Integrand size = 10, antiderivative size = 487 \[ \int \frac {1}{a+b \sin ^4(x)} \, dx=-\frac {\left (\sqrt {a}+\sqrt {a+b}\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}-\sqrt {2} (a+b)^{3/4} \tan (x)}{\sqrt [4]{a} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}+\frac {\left (\sqrt {a}+\sqrt {a+b}\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}+\sqrt {2} (a+b)^{3/4} \tan (x)}{\sqrt [4]{a} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}+\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}-\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \tan (x)+(a+b)^{3/4} \tan ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}-\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}+\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \tan (x)+(a+b)^{3/4} \tan ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}} \]
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Time = 0.72 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3288, 1183, 648, 632, 210, 642} \[ \int \frac {1}{a+b \sin ^4(x)} \, dx=-\frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}-\sqrt {2} (a+b)^{3/4} \tan (x)}{\sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}+\frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \arctan \left (\frac {\sqrt {2} (a+b)^{3/4} \tan (x)+\sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}{\sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}+\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left ((a+b)^{3/4} \tan ^2(x)-\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \tan (x)+\sqrt {a} \sqrt [4]{a+b}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}-\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left ((a+b)^{3/4} \tan ^2(x)+\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \tan (x)+\sqrt {a} \sqrt [4]{a+b}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}} \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1183
Rule 3288
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1+x^2}{a+2 a x^2+(a+b) x^4} \, dx,x,\tan (x)\right ) \\ & = \frac {\sqrt [4]{a+b} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}-\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) x}{\frac {\sqrt {a}}{\sqrt {a+b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\tan (x)\right )}{2 \sqrt {2} a^{3/4} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}+\frac {\sqrt [4]{a+b} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}+\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) x}{\frac {\sqrt {a}}{\sqrt {a+b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\tan (x)\right )}{2 \sqrt {2} a^{3/4} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}} \\ & = \frac {\left (1+\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {a+b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\tan (x)\right )}{4 (a+b)}+\frac {\left (1+\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {a+b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\tan (x)\right )}{4 (a+b)}+\frac {\left (\sqrt [4]{a+b} \left (-1+\frac {\sqrt {a}}{\sqrt {a+b}}\right )\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}+2 x}{\frac {\sqrt {a}}{\sqrt {a+b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\tan (x)\right )}{4 \sqrt {2} a^{3/4} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}-\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}+2 x}{\frac {\sqrt {a}}{\sqrt {a+b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\tan (x)\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}} \\ & = -\frac {\sqrt [4]{a+b} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}-\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \tan (x)+(a+b)^{3/4} \tan ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}-\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}+\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \tan (x)+(a+b)^{3/4} \tan ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}-\frac {\left (1+\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {a} \left (a+b+\sqrt {a} \sqrt {a+b}\right )}{(a+b)^{3/2}}-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}+2 \tan (x)\right )}{2 (a+b)}-\frac {\left (1+\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {a} \left (a+b+\sqrt {a} \sqrt {a+b}\right )}{(a+b)^{3/2}}-x^2} \, dx,x,\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}+2 \tan (x)\right )}{2 (a+b)} \\ & = -\frac {\left (\sqrt {a}+\sqrt {a+b}\right ) \arctan \left (\frac {(a+b)^{3/4} \left (\frac {\sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}-\sqrt {2} \tan (x)\right )}{\sqrt [4]{a} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}+\frac {\left (\sqrt {a}+\sqrt {a+b}\right ) \arctan \left (\frac {(a+b)^{3/4} \left (\frac {\sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}+\sqrt {2} \tan (x)\right )}{\sqrt [4]{a} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}-\frac {\sqrt [4]{a+b} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}-\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \tan (x)+(a+b)^{3/4} \tan ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}-\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}+\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \tan (x)+(a+b)^{3/4} \tan ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 4.01 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.30 \[ \int \frac {1}{a+b \sin ^4(x)} \, dx=\frac {\left (\sqrt {a}-i \sqrt {b}\right ) \sqrt {a+i \sqrt {a} \sqrt {b}} \arctan \left (\frac {\sqrt {a+i \sqrt {a} \sqrt {b}} \tan (x)}{\sqrt {a}}\right )-\left (\sqrt {a}+i \sqrt {b}\right ) \sqrt {-a+i \sqrt {a} \sqrt {b}} \text {arctanh}\left (\frac {\sqrt {-a+i \sqrt {a} \sqrt {b}} \tan (x)}{\sqrt {a}}\right )}{2 a (a+b)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.62 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.21
method | result | size |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (256 a^{4}+256 a^{3} b \right ) \textit {\_Z}^{4}+32 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-\frac {128 i a^{4}}{b}-128 i a^{3}\right ) \textit {\_R}^{3}+\left (\frac {32 a^{3}}{b}+32 a^{2}\right ) \textit {\_R}^{2}+\left (-\frac {8 i a^{2}}{b}+8 i a \right ) \textit {\_R} +\frac {2 a}{b}-1\right )\) | \(101\) |
default | \(\frac {-\frac {\left (-a^{\frac {5}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}-a^{\frac {3}{2}} \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}+\sqrt {a +b}\, \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a +\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}\right ) \ln \left (-\sqrt {a +b}\, \left (\tan ^{2}\left (x \right )\right )+\tan \left (x \right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}-\sqrt {a}\right )}{2 \sqrt {a +b}}+\frac {2 \left (-2 a^{2} b +\frac {\left (-a^{\frac {5}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}-a^{\frac {3}{2}} \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}+\sqrt {a +b}\, \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a +\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}\right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{2 \sqrt {a +b}}\right ) \arctan \left (\frac {-2 \sqrt {a +b}\, \tan \left (x \right )+\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}}{4 a^{\frac {5}{2}} b}+\frac {\frac {\left (-a^{\frac {5}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}-a^{\frac {3}{2}} \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}+\sqrt {a +b}\, \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a +\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}\right ) \ln \left (\sqrt {a +b}\, \left (\tan ^{2}\left (x \right )\right )+\tan \left (x \right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}+\sqrt {a}\right )}{2 \sqrt {a +b}}+\frac {2 \left (2 a^{2} b -\frac {\left (-a^{\frac {5}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}-a^{\frac {3}{2}} \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}+\sqrt {a +b}\, \sqrt {a^{2}+a b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a +\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}\right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{2 \sqrt {a +b}}\right ) \arctan \left (\frac {2 \sqrt {a +b}\, \tan \left (x \right )+\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}}{4 a^{\frac {5}{2}} b}\) | \(771\) |
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Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (344) = 688\).
Time = 0.38 (sec) , antiderivative size = 823, normalized size of antiderivative = 1.69 \[ \int \frac {1}{a+b \sin ^4(x)} \, dx=-\frac {1}{8} \, \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} \log \left (\frac {1}{4} \, b \cos \left (x\right )^{2} + \frac {1}{2} \, {\left (a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} - \frac {1}{4} \, {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - \frac {1}{4} \, b\right ) + \frac {1}{8} \, \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} \log \left (\frac {1}{4} \, b \cos \left (x\right )^{2} - \frac {1}{2} \, {\left (a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} - \frac {1}{4} \, {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - \frac {1}{4} \, b\right ) + \frac {1}{8} \, \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} \log \left (-\frac {1}{4} \, b \cos \left (x\right )^{2} + \frac {1}{2} \, {\left (a b \cos \left (x\right ) \sin \left (x\right ) - {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} - \frac {1}{4} \, {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + \frac {1}{4} \, b\right ) - \frac {1}{8} \, \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} \log \left (-\frac {1}{4} \, b \cos \left (x\right )^{2} - \frac {1}{2} \, {\left (a b \cos \left (x\right ) \sin \left (x\right ) - {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} - \frac {1}{4} \, {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + \frac {1}{4} \, b\right ) \]
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Timed out. \[ \int \frac {1}{a+b \sin ^4(x)} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{a+b \sin ^4(x)} \, dx=\int { \frac {1}{b \sin \left (x\right )^{4} + a} \,d x } \]
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none
Time = 0.84 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.65 \[ \int \frac {1}{a+b \sin ^4(x)} \, dx=\frac {{\left (3 \, \sqrt {a^{2} + a b + \sqrt {-a b} {\left (a + b\right )}} a^{2} + 6 \, \sqrt {a^{2} + a b + \sqrt {-a b} {\left (a + b\right )}} a b - \sqrt {a^{2} + a b + \sqrt {-a b} {\left (a + b\right )}} b^{2}\right )} {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \left (x\right )}{\sqrt {\frac {4 \, a + \sqrt {-16 \, {\left (a + b\right )} a + 16 \, a^{2}}}{a + b}}}\right )\right )} {\left | a + b \right |}}{2 \, {\left (3 \, a^{5} + 12 \, a^{4} b + 14 \, a^{3} b^{2} + 4 \, a^{2} b^{3} - a b^{4}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} + a b - \sqrt {-a b} {\left (a + b\right )}} a^{2} + 6 \, \sqrt {a^{2} + a b - \sqrt {-a b} {\left (a + b\right )}} a b - \sqrt {a^{2} + a b - \sqrt {-a b} {\left (a + b\right )}} b^{2}\right )} {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \left (x\right )}{\sqrt {\frac {4 \, a - \sqrt {-16 \, {\left (a + b\right )} a + 16 \, a^{2}}}{a + b}}}\right )\right )} {\left | a + b \right |}}{2 \, {\left (3 \, a^{5} + 12 \, a^{4} b + 14 \, a^{3} b^{2} + 4 \, a^{2} b^{3} - a b^{4}\right )}} \]
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Time = 14.97 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.84 \[ \int \frac {1}{a+b \sin ^4(x)} \, dx=\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {a^2-\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}}\,4{}\mathrm {i}+a^5\,\mathrm {tan}\left (x\right )\,{\left (-\frac {a^2-\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}\right )}^{3/2}\,64{}\mathrm {i}-a^2\,b\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {a^2-\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}}\,4{}\mathrm {i}+a^4\,b\,\mathrm {tan}\left (x\right )\,{\left (-\frac {a^2-\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}\right )}^{3/2}\,64{}\mathrm {i}}{\sqrt {-a^3\,b}}\right )\,\sqrt {-\frac {a^2-\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {a^2+\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}}\,4{}\mathrm {i}+a^5\,\mathrm {tan}\left (x\right )\,{\left (-\frac {a^2+\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}\right )}^{3/2}\,64{}\mathrm {i}-a^2\,b\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {a^2+\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}}\,4{}\mathrm {i}+a^4\,b\,\mathrm {tan}\left (x\right )\,{\left (-\frac {a^2+\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}\right )}^{3/2}\,64{}\mathrm {i}}{\sqrt {-a^3\,b}}\right )\,\sqrt {-\frac {a^2+\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}}\,2{}\mathrm {i} \]
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