Optimal. Leaf size=245 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}-\sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}+\sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+\sqrt [4]{b}}} \]
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Rubi [A] time = 0.53, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3211, 3181, 203} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}+\sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+\sqrt [4]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {a \sqrt [4]{b}+(-a)^{5/4}} \tan (x)}{(-a)^{5/8}}\right )}{4 (-a)^{3/8} \sqrt {a \sqrt [4]{b}+(-a)^{5/4}}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3181
Rule 3211
Rubi steps
\begin {align*} \int \frac {1}{a+b \sin ^8(x)} \, dx &=\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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\tan ^{-1}\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 {\tan ^{-1}\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 {\tan ^{-1}\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 {\tan ^{-1}\left (\frac {\sqrt {(-a)^{5/4}+a \sqrt [4]{b}} \tan (x)}{(-a)^{5/8}}\right )}{4 (-a)^{3/8} \sqrt {(-a)^{5/4}+a \sqrt [4]{b}}}\\ \end {align*}
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Mathematica [C] time = 0.26, size = 174, normalized size = 0.71 \[ 8 \text {RootSum}\left [\text {$\#$1}^8 b-8 \text {$\#$1}^7 b+28 \text {$\#$1}^6 b-56 \text {$\#$1}^5 b+256 \text {$\#$1}^4 a+70 \text {$\#$1}^4 b-56 \text {$\#$1}^3 b+28 \text {$\#$1}^2 b-8 \text {$\#$1} b+b\& ,\frac {2 \text {$\#$1}^3 \tan ^{-1}\left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right )-i \text {$\#$1}^3 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (2 x)+1\right )}{\text {$\#$1}^7 b-7 \text {$\#$1}^6 b+21 \text {$\#$1}^5 b-35 \text {$\#$1}^4 b+128 \text {$\#$1}^3 a+35 \text {$\#$1}^3 b-21 \text {$\#$1}^2 b+7 \text {$\#$1} b-b}\& \right ] \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{b \sin \relax (x)^{8} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 85, normalized size = 0.35 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a +b \right ) \textit {\_Z}^{8}+4 a \,\textit {\_Z}^{6}+6 a \,\textit {\_Z}^{4}+4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{6}+3 \textit {\_R}^{4}+3 \textit {\_R}^{2}+1\right ) \ln \left (\tan \relax (x )-\textit {\_R} \right )}{\textit {\_R}^{7} a +\textit {\_R}^{7} b +3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a +\textit {\_R} a}\right )}{8} \]
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 \[ \int \frac {1}{b \sin \relax (x)^{8} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 16.95, size = 816, normalized size = 3.33 \[ \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}\relax (x)\,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}\relax (x)\,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}\relax (x)\,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}\relax (x)\,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}\relax (x)\,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}\relax (x)\,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}\relax (x)\,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}\relax (x)\,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 ) \]
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 \[ \int \frac {1}{a + b \sin ^{8}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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