Optimal. Leaf size=384 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{b}+\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-b^{2/5}}}+\frac {2 \tan ^{-1}\left (\frac {(-1)^{2/5} \sqrt [5]{b}+\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}+\frac {2 \tan ^{-1}\left (\frac {(-1)^{4/5} \sqrt [5]{b}+\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}-\frac {2 \tan ^{-1}\left (\frac {(-1)^{3/5} \left (\sqrt [5]{b}+(-1)^{2/5} \sqrt [5]{a} \tan \left (\frac {x}{2}\right )\right )}{\sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{-1} \left (\sqrt [5]{b}+(-1)^{4/5} \sqrt [5]{a} \tan \left (\frac {x}{2}\right )\right )}{\sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}} \]
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Rubi [A]
time = 0.49, antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3292, 2739,
632, 210} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sqrt [5]{a} \tan \left (\frac {x}{2}\right )+\sqrt [5]{b}}{\sqrt {a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-b^{2/5}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt [5]{a} \tan \left (\frac {x}{2}\right )+(-1)^{2/5} \sqrt [5]{b}}{\sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt [5]{a} \tan \left (\frac {x}{2}\right )+(-1)^{4/5} \sqrt [5]{b}}{\sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}-\frac {2 \text {ArcTan}\left (\frac {(-1)^{3/5} \left ((-1)^{2/5} \sqrt [5]{a} \tan \left (\frac {x}{2}\right )+\sqrt [5]{b}\right )}{\sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}}-\frac {2 \text {ArcTan}\left (\frac {\sqrt [5]{-1} \left ((-1)^{4/5} \sqrt [5]{a} \tan \left (\frac {x}{2}\right )+\sqrt [5]{b}\right )}{\sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 3292
Rubi steps
\begin {align*} \int \frac {1}{a+b \sin ^5(x)} \, dx &=\int \left (-\frac {1}{5 a^{4/5} \left (-\sqrt [5]{a}-\sqrt [5]{b} \sin (x)\right )}-\frac {1}{5 a^{4/5} \left (-\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b} \sin (x)\right )}-\frac {1}{5 a^{4/5} \left (-\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b} \sin (x)\right )}-\frac {1}{5 a^{4/5} \left (-\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b} \sin (x)\right )}-\frac {1}{5 a^{4/5} \left (-\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b} \sin (x)\right )}\right ) \, dx\\ &=-\frac {\int \frac {1}{-\sqrt [5]{a}-\sqrt [5]{b} \sin (x)} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{-\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b} \sin (x)} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{-\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b} \sin (x)} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{-\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b} \sin (x)} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{-\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b} \sin (x)} \, dx}{5 a^{4/5}}\\ &=-\frac {2 \text {Subst}\left (\int \frac {1}{-\sqrt [5]{a}-2 \sqrt [5]{b} x-\sqrt [5]{a} 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}+2 \sqrt [5]{-1} \sqrt [5]{b} x-\sqrt [5]{a} 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}-2 (-1)^{2/5} \sqrt [5]{b} x-\sqrt [5]{a} 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}+2 (-1)^{3/5} \sqrt [5]{b} x-\sqrt [5]{a} 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}-2 (-1)^{4/5} \sqrt [5]{b} x-\sqrt [5]{a} x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}\\ &=\frac {4 \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/5}-b^{2/5}\right )-x^2} \, dx,x,-2 \sqrt [5]{b}-2 \sqrt [5]{a} \tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {4 \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/5}+\sqrt [5]{-1} b^{2/5}\right )-x^2} \, dx,x,2 (-1)^{3/5} \sqrt [5]{b}-2 \sqrt [5]{a} \tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {4 \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/5}-(-1)^{2/5} b^{2/5}\right )-x^2} \, dx,x,2 \sqrt [5]{-1} \sqrt [5]{b}-2 \sqrt [5]{a} \tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {4 \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/5}+(-1)^{3/5} b^{2/5}\right )-x^2} \, dx,x,-2 (-1)^{4/5} \sqrt [5]{b}-2 \sqrt [5]{a} \tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {4 \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/5}-(-1)^{4/5} b^{2/5}\right )-x^2} \, dx,x,-2 (-1)^{2/5} \sqrt [5]{b}-2 \sqrt [5]{a} \tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{-1} \sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}}-\frac {2 \tan ^{-1}\left (\frac {(-1)^{3/5} \sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{b}+\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-b^{2/5}}}+\frac {2 \tan ^{-1}\left (\frac {(-1)^{2/5} \sqrt [5]{b}+\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}+\frac {2 \tan ^{-1}\left (\frac {(-1)^{4/5} \sqrt [5]{b}+\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.15, size = 149, normalized size = 0.39 \begin {gather*} \frac {8}{5} i \text {RootSum}\left [i b-5 i b \text {$\#$1}^2+10 i b \text {$\#$1}^4+32 a \text {$\#$1}^5-10 i b \text {$\#$1}^6+5 i b \text {$\#$1}^8-i b \text {$\#$1}^{10}\&,\frac {2 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{b-4 b \text {$\#$1}^2+16 i a \text {$\#$1}^3+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$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 = 0.71, size = 109, normalized size = 0.28
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{10}+5 a \,\textit {\_Z}^{8}+10 a \,\textit {\_Z}^{6}+32 b \,\textit {\_Z}^{5}+10 a \,\textit {\_Z}^{4}+5 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{8}+4 \textit {\_R}^{6}+6 \textit {\_R}^{4}+4 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{9} a +4 \textit {\_R}^{7} a +6 \textit {\_R}^{5} a +16 \textit {\_R}^{4} b +4 \textit {\_R}^{3} a +\textit {\_R} a}\right )}{5}\) | \(109\) |
risch | \(\munderset {\textit {\_R} =\RootOf \left (1+\left (9765625 a^{10}-9765625 a^{8} b^{2}\right ) \textit {\_Z}^{10}+1953125 a^{8} \textit {\_Z}^{8}+156250 a^{6} \textit {\_Z}^{6}+6250 a^{4} \textit {\_Z}^{4}+125 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+\left (\frac {11718750 a^{10}}{b}-11718750 a^{8} b \right ) \textit {\_R}^{9}+\left (\frac {1171875 i a^{9}}{b}-1171875 i a^{7} b \right ) \textit {\_R}^{8}+\left (\frac {2109375 a^{8}}{b}+234375 a^{6} b \right ) \textit {\_R}^{7}+\left (\frac {218750 i a^{7}}{b}+15625 i a^{5} b \right ) \textit {\_R}^{6}+\left (\frac {143750 a^{6}}{b}-3125 a^{4} b \right ) \textit {\_R}^{5}+\frac {15625 i a^{5} \textit {\_R}^{4}}{b}+\frac {4375 a^{4} \textit {\_R}^{3}}{b}+\frac {500 i a^{3} \textit {\_R}^{2}}{b}+\frac {50 a^{2} \textit {\_R}}{b}+\frac {6 i a}{b}\right )\) | \(216\) |
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 \sin ^{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 = 19.75, size = 1515, normalized size = 3.95 \begin {gather*} \sum _{k=1}^{10}\ln \left (-a\,b^7\,\left (16\,\mathrm {tan}\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 )}^3\,a^3\,5425+{\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\,196875+{\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\,3171875+{\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\,19140625+{\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 {tan}\left (\frac {x}{2}\right )\,1560+{\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 {tan}\left (\frac {x}{2}\right )\,57000+{\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 {tan}\left (\frac {x}{2}\right )\,925000+{\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 {tan}\left (\frac {x}{2}\right )\,5625000+{\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\,14000+{\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\,175000+{\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\,546875+\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 {tan}\left (\frac {x}{2}\right )\,128+{\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\,1000000-{\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\,18750000+{\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\,320+{\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\,\mathrm {tan}\left (\frac {x}{2}\right )\,6400+{\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\,\mathrm {tan}\left (\frac {x}{2}\right )\,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\,\mathrm {tan}\left (\frac {x}{2}\right )\,500000+{\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\,\mathrm {tan}\left (\frac {x}{2}\right )\,390625+{\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 {tan}\left (\frac {x}{2}\right )\,400000-{\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 {tan}\left (\frac {x}{2}\right )\,5000000\right )\,10995116277760\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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