Integrand size = 11, antiderivative size = 175 \[ \int \frac {1}{a-b \sin ^6(x)} \, dx=\frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \]
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Time = 0.34 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3290, 3260, 209} \[ \int \frac {1}{a-b \sin ^6(x)} \, dx=\frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \]
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Rule 209
Rule 3260
Rule 3290
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{1-\frac {\sqrt [3]{b} \sin ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac {\int \frac {1}{1+\frac {\sqrt [3]{-1} \sqrt [3]{b} \sin ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac {\int \frac {1}{1-\frac {(-1)^{2/3} \sqrt [3]{b} \sin ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a} \\ & = \frac {\text {Subst}\left (\int \frac {1}{1+\left (1-\frac {\sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\tan (x)\right )}{3 a}+\frac {\text {Subst}\left (\int \frac {1}{1+\left (1+\frac {\sqrt [3]{-1} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\tan (x)\right )}{3 a}+\frac {\text {Subst}\left (\int \frac {1}{1+\left (1-\frac {(-1)^{2/3} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\tan (x)\right )}{3 a} \\ & = \frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.85 \[ \int \frac {1}{a-b \sin ^6(x)} \, dx=\frac {8}{3} \text {RootSum}\left [b-6 b \text {$\#$1}+15 b \text {$\#$1}^2+64 a \text {$\#$1}^3-20 b \text {$\#$1}^3+15 b \text {$\#$1}^4-6 b \text {$\#$1}^5+b \text {$\#$1}^6\&,\frac {2 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^2-i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2}{-b+5 b \text {$\#$1}+32 a \text {$\#$1}^2-10 b \text {$\#$1}^2+10 b \text {$\#$1}^3-5 b \text {$\#$1}^4+b \text {$\#$1}^5}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.41
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a -b \right ) \textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (x \right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -\textit {\_R}^{5} b +2 \textit {\_R}^{3} a +\textit {\_R} a}\right )}{6}\) | \(71\) |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (46656 a^{6}-46656 a^{5} b \right ) \textit {\_Z}^{6}+3888 a^{4} \textit {\_Z}^{4}+108 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-\frac {15552 i a^{6}}{b}+15552 i a^{5}\right ) \textit {\_R}^{5}+\left (\frac {2592 a^{5}}{b}-2592 a^{4}\right ) \textit {\_R}^{4}+\left (-\frac {864 i a^{4}}{b}-432 i a^{3}\right ) \textit {\_R}^{3}+\left (\frac {144 a^{3}}{b}+72 a^{2}\right ) \textit {\_R}^{2}+\left (-\frac {12 i a^{2}}{b}+12 i a \right ) \textit {\_R} +\frac {2 a}{b}-1\right )\) | \(147\) |
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Result contains complex when optimal does not.
Time = 1.76 (sec) , antiderivative size = 16697, normalized size of antiderivative = 95.41 \[ \int \frac {1}{a-b \sin ^6(x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{a-b \sin ^6(x)} \, dx=\int \frac {1}{a - b \sin ^{6}{\left (x \right )}}\, dx \]
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\[ \int \frac {1}{a-b \sin ^6(x)} \, dx=\int { -\frac {1}{b \sin \left (x\right )^{6} - a} \,d x } \]
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\[ \int \frac {1}{a-b \sin ^6(x)} \, dx=\int { -\frac {1}{b \sin \left (x\right )^{6} - a} \,d x } \]
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Time = 14.75 (sec) , antiderivative size = 513, normalized size of antiderivative = 2.93 \[ \int \frac {1}{a-b \sin ^6(x)} \, dx=\sum _{k=1}^6\ln \left (-\frac {b^3\,\left (a-b\right )\,\left (\mathrm {cot}\left (x\right )-\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )\,a\,8+\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )\,b\,2-{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^3\,a^3\,504-{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^5\,a^5\,7776-{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^3\,a^2\,b\,144+{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^5\,a^4\,b\,7776+{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^2\,a^2\,\mathrm {cot}\left (x\right )\,60+{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^4\,a^4\,\mathrm {cot}\left (x\right )\,864-{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^4\,a^3\,b\,\mathrm {cot}\left (x\right )\,864+{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^2\,a\,b\,\mathrm {cot}\left (x\right )\,12\right )\,3}{\mathrm {cot}\left (x\right )}\right )\,\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right ) \]
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