Optimal. Leaf size=210 \[ -\frac {3 a b \left (a^4-6 a^2 b^2+b^4\right ) x}{4 \left (a^2+b^2\right )^4}-\frac {b^2 \cos ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac {3 a^2 b^2 \left (a^2-b^2\right ) \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4}+\frac {a b \left (5 a^2-3 b^2\right ) \cos (x) \sin (x)}{4 \left (a^2+b^2\right )^3}-\frac {a b \cos ^3(x) \sin (x)}{2 \left (a^2+b^2\right )^2}-\frac {2 a^2 b^2 \sin ^2(x)}{\left (a^2+b^2\right )^3}+\frac {a^2 \sin ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac {a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))} \]
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Rubi [A]
time = 0.86, antiderivative size = 289, normalized size of antiderivative = 1.38, number of steps
used = 48, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3190, 3188,
2645, 30, 2648, 2715, 8, 2644, 3177, 3212, 3176, 3154} \begin {gather*} \frac {a b x}{4 \left (a^2+b^2\right )^2}+\frac {a^2 \sin ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac {2 a^2 b^2 \sin ^2(x)}{\left (a^2+b^2\right )^3}-\frac {b^2 \cos ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac {a b \sin (x) \cos ^3(x)}{2 \left (a^2+b^2\right )^2}+\frac {a b \sin (x) \cos (x)}{4 \left (a^2+b^2\right )^2}+\frac {3 a^2 b^4 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4}-\frac {a b^3 x}{\left (a^2+b^2\right )^3}-\frac {a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))}-\frac {a b^3 \sin (x) \cos (x)}{\left (a^2+b^2\right )^3}-\frac {3 a^4 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4}-\frac {a^3 b x}{\left (a^2+b^2\right )^3}+\frac {a^3 b \sin (x) \cos (x)}{\left (a^2+b^2\right )^3}+\frac {6 a^3 b^3 x}{\left (a^2+b^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2644
Rule 2645
Rule 2648
Rule 2715
Rule 3154
Rule 3176
Rule 3177
Rule 3188
Rule 3190
Rule 3212
Rubi steps
\begin {align*} \int \frac {\cos ^3(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx &=\frac {a \int \frac {\cos ^2(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}+\frac {b \int \frac {\cos ^3(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}-\frac {(a b) \int \frac {\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx}{a^2+b^2}\\ &=\frac {a^2 \int \cos (x) \sin ^3(x) \, dx}{\left (a^2+b^2\right )^2}+2 \frac {(a b) \int \cos ^2(x) \sin ^2(x) \, dx}{\left (a^2+b^2\right )^2}-2 \frac {\left (a^2 b\right ) \int \frac {\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac {b^2 \int \cos ^3(x) \sin (x) \, dx}{\left (a^2+b^2\right )^2}-2 \frac {\left (a b^2\right ) \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a^2 b^2\right ) \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx}{\left (a^2+b^2\right )^2}\\ &=-2 \left (\frac {\left (a^3 b\right ) \int \sin ^2(x) \, dx}{\left (a^2+b^2\right )^3}+\frac {\left (a^2 b^2\right ) \int \cos (x) \sin (x) \, dx}{\left (a^2+b^2\right )^3}-\frac {\left (a^3 b^2\right ) \int \frac {\sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}\right )+\frac {\left (a^3 b^2\right ) \int \frac {\sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}-2 \left (\frac {\left (a^2 b^2\right ) \int \cos (x) \sin (x) \, dx}{\left (a^2+b^2\right )^3}+\frac {\left (a b^3\right ) \int \cos ^2(x) \, dx}{\left (a^2+b^2\right )^3}-\frac {\left (a^2 b^3\right ) \int \frac {\cos (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}\right )+\frac {\left (a^2 b^3\right ) \int \frac {\cos (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}-\frac {\left (a^3 b^3\right ) \int \frac {1}{(a \cos (x)+b \sin (x))^2} \, dx}{\left (a^2+b^2\right )^3}+\frac {a^2 \text {Subst}\left (\int x^3 \, dx,x,\sin (x)\right )}{\left (a^2+b^2\right )^2}+2 \left (-\frac {a b \cos ^3(x) \sin (x)}{4 \left (a^2+b^2\right )^2}+\frac {(a b) \int \cos ^2(x) \, dx}{4 \left (a^2+b^2\right )^2}\right )-\frac {b^2 \text {Subst}\left (\int x^3 \, dx,x,\cos (x)\right )}{\left (a^2+b^2\right )^2}\\ &=\frac {2 a^3 b^3 x}{\left (a^2+b^2\right )^4}-\frac {b^2 \cos ^4(x)}{4 \left (a^2+b^2\right )^2}+\frac {a^2 \sin ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac {a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))}-\frac {\left (a^4 b^2\right ) \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^4}+\frac {\left (a^2 b^4\right ) \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^4}-2 \left (-\frac {a^3 b^3 x}{\left (a^2+b^2\right )^4}-\frac {a^3 b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^3}+\frac {\left (a^4 b^2\right ) \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^4}+\frac {\left (a^3 b\right ) \int 1 \, dx}{2 \left (a^2+b^2\right )^3}+\frac {\left (a^2 b^2\right ) \text {Subst}(\int x \, dx,x,\sin (x))}{\left (a^2+b^2\right )^3}\right )-2 \left (-\frac {a^3 b^3 x}{\left (a^2+b^2\right )^4}+\frac {a b^3 \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^3}-\frac {\left (a^2 b^4\right ) \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^4}+\frac {\left (a^2 b^2\right ) \text {Subst}(\int x \, dx,x,\sin (x))}{\left (a^2+b^2\right )^3}+\frac {\left (a b^3\right ) \int 1 \, dx}{2 \left (a^2+b^2\right )^3}\right )+2 \left (\frac {a b \cos (x) \sin (x)}{8 \left (a^2+b^2\right )^2}-\frac {a b \cos ^3(x) \sin (x)}{4 \left (a^2+b^2\right )^2}+\frac {(a b) \int 1 \, dx}{8 \left (a^2+b^2\right )^2}\right )\\ &=\frac {2 a^3 b^3 x}{\left (a^2+b^2\right )^4}-\frac {b^2 \cos ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac {a^4 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4}+\frac {a^2 b^4 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4}+\frac {a^2 \sin ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac {a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))}+2 \left (\frac {a b x}{8 \left (a^2+b^2\right )^2}+\frac {a b \cos (x) \sin (x)}{8 \left (a^2+b^2\right )^2}-\frac {a b \cos ^3(x) \sin (x)}{4 \left (a^2+b^2\right )^2}\right )-2 \left (-\frac {a^3 b^3 x}{\left (a^2+b^2\right )^4}+\frac {a^3 b x}{2 \left (a^2+b^2\right )^3}+\frac {a^4 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4}-\frac {a^3 b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^3}+\frac {a^2 b^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^3}\right )-2 \left (-\frac {a^3 b^3 x}{\left (a^2+b^2\right )^4}+\frac {a b^3 x}{2 \left (a^2+b^2\right )^3}-\frac {a^2 b^4 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4}+\frac {a b^3 \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^3}+\frac {a^2 b^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^3}\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.02, size = 409, normalized size = 1.95 \begin {gather*} \frac {-12 a b \left (a^2-3 b^2\right ) \left (3 a^2-b^2\right ) x+6 i \left (a^6-15 a^4 b^2+15 a^2 b^4-b^6\right ) x-6 i \left (a^6-15 a^4 b^2+15 a^2 b^4-b^6\right ) \text {ArcTan}(\tan (x))-4 \left (a^2+b^2\right ) \left (a^4-6 a^2 b^2+b^4\right ) \cos (2 x)+\left (a^2-b^2\right ) \left (a^2+b^2\right )^2 \cos (4 x)+3 \left (a^6-15 a^4 b^2+15 a^2 b^4-b^6\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )+\frac {2 b \left (a^2+b^2\right ) \left (3 a^4-10 a^2 b^2+3 b^4\right ) \sin (x)}{a \cos (x)+b \sin (x)}+\frac {3 \left (a^2+b^2\right )^2 \left (a \cos (x) \left (-2 i (a+i b)^2 x+\left (-a^2+b^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )\right )+b \left (2 (a+i b) (a (-1-i x)+b (i+x))+\left (-a^2+b^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )\right ) \sin (x)+2 i \left (a^2-b^2\right ) \text {ArcTan}(\tan (x)) (a \cos (x)+b \sin (x))\right )}{a \cos (x)+b \sin (x)}+16 a b \left (a^4-b^4\right ) \sin (2 x)-2 a b \left (a^2+b^2\right )^2 \sin (4 x)}{32 \left (a^2+b^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 232, normalized size = 1.10
method | result | size |
default | \(\frac {a^{3} b^{2}}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (x \right )\right )}-\frac {3 a^{2} b^{2} \left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (x \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {\frac {\left (\frac {1}{2} a^{3} b^{3}-\frac {3}{4} a \,b^{5}+\frac {5}{4} a^{5} b \right ) \left (\tan ^{3}\left (x \right )\right )+\left (-\frac {1}{2} a^{6}+a^{4} b^{2}+\frac {3}{2} a^{2} b^{4}\right ) \left (\tan ^{2}\left (x \right )\right )+\left (\frac {3}{4} a^{5} b -\frac {1}{2} a^{3} b^{3}-\frac {5}{4} a \,b^{5}\right ) \tan \left (x \right )-\frac {a^{6}}{4}+\frac {5 a^{4} b^{2}}{4}+\frac {5 a^{2} b^{4}}{4}-\frac {b^{6}}{4}}{\left (\tan ^{2}\left (x \right )+1\right )^{2}}+\frac {3 a b \left (\frac {\left (4 a^{3} b -4 a \,b^{3}\right ) \ln \left (\tan ^{2}\left (x \right )+1\right )}{2}+\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (x \right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{4}}\) | \(232\) |
risch | \(\frac {3 a b x}{4 \left (4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}\right )}+\frac {{\mathrm e}^{4 i x}}{-128 i a b +64 a^{2}-64 b^{2}}-\frac {i {\mathrm e}^{2 i x} b}{16 \left (-3 i a^{2} b +i b^{3}+a^{3}-3 a \,b^{2}\right )}-\frac {{\mathrm e}^{2 i x} a}{16 \left (-3 i a^{2} b +i b^{3}+a^{3}-3 a \,b^{2}\right )}+\frac {i {\mathrm e}^{-2 i x} b}{16 \left (2 i a b +a^{2}-b^{2}\right ) \left (i b +a \right )}-\frac {{\mathrm e}^{-2 i x} a}{16 \left (2 i a b +a^{2}-b^{2}\right ) \left (i b +a \right )}+\frac {{\mathrm e}^{-4 i x}}{128 i a b +64 a^{2}-64 b^{2}}+\frac {6 i a^{4} b^{2} x}{a^{8}+4 b^{2} a^{6}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {6 i a^{2} b^{4} x}{a^{8}+4 b^{2} a^{6}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {2 i a^{3} b^{3}}{\left (i b +a \right )^{3} \left (-i b +a \right )^{4} \left (-i b \,{\mathrm e}^{2 i x}+a \,{\mathrm e}^{2 i x}+i b +a \right )}-\frac {3 a^{4} b^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{8}+4 b^{2} a^{6}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}+\frac {3 a^{2} b^{4} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{8}+4 b^{2} a^{6}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}\) | \(487\) |
norman | \(\text {Expression too large to display}\) | \(1658\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 456 vs.
\(2 (200) = 400\).
time = 0.49, size = 456, normalized size = 2.17 \begin {gather*} -\frac {3 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} x}{4 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} - \frac {3 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} \log \left (b \tan \left (x\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} - \frac {a^{5} - 10 \, a^{3} b^{2} + a b^{4} - 3 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} \tan \left (x\right )^{4} - 3 \, {\left (a^{4} b + a^{2} b^{3}\right )} \tan \left (x\right )^{3} + {\left (2 \, a^{5} - 17 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \tan \left (x\right )^{2} - {\left (2 \, a^{4} b + a^{2} b^{3} - b^{5}\right )} \tan \left (x\right )}{4 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6} + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (x\right )^{5} + {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (x\right )^{4} + 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (x\right )^{3} + 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (x\right )^{2} + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.69, size = 371, normalized size = 1.77 \begin {gather*} \frac {8 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right )^{5} - 8 \, {\left (2 \, a^{7} + 3 \, a^{5} b^{2} - a b^{6}\right )} \cos \left (x\right )^{3} + {\left (5 \, a^{7} + 21 \, a^{5} b^{2} + 27 \, a^{3} b^{4} - 21 \, a b^{6} - 24 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} b^{5}\right )} x\right )} \cos \left (x\right ) - 48 \, {\left ({\left (a^{5} b^{2} - a^{3} b^{4}\right )} \cos \left (x\right ) + {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \sin \left (x\right )\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) + {\left (5 \, a^{6} b - 51 \, a^{4} b^{3} - 21 \, a^{2} b^{5} + 3 \, b^{7} - 8 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (x\right )^{4} + 24 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (x\right )^{2} - 24 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} x\right )} \sin \left (x\right )}{32 \, {\left ({\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (x\right ) + {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \sin \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 435 vs.
\(2 (200) = 400\).
time = 0.44, size = 435, normalized size = 2.07 \begin {gather*} -\frac {3 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} x}{4 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} + \frac {3 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} - \frac {3 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} + \frac {3 \, a^{4} b^{3} \tan \left (x\right ) - 3 \, a^{2} b^{5} \tan \left (x\right ) + 4 \, a^{5} b^{2} - 2 \, a^{3} b^{4}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (x\right ) + a\right )}} - \frac {9 \, a^{4} b^{2} \tan \left (x\right )^{4} - 9 \, a^{2} b^{4} \tan \left (x\right )^{4} - 5 \, a^{5} b \tan \left (x\right )^{3} - 2 \, a^{3} b^{3} \tan \left (x\right )^{3} + 3 \, a b^{5} \tan \left (x\right )^{3} + 2 \, a^{6} \tan \left (x\right )^{2} + 14 \, a^{4} b^{2} \tan \left (x\right )^{2} - 24 \, a^{2} b^{4} \tan \left (x\right )^{2} - 3 \, a^{5} b \tan \left (x\right ) + 2 \, a^{3} b^{3} \tan \left (x\right ) + 5 \, a b^{5} \tan \left (x\right ) + a^{6} + 4 \, a^{4} b^{2} - 14 \, a^{2} b^{4} + b^{6}}{4 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (\tan \left (x\right )^{2} + 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.92, size = 2500, normalized size = 11.90 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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