Optimal. Leaf size=156 \[ -\frac {3 (b \cos (c+d x)-a \sin (c+d x))}{8 d \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))^2}-\frac {b \cos (c+d x)-a \sin (c+d x)}{4 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {3 \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{8 d \left (a^2+b^2\right )^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3076, 3074, 206} \[ -\frac {3 (b \cos (c+d x)-a \sin (c+d x))}{8 d \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))^2}-\frac {b \cos (c+d x)-a \sin (c+d x)}{4 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {3 \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{8 d \left (a^2+b^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3074
Rule 3076
Rubi steps
\begin {align*} \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^5} \, dx &=-\frac {b \cos (c+d x)-a \sin (c+d x)}{4 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {3 \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx}{4 \left (a^2+b^2\right )}\\ &=-\frac {b \cos (c+d x)-a \sin (c+d x)}{4 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {3 (b \cos (c+d x)-a \sin (c+d x))}{8 \left (a^2+b^2\right )^2 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac {3 \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{8 \left (a^2+b^2\right )^2}\\ &=-\frac {b \cos (c+d x)-a \sin (c+d x)}{4 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {3 (b \cos (c+d x)-a \sin (c+d x))}{8 \left (a^2+b^2\right )^2 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}\\ &=-\frac {3 \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{8 \left (a^2+b^2\right )^{5/2} d}-\frac {b \cos (c+d x)-a \sin (c+d x)}{4 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {3 (b \cos (c+d x)-a \sin (c+d x))}{8 \left (a^2+b^2\right )^2 d (a \cos (c+d x)+b \sin (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 1.15, size = 157, normalized size = 1.01 \[ \frac {\frac {6 \tanh ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )-b}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}+\frac {\left (3 b^3-9 a^2 b\right ) \cos (3 (c+d x))-11 b \left (a^2+b^2\right ) \cos (c+d x)+2 a \sin (c+d x) \left (3 \left (a^2-3 b^2\right ) \cos (2 (c+d x))+7 a^2+b^2\right )}{4 \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))^4}}{8 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.13, size = 544, normalized size = 3.49 \[ -\frac {6 \, {\left (3 \, a^{4} b + 2 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left ({\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + b^{4} + 2 \, {\left (3 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (a b^{3} \cos \left (d x + c\right ) + {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 2 \, {\left (4 \, a^{4} b - a^{2} b^{3} - 5 \, b^{5}\right )} \cos \left (d x + c\right ) - 2 \, {\left (2 \, a^{5} + 7 \, a^{3} b^{2} + 5 \, a b^{4} + 3 \, {\left (a^{5} - 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{10} - 3 \, a^{8} b^{2} - 14 \, a^{6} b^{4} - 14 \, a^{4} b^{6} - 3 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{4} + 2 \, {\left (3 \, a^{8} b^{2} + 8 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - b^{10}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} d + 4 \, {\left ({\left (a^{9} b + 2 \, a^{7} b^{3} - 2 \, a^{3} b^{7} - a b^{9}\right )} d \cos \left (d x + c\right )^{3} + {\left (a^{7} b^{3} + 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 588, normalized size = 3.77 \[ -\frac {\frac {3 \, \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (5 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 16 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 24 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 56 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 32 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 114 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16 \, b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 84 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 56 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 32 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 23 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 64 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{6} b - 2 \, a^{4} b^{3}\right )}}{{\left (a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}^{4}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.60, size = 514, normalized size = 3.29 \[ \frac {-\frac {2 \left (-\frac {\left (5 a^{4}+16 a^{2} b^{2}+8 b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {3 b \left (a^{4}+16 a^{2} b^{2}+8 b^{4}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (3 a^{6}-36 a^{4} b^{2}+56 a^{2} b^{4}+32 b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (15 a^{6}-114 a^{4} b^{2}-8 a^{2} b^{4}+16 b^{6}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (3 a^{6}+84 a^{4} b^{2}-56 a^{2} b^{4}-32 b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (23 a^{4}-64 a^{2} b^{2}-24 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (5 a^{4}-24 a^{2} b^{2}-8 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (5 a^{2}+2 b^{2}\right )}{8 a^{4}+16 a^{2} b^{2}+8 b^{4}}\right )}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{4}}+\frac {3 \arctanh \left (\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a -2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{4 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}+b^{2}}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.04, size = 822, normalized size = 5.27 \[ -\frac {\frac {2 \, {\left (5 \, a^{6} b + 2 \, a^{4} b^{3} - \frac {{\left (5 \, a^{7} - 24 \, a^{5} b^{2} - 8 \, a^{3} b^{4}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {{\left (23 \, a^{6} b - 64 \, a^{4} b^{3} - 24 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {{\left (3 \, a^{7} + 84 \, a^{5} b^{2} - 56 \, a^{3} b^{4} - 32 \, a b^{6}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {{\left (15 \, a^{6} b - 114 \, a^{4} b^{3} - 8 \, a^{2} b^{5} + 16 \, b^{7}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {{\left (3 \, a^{7} - 36 \, a^{5} b^{2} + 56 \, a^{3} b^{4} + 32 \, a b^{6}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, {\left (a^{6} b + 16 \, a^{4} b^{3} + 8 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {{\left (5 \, a^{7} + 16 \, a^{5} b^{2} + 8 \, a^{3} b^{4}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{12} + 2 \, a^{10} b^{2} + a^{8} b^{4} + \frac {8 \, {\left (a^{11} b + 2 \, a^{9} b^{3} + a^{7} b^{5}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {4 \, {\left (a^{12} - 4 \, a^{10} b^{2} - 11 \, a^{8} b^{4} - 6 \, a^{6} b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, {\left (3 \, a^{11} b + 2 \, a^{9} b^{3} - 5 \, a^{7} b^{5} - 4 \, a^{5} b^{7}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2 \, {\left (3 \, a^{12} - 18 \, a^{10} b^{2} - 37 \, a^{8} b^{4} - 8 \, a^{6} b^{6} + 8 \, a^{4} b^{8}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {8 \, {\left (3 \, a^{11} b + 2 \, a^{9} b^{3} - 5 \, a^{7} b^{5} - 4 \, a^{5} b^{7}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {4 \, {\left (a^{12} - 4 \, a^{10} b^{2} - 11 \, a^{8} b^{4} - 6 \, a^{6} b^{6}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {8 \, {\left (a^{11} b + 2 \, a^{9} b^{3} + a^{7} b^{5}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {{\left (a^{12} + 2 \, a^{10} b^{2} + a^{8} b^{4}\right )} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac {3 \, \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.09, size = 719, normalized size = 4.61 \[ -\frac {\frac {5\,a^2\,b+2\,b^3}{4\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (a^4\,b+16\,a^2\,b^3+8\,b^5\right )}{4\,a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-23\,a^4\,b+64\,a^2\,b^3+24\,b^5\right )}{4\,a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-5\,a^4+24\,a^2\,b^2+8\,b^4\right )}{4\,a\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,a^6-36\,a^4\,b^2+56\,a^2\,b^4+32\,b^6\right )}{4\,a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,a^6+84\,a^4\,b^2-56\,a^2\,b^4-32\,b^6\right )}{4\,a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (5\,a^4+16\,a^2\,b^2+8\,b^4\right )}{4\,a\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (5\,a^2\,b+2\,b^3\right )\,\left (3\,a^4-24\,a^2\,b^2+8\,b^4\right )}{4\,a^4\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (6\,a^4-48\,a^2\,b^2+16\,b^4\right )+a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+a^4-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,a^4-24\,a^2\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (4\,a^4-24\,a^2\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (32\,a\,b^3-24\,a^3\,b\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (32\,a\,b^3-24\,a^3\,b\right )+8\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-8\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}+\frac {\mathrm {atan}\left (\frac {-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^5+a^4\,b\,1{}\mathrm {i}-2{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b^2+a^2\,b^3\,2{}\mathrm {i}-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^4+b^5\,1{}\mathrm {i}}{{\left (a^2+b^2\right )}^{5/2}}\right )\,3{}\mathrm {i}}{4\,d\,{\left (a^2+b^2\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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