6 a^5 \left (a^2-3 b^2\right ) \tan (c+d x)+6 b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 a^4 \left (a^2+b^2\right )^3}\\ &=-\frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {(4 b) \int \cot (c+d x) \, dx}{a^5}+\frac {\left (4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^5 \left (a^2+b^2\right )^4}\\ &=-\frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac {4 b \log (\sin (c+d x))}{a^5 d}+\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 \left (a^2+b^2\right )^4 d}-\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 3.55, size = 241, normalized size = 0.87 \[ -\frac {\frac {\cot (c+d x)}{a^4}-\frac {b^6}{3 a^5 \left (a^2+b^2\right ) (a \cot (c+d x)+b)^3}+\frac {b^5 \left (3 a^2+2 b^2\right )}{a^5 \left (a^2+b^2\right )^2 (a \cot (c+d x)+b)^2}-\frac {b^4 \left (15 a^4+17 a^2 b^2+6 b^4\right )}{a^5 \left (a^2+b^2\right )^3 (a \cot (c+d x)+b)}-\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cot (c+d x)+b)}{a^5 \left (a^2+b^2\right )^4}+\frac {i \log (-\cot (c+d x)+i)}{2 (a-i b)^4}-\frac {i \log (\cot (c+d x)+i)}{2 (a+i b)^4}}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 925, normalized size = 3.33 \[ -\frac {3 \, a^{12} + 12 \, a^{10} b^{2} + 18 \, a^{8} b^{4} + 12 \, a^{6} b^{6} + 3 \, a^{4} b^{8} - {\left (37 \, a^{6} b^{6} + 21 \, a^{4} b^{8} + 6 \, a^{2} b^{10} - 3 \, {\left (a^{9} b^{3} - 6 \, a^{7} b^{5} + a^{5} b^{7}\right )} d x\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{9} b^{3} - 23 \, a^{7} b^{5} + 4 \, a^{5} b^{7} + 10 \, a^{3} b^{9} + 4 \, a b^{11} + 3 \, {\left (a^{10} b^{2} - 6 \, a^{8} b^{4} + a^{6} b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (3 \, a^{10} b^{2} - 3 \, a^{8} b^{4} + 40 \, a^{6} b^{6} + 34 \, a^{4} b^{8} + 10 \, a^{2} b^{10} + 3 \, {\left (a^{11} b - 6 \, a^{9} b^{3} + a^{7} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 6 \, {\left ({\left (a^{8} b^{4} + 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} + 4 \, a^{2} b^{10} + b^{12}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{9} b^{3} + 4 \, a^{7} b^{5} + 6 \, a^{5} b^{7} + 4 \, a^{3} b^{9} + a b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{10} b^{2} + 4 \, a^{8} b^{4} + 6 \, a^{6} b^{6} + 4 \, a^{4} b^{8} + a^{2} b^{10}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{11} b + 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} + 4 \, a^{5} b^{7} + a^{3} b^{9}\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 ) - 6 \, {\left ({\left (5 \, a^{6} b^{6} + 6 \, a^{4} b^{8} + 4 \, a^{2} b^{10} + b^{12}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (5 \, a^{7} b^{5} + 6 \, a^{5} b^{7} + 4 \, a^{3} b^{9} + a b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (5 \, a^{8} b^{4} + 6 \, a^{6} b^{6} + 4 \, a^{4} b^{8} + a^{2} b^{10}\right )} \tan \left (d x + c\right )^{2} + {\left (5 \, a^{9} b^{3} + 6 \, a^{7} b^{5} + 4 \, a^{5} b^{7} + a^{3} b^{9}\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 (9 \, a^{11} b + 36 \, a^{9} b^{3} + 108 \, a^{7} b^{5} + 81 \, a^{5} b^{7} + 22 \, a^{3} b^{9} + 3 \, {\left (a^{12} - 6 \, a^{10} b^{2} + a^{8} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{3 \, {\left ({\left (a^{13} b^{3} + 4 \, a^{11} b^{5} + 6 \, a^{9} b^{7} + 4 \, a^{7} b^{9} + a^{5} b^{11}\right )} d \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{14} b^{2} + 4 \, a^{12} b^{4} + 6 \, a^{10} b^{6} + 4 \, a^{8} b^{8} + a^{6} b^{10}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{15} b + 4 \, a^{13} b^{3} + 6 \, a^{11} b^{5} + 4 \, a^{9} b^{7} + a^{7} b^{9}\right )} d \tan \left (d x + c\right )^{2} + {\left (a^{16} + 4 \, a^{14} b^{2} + 6 \, a^{12} b^{4} + 4 \, a^{10} b^{6} + a^{8} b^{8}\right )} d \tan \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.46, size = 502, normalized size = 1.81 \[ -\frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {12 \, {\left (5 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{13} b + 4 \, a^{11} b^{3} + 6 \, a^{9} b^{5} + 4 \, a^{7} b^{7} + a^{5} b^{9}} + \frac {110 \, a^{6} b^{6} \tan \left (d x + c\right )^{3} + 132 \, a^{4} b^{8} \tan \left (d x + c\right )^{3} + 88 \, a^{2} b^{10} \tan \left (d x + c\right )^{3} + 22 \, b^{12} \tan \left (d x + c\right )^{3} + 360 \, a^{7} b^{5} \tan \left (d x + c\right )^{2} + 453 \, a^{5} b^{7} \tan \left (d x + c\right )^{2} + 300 \, a^{3} b^{9} \tan \left (d x + c\right )^{2} + 75 \, a b^{11} \tan \left (d x + c\right )^{2} + 396 \, a^{8} b^{4} \tan \left (d x + c\right ) + 525 \, a^{6} b^{6} \tan \left (d x + c\right ) + 348 \, a^{4} b^{8} \tan \left (d x + c\right ) + 87 \, a^{2} b^{10} \tan \left (d x + c\right ) + 147 \, a^{9} b^{3} + 207 \, a^{7} b^{5} + 139 \, a^{5} b^{7} + 35 \, a^{3} b^{9}}{{\left (a^{13} + 4 \, a^{11} b^{2} + 6 \, a^{9} b^{4} + 4 \, a^{7} b^{6} + a^{5} b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}} + \frac {12 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{5}} - \frac {3 \, {\left (4 \, b \tan \left (d x + c\right ) - a\right )}}{a^{5} \tan \left (d x + c\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 478, normalized size = 1.72 \[ -\frac {b^{3}}{3 d \left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {10 b^{3}}{d \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {9 b^{5}}{d \left (a^{2}+b^{2}\right )^{3} a^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {3 b^{7}}{d \left (a^{2}+b^{2}\right )^{3} a^{4} \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 b^{3}}{d \left (a^{2}+b^{2}\right )^{2} a \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {b^{5}}{d \left (a^{2}+b^{2}\right )^{2} a^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {20 a \,b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {24 b^{5} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{4} a}+\frac {16 b^{7} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{4} a^{3}}+\frac {4 b^{9} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{4} a^{5}}-\frac {1}{d \,a^{4} \tan \left (d x +c \right )}-\frac {4 b \ln \left (\tan \left (d x +c \right )\right )}{a^{5} d}+\frac {2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} b}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a \,b^{3}}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {\arctan \left (\tan \left (d x +c \right )\right ) a^{4}}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {6 \arctan \left (\tan \left (d x +c \right )\right ) a^{2} b^{2}}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {\arctan \left (\tan \left (d x +c \right )\right ) b^{4}}{d \left (a^{2}+b^{2}\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 516, normalized size = 1.86 \[ -\frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {12 \, {\left (5 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{13} + 4 \, a^{11} b^{2} + 6 \, a^{9} b^{4} + 4 \, a^{7} b^{6} + a^{5} b^{8}} - \frac {6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, a^{9} + 9 \, a^{7} b^{2} + 9 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + 3 \, {\left (a^{6} b^{3} + 13 \, a^{4} b^{5} + 12 \, a^{2} b^{7} + 4 \, b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (3 \, a^{7} b^{2} + 31 \, a^{5} b^{4} + 30 \, a^{3} b^{6} + 10 \, a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (9 \, a^{8} b + 64 \, a^{6} b^{3} + 65 \, a^{4} b^{5} + 22 \, a^{2} b^{7}\right )} \tan \left (d x + c\right )}{{\left (a^{10} b^{3} + 3 \, a^{8} b^{5} + 3 \, a^{6} b^{7} + a^{4} b^{9}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{11} b^{2} + 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} + a^{5} b^{8}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{12} b + 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} + a^{6} b^{7}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{13} + 3 \, a^{11} b^{2} + 3 \, a^{9} b^{4} + a^{7} b^{6}\right )} \tan \left (d x + c\right )} + \frac {12 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{5}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.55, size = 430, normalized size = 1.55 \[ \frac {4\,b^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (5\,a^6+6\,a^4\,b^2+4\,a^2\,b^4+b^6\right )}{a^5\,d\,{\left (a^2+b^2\right )}^4}-\frac {\frac {1}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (a^6\,b^3+13\,a^4\,b^5+12\,a^2\,b^7+4\,b^9\right )}{a^4\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,a^6\,b^2+31\,a^4\,b^4+30\,a^2\,b^6+10\,b^8\right )}{a^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (9\,a^6\,b+64\,a^4\,b^3+65\,a^2\,b^5+22\,b^7\right )}{3\,a^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3\,\mathrm {tan}\left (c+d\,x\right )+3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^4\right )}-\frac {4\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^5\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )} \]
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 \[ \text {Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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