Integrand size = 11, antiderivative size = 213 \[ \int \frac {1}{a-b \sin ^8(x)} \, dx=\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}-\sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}-\sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}-i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}-i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}+i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}+i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}+\sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}+\sqrt [4]{b}}} \]
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Time = 0.34 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3290, 3260, 209} \[ \int \frac {1}{a-b \sin ^8(x)} \, dx=\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}-\sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}-\sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}-i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}-i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}+i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}+i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}+\sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}+\sqrt [4]{b}}} \]
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Rule 209
Rule 3260
Rule 3290
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{1-\frac {\sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{a}}} \, dx}{4 a}+\frac {\int \frac {1}{1-\frac {i \sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{a}}} \, dx}{4 a}+\frac {\int \frac {1}{1+\frac {i \sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{a}}} \, dx}{4 a}+\frac {\int \frac {1}{1+\frac {\sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{a}}} \, dx}{4 a} \\ & = \frac {\text {Subst}\left (\int \frac {1}{1+\left (1-\frac {\sqrt [4]{b}}{\sqrt [4]{a}}\right ) x^2} \, dx,x,\tan (x)\right )}{4 a}+\frac {\text {Subst}\left (\int \frac {1}{1+\left (1-\frac {i \sqrt [4]{b}}{\sqrt [4]{a}}\right ) x^2} \, dx,x,\tan (x)\right )}{4 a}+\frac {\text {Subst}\left (\int \frac {1}{1+\left (1+\frac {i \sqrt [4]{b}}{\sqrt [4]{a}}\right ) x^2} \, dx,x,\tan (x)\right )}{4 a}+\frac {\text {Subst}\left (\int \frac {1}{1+\left (1+\frac {\sqrt [4]{b}}{\sqrt [4]{a}}\right ) x^2} \, dx,x,\tan (x)\right )}{4 a} \\ & = \frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}-\sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}-\sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}-i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}-i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}+i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}+i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}+\sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}+\sqrt [4]{b}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.02 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.82 \[ \int \frac {1}{a-b \sin ^8(x)} \, dx=-8 \text {RootSum}\left [b-8 b \text {$\#$1}+28 b \text {$\#$1}^2-56 b \text {$\#$1}^3-256 a \text {$\#$1}^4+70 b \text {$\#$1}^4-56 b \text {$\#$1}^5+28 b \text {$\#$1}^6-8 b \text {$\#$1}^7+b \text {$\#$1}^8\&,\frac {2 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{-b+7 b \text {$\#$1}-21 b \text {$\#$1}^2-128 a \text {$\#$1}^3+35 b \text {$\#$1}^3-35 b \text {$\#$1}^4+21 b \text {$\#$1}^5-7 b \text {$\#$1}^6+b \text {$\#$1}^7}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.73 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.41
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a -b \right ) \textit {\_Z}^{8}+4 a \,\textit {\_Z}^{6}+6 a \,\textit {\_Z}^{4}+4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{6}+3 \textit {\_R}^{4}+3 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (x \right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -\textit {\_R}^{7} b +3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a +\textit {\_R} a}\right )}{8}\) | \(88\) |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (16777216 a^{8}-16777216 a^{7} b \right ) \textit {\_Z}^{8}+1048576 a^{6} \textit {\_Z}^{6}+24576 a^{4} \textit {\_Z}^{4}+256 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (\frac {4194304 i a^{8}}{b}-4194304 i a^{7}\right ) \textit {\_R}^{7}+\left (-\frac {524288 a^{7}}{b}+524288 a^{6}\right ) \textit {\_R}^{6}+\left (\frac {196608 i a^{6}}{b}+65536 i a^{5}\right ) \textit {\_R}^{5}+\left (-\frac {24576 a^{5}}{b}-8192 a^{4}\right ) \textit {\_R}^{4}+\left (\frac {3072 i a^{4}}{b}-1024 i a^{3}\right ) \textit {\_R}^{3}+\left (-\frac {384 a^{3}}{b}+128 a^{2}\right ) \textit {\_R}^{2}+\left (\frac {16 i a^{2}}{b}+16 i a \right ) \textit {\_R} -\frac {2 a}{b}-1\right )\) | \(193\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 643307 vs. \(2 (133) = 266\).
Time = 6.89 (sec) , antiderivative size = 643307, normalized size of antiderivative = 3020.22 \[ \int \frac {1}{a-b \sin ^8(x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{a-b \sin ^8(x)} \, dx=\int \frac {1}{a - b \sin ^{8}{\left (x \right )}}\, dx \]
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\[ \int \frac {1}{a-b \sin ^8(x)} \, dx=\int { -\frac {1}{b \sin \left (x\right )^{8} - a} \,d x } \]
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\[ \int \frac {1}{a-b \sin ^8(x)} \, dx=\int { -\frac {1}{b \sin \left (x\right )^{8} - a} \,d x } \]
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Time = 16.36 (sec) , antiderivative size = 818, normalized size of antiderivative = 3.84 \[ \int \frac {1}{a-b \sin ^8(x)} \, dx=\sum _{k=1}^8\ln \left (-b^5\,\left (a-b\right )\,\left (-{\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )}^2\,a^2\,800-{\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )}^4\,a^4\,43008-{\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )}^6\,a^6\,786432+\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )\,b\,\mathrm {tan}\left (x\right )\,4-{\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )}^4\,a^3\,b\,6144+{\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )}^6\,a^5\,b\,786432+{\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )}^3\,a^3\,\mathrm {tan}\left (x\right )\,9984+{\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )}^5\,a^5\,\mathrm {tan}\left (x\right )\,557056+{\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )}^7\,a^7\,\mathrm {tan}\left (x\right )\,10485760+{\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )}^2\,a\,b\,32+\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )\,a\,\mathrm {tan}\left (x\right )\,60-{\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )}^3\,a^2\,b\,\mathrm {tan}\left (x\right )\,768+{\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )}^5\,a^4\,b\,\mathrm {tan}\left (x\right )\,98304-{\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right )}^7\,a^6\,b\,\mathrm {tan}\left (x\right )\,10485760-5\right )\,2\right )\,\mathrm {root}\left (16777216\,a^7\,b\,d^8-16777216\,a^8\,d^8-1048576\,a^6\,d^6-24576\,a^4\,d^4-256\,a^2\,d^2-1,d,k\right ) \]
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