Integrand size = 11, antiderivative size = 379 \[ \int \frac {1}{a-b \sin ^5(x)} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-b^{2/5}}}-\frac {2 \arctan \left (\frac {(-1)^{2/5} \sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}-\frac {2 \arctan \left (\frac {(-1)^{4/5} \sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}+\frac {2 \arctan \left (\frac {\sqrt [5]{-1} \sqrt [5]{b}+\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}}+\frac {2 \arctan \left (\frac {(-1)^{3/5} \sqrt [5]{b}+\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}} \]
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Time = 0.52 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3292, 2739, 632, 210} \[ \int \frac {1}{a-b \sin ^5(x)} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-b^{2/5}}}-\frac {2 \arctan \left (\frac {(-1)^{2/5} \sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}-\frac {2 \arctan \left (\frac {(-1)^{4/5} \sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}+\frac {2 \arctan \left (\frac {\sqrt [5]{a} \tan \left (\frac {x}{2}\right )+\sqrt [5]{-1} \sqrt [5]{b}}{\sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}}+\frac {2 \arctan \left (\frac {\sqrt [5]{a} \tan \left (\frac {x}{2}\right )+(-1)^{3/5} \sqrt [5]{b}}{\sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}} \]
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Rule 210
Rule 632
Rule 2739
Rule 3292
Rubi steps \begin{align*} \text {integral}& = \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 \arctan \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 \arctan \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 \arctan \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 \arctan \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 \arctan \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}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.12 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.39 \[ \int \frac {1}{a-b \sin ^5(x)} \, dx=-\frac {8}{5} i \text {RootSum}\left [-i b+5 i b \text {$\#$1}^2-10 i b \text {$\#$1}^4+32 a \text {$\#$1}^5+10 i b \text {$\#$1}^6-5 i b \text {$\#$1}^8+i b \text {$\#$1}^{10}\&,\frac {2 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{b-4 b \text {$\#$1}^2-16 i a \text {$\#$1}^3+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.01 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.29
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{10}+5 a \,\textit {\_Z}^{8}+10 a \,\textit {\_Z}^{6}-32 b \,\textit {\_Z}^{5}+10 a \,\textit {\_Z}^{4}+5 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{8}+4 \textit {\_R}^{6}+6 \textit {\_R}^{4}+4 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{9} a +4 \textit {\_R}^{7} a +6 \textit {\_R}^{5} a -16 \textit {\_R}^{4} b +4 \textit {\_R}^{3} a +\textit {\_R} a}\right )}{5}\) | \(109\) |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (9765625 a^{10}-9765625 a^{8} b^{2}\right ) \textit {\_Z}^{10}+1953125 a^{8} \textit {\_Z}^{8}+156250 a^{6} \textit {\_Z}^{6}+6250 a^{4} \textit {\_Z}^{4}+125 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+\left (-\frac {11718750 a^{10}}{b}+11718750 a^{8} b \right ) \textit {\_R}^{9}+\left (-\frac {1171875 i a^{9}}{b}+1171875 i a^{7} b \right ) \textit {\_R}^{8}+\left (-\frac {2109375 a^{8}}{b}-234375 a^{6} b \right ) \textit {\_R}^{7}+\left (-\frac {218750 i a^{7}}{b}-15625 i a^{5} b \right ) \textit {\_R}^{6}+\left (-\frac {143750 a^{6}}{b}+3125 a^{4} b \right ) \textit {\_R}^{5}-\frac {15625 i a^{5} \textit {\_R}^{4}}{b}-\frac {4375 a^{4} \textit {\_R}^{3}}{b}-\frac {500 i a^{3} \textit {\_R}^{2}}{b}-\frac {50 a^{2} \textit {\_R}}{b}-\frac {6 i a}{b}\right )\) | \(216\) |
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Exception generated. \[ \int \frac {1}{a-b \sin ^5(x)} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{a-b \sin ^5(x)} \, dx=\int \frac {1}{a - b \sin ^{5}{\left (x \right )}}\, dx \]
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\[ \int \frac {1}{a-b \sin ^5(x)} \, dx=\int { -\frac {1}{b \sin \left (x\right )^{5} - a} \,d x } \]
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\[ \int \frac {1}{a-b \sin ^5(x)} \, dx=\int { -\frac {1}{b \sin \left (x\right )^{5} - a} \,d x } \]
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Time = 18.73 (sec) , antiderivative size = 1515, normalized size of antiderivative = 4.00 \[ \int \frac {1}{a-b \sin ^5(x)} \, dx=\text {Too large to display} \]
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