Optimal. Leaf size=494 \[ \frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \]
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Rubi [A]
time = 0.65, antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3292, 2738,
211} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 2738
Rule 3292
Rubi steps
\begin {align*} \int \frac {1}{a+b \cos ^5(x)} \, dx &=\int \left (-\frac {1}{5 a^{4/5} \left (-\sqrt [5]{a}-\sqrt [5]{b} \cos (x)\right )}-\frac {1}{5 a^{4/5} \left (-\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b} \cos (x)\right )}-\frac {1}{5 a^{4/5} \left (-\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b} \cos (x)\right )}-\frac {1}{5 a^{4/5} \left (-\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b} \cos (x)\right )}-\frac {1}{5 a^{4/5} \left (-\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b} \cos (x)\right )}\right ) \, dx\\ &=-\frac {\int \frac {1}{-\sqrt [5]{a}-\sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{-\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{-\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{-\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{-\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}\\ &=-\frac {2 \text {Subst}\left (\int \frac {1}{-\sqrt [5]{a}-\sqrt [5]{b}+\left (-\sqrt [5]{a}+\sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}-\frac {2 \text {Subst}\left (\int \frac {1}{-\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}+\left (-\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}-\frac {2 \text {Subst}\left (\int \frac {1}{-\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}+\left (-\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}-\frac {2 \text {Subst}\left (\int \frac {1}{-\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}+\left (-\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}-\frac {2 \text {Subst}\left (\int \frac {1}{-\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}+\left (-\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.22, size = 130, normalized size = 0.26 \begin {gather*} \frac {8}{5} \text {RootSum}\left [b+5 b \text {$\#$1}^2+10 b \text {$\#$1}^4+32 a \text {$\#$1}^5+10 b \text {$\#$1}^6+5 b \text {$\#$1}^8+b \text {$\#$1}^{10}\&,\frac {2 \text {ArcTan}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{b+4 b \text {$\#$1}^2+16 a \text {$\#$1}^3+6 b \text {$\#$1}^4+4 b \text {$\#$1}^6+b \text {$\#$1}^8}\&\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.52, size = 150, normalized size = 0.30
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a -b \right ) \textit {\_Z}^{10}+\left (5 a +5 b \right ) \textit {\_Z}^{8}+\left (10 a -10 b \right ) \textit {\_Z}^{6}+\left (10 a +10 b \right ) \textit {\_Z}^{4}+\left (5 a -5 b \right ) \textit {\_Z}^{2}+a +b \right )}{\sum }\frac {\left (\textit {\_R}^{8}+4 \textit {\_R}^{6}+6 \textit {\_R}^{4}+4 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{9} a -\textit {\_R}^{9} b +4 \textit {\_R}^{7} a +4 \textit {\_R}^{7} b +6 \textit {\_R}^{5} a -6 \textit {\_R}^{5} b +4 \textit {\_R}^{3} a +4 \textit {\_R}^{3} b +\textit {\_R} a -\textit {\_R} b}\right )}{5}\) | \(150\) |
risch | \(\munderset {\textit {\_R} =\RootOf \left (1+\left (9765625 a^{10}-9765625 a^{8} b^{2}\right ) \textit {\_Z}^{10}+1953125 a^{8} \textit {\_Z}^{8}+156250 a^{6} \textit {\_Z}^{6}+6250 a^{4} \textit {\_Z}^{4}+125 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+\left (-\frac {11718750 i a^{10}}{b}+11718750 i b \,a^{8}\right ) \textit {\_R}^{9}+\left (\frac {1171875 a^{9}}{b}-1171875 a^{7} b \right ) \textit {\_R}^{8}+\left (-\frac {2109375 i a^{8}}{b}-234375 i a^{6} b \right ) \textit {\_R}^{7}+\left (\frac {218750 a^{7}}{b}+15625 a^{5} b \right ) \textit {\_R}^{6}+\left (-\frac {143750 i a^{6}}{b}+3125 i a^{4} b \right ) \textit {\_R}^{5}+\frac {15625 a^{5} \textit {\_R}^{4}}{b}-\frac {4375 i a^{4} \textit {\_R}^{3}}{b}+\frac {500 a^{3} \textit {\_R}^{2}}{b}-\frac {50 i a^{2} \textit {\_R}}{b}+\frac {6 a}{b}\right )\) | \(217\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a + b \cos ^{5}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.82, size = 1520, normalized size = 3.08 \begin {gather*} \sum _{k=1}^{10}\ln \left (-\frac {b^7\,\left (a-b\right )\,\left (7\,\mathrm {cot}\left (\frac {x}{2}\right )-\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )\,a\,56+\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )\,b-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^3\,a^3\,5800-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^5\,a^5\,225000-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^7\,a^7\,3875000-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^9\,a^9\,25000000+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^2\,a^2\,\mathrm {cot}\left (\frac {x}{2}\right )\,735+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^4\,a^4\,\mathrm {cot}\left (\frac {x}{2}\right )\,28875+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^6\,a^6\,\mathrm {cot}\left (\frac {x}{2}\right )\,503125+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^8\,a^8\,\mathrm {cot}\left (\frac {x}{2}\right )\,3281250+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^3\,a^2\,b\,800+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^5\,a^4\,b\,100000+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^7\,a^6\,b\,4000000+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^9\,a^8\,b\,50000000-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^7\,a^5\,b^2\,125000-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^9\,a^7\,b^2\,25000000-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^2\,a\,b\,\mathrm {cot}\left (\frac {x}{2}\right )\,35-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^4\,a^3\,b\,\mathrm {cot}\left (\frac {x}{2}\right )\,7000-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^6\,a^5\,b\,\mathrm {cot}\left (\frac {x}{2}\right )\,350000-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^8\,a^7\,b\,\mathrm {cot}\left (\frac {x}{2}\right )\,5000000+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^6\,a^4\,b^2\,\mathrm {cot}\left (\frac {x}{2}\right )\,3125+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^8\,a^6\,b^2\,\mathrm {cot}\left (\frac {x}{2}\right )\,1718750\right )\,10995116277760}{\mathrm {cot}\left (\frac {x}{2}\right )}\right )\,\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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