Integrand size = 24, antiderivative size = 125 \[ \int \frac {\cos ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\sqrt {\sqrt {a}-\sqrt {b}} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {b} d}+\frac {\sqrt {\sqrt {a}+\sqrt {b}} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {b} d} \]
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Time = 0.17 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3303, 1107, 211} \[ \int \frac {\cos ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\sqrt {\sqrt {a}+\sqrt {b}} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {b} d}-\frac {\sqrt {\sqrt {a}-\sqrt {b}} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {b} d} \]
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Rule 211
Rule 1107
Rule 3303
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {(a-b) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 \sqrt {a} \sqrt {b} d}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 \sqrt {a} \sqrt {b} d} \\ & = -\frac {\sqrt {\sqrt {a}-\sqrt {b}} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {b} d}+\frac {\sqrt {\sqrt {a}+\sqrt {b}} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {b} d} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.26 \[ \int \frac {\cos ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {\left (\sqrt {a} \sqrt {b}+b\right ) \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {\left (\sqrt {a} \sqrt {b}-b\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}}{2 \sqrt {a} b d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.72 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.58
method | result | size |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (256 a^{3} b^{2} d^{4} \textit {\_Z}^{4}+32 a^{2} b \,d^{2} \textit {\_Z}^{2}+a -b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+32 a^{2} d^{2} \textit {\_R}^{2}+8 i a d \textit {\_R} +\frac {2 a}{b}-1\right )\) | \(73\) |
derivativedivides | \(\frac {\left (a -b \right ) \left (\frac {\operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}-\frac {\arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{d}\) | \(115\) |
default | \(\frac {\left (a -b \right ) \left (\frac {\operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}-\frac {\arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{d}\) | \(115\) |
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Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (85) = 170\).
Time = 0.36 (sec) , antiderivative size = 541, normalized size of antiderivative = 4.33 \[ \int \frac {\cos ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {1}{8} \, \sqrt {-\frac {a b d^{2} \sqrt {\frac {1}{a^{3} b d^{4}}} + 1}{a b d^{2}}} \log \left (\frac {1}{2} \, a d \sqrt {-\frac {a b d^{2} \sqrt {\frac {1}{a^{3} b d^{4}}} + 1}{a b d^{2}}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4} \, {\left (2 \, a^{2} d^{2} \cos \left (d x + c\right )^{2} - a^{2} d^{2}\right )} \sqrt {\frac {1}{a^{3} b d^{4}}} - \frac {1}{4}\right ) + \frac {1}{8} \, \sqrt {-\frac {a b d^{2} \sqrt {\frac {1}{a^{3} b d^{4}}} + 1}{a b d^{2}}} \log \left (-\frac {1}{2} \, a d \sqrt {-\frac {a b d^{2} \sqrt {\frac {1}{a^{3} b d^{4}}} + 1}{a b d^{2}}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4} \, {\left (2 \, a^{2} d^{2} \cos \left (d x + c\right )^{2} - a^{2} d^{2}\right )} \sqrt {\frac {1}{a^{3} b d^{4}}} - \frac {1}{4}\right ) + \frac {1}{8} \, \sqrt {\frac {a b d^{2} \sqrt {\frac {1}{a^{3} b d^{4}}} - 1}{a b d^{2}}} \log \left (\frac {1}{2} \, a d \sqrt {\frac {a b d^{2} \sqrt {\frac {1}{a^{3} b d^{4}}} - 1}{a b d^{2}}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - \frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4} \, {\left (2 \, a^{2} d^{2} \cos \left (d x + c\right )^{2} - a^{2} d^{2}\right )} \sqrt {\frac {1}{a^{3} b d^{4}}} + \frac {1}{4}\right ) - \frac {1}{8} \, \sqrt {\frac {a b d^{2} \sqrt {\frac {1}{a^{3} b d^{4}}} - 1}{a b d^{2}}} \log \left (-\frac {1}{2} \, a d \sqrt {\frac {a b d^{2} \sqrt {\frac {1}{a^{3} b d^{4}}} - 1}{a b d^{2}}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - \frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4} \, {\left (2 \, a^{2} d^{2} \cos \left (d x + c\right )^{2} - a^{2} d^{2}\right )} \sqrt {\frac {1}{a^{3} b d^{4}}} + \frac {1}{4}\right ) \]
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Timed out. \[ \int \frac {\cos ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\cos \left (d x + c\right )^{2}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (85) = 170\).
Time = 0.80 (sec) , antiderivative size = 559, normalized size of antiderivative = 4.47 \[ \int \frac {\cos ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {{\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{2} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} b^{3} + 3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \left (d x + c\right )}{\sqrt {\frac {4 \, a + \sqrt {-16 \, {\left (a - b\right )} a + 16 \, a^{2}}}{a - b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{5} b - 12 \, a^{4} b^{2} + 14 \, a^{3} b^{3} - 4 \, a^{2} b^{4} - a b^{5}} + \frac {{\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{2} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} b^{3} - 3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} + 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b + \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \left (d x + c\right )}{\sqrt {\frac {4 \, a - \sqrt {-16 \, {\left (a - b\right )} a + 16 \, a^{2}}}{a - b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{5} b - 12 \, a^{4} b^{2} + 14 \, a^{3} b^{3} - 4 \, a^{2} b^{4} - a b^{5}}}{2 \, d} \]
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Time = 14.75 (sec) , antiderivative size = 1409, normalized size of antiderivative = 11.27 \[ \int \frac {\cos ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]
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