Integrand size = 27, antiderivative size = 183 \[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=\frac {2 \sqrt {2} \sqrt [4]{-a+b} \sqrt {c \cos (e+f x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b} \sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}}}{\sqrt [4]{-a+b}}\right ),-1\right ) \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}}}{\sqrt [4]{a+b} c f \sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}} \sqrt {a+b \sin (e+f x)}} \]
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Leaf count is larger than twice the leaf count of optimal. \(374\) vs. \(2(183)=366\).
Time = 0.32 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2776, 226} \[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=\frac {\sqrt {2} \sqrt [4]{a-b} \sqrt {c \cos (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (\sin (e) (-\cos (f x))-\cos (e) \sin (f x)+1) \left (\frac {\sqrt {a+b} (\sin (e+f x)+\cos (e+f x)+1)}{\sqrt {a-b} (-\sin (e+f x)+\cos (e+f x)+1)}+1\right )^2}} \left (\frac {\sqrt {a+b} (\sin (e+f x)+\cos (e+f x)+1)}{\sqrt {a-b} (-\sin (e+f x)+\cos (e+f x)+1)}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a+b} \sqrt {\frac {\cos (e+f x)+\sin (e+f x)+1}{\cos (e+f x)-\sin (e+f x)+1}}}{\sqrt [4]{a-b}}\right ),\frac {1}{2}\right )}{c f \sqrt [4]{a+b} \sqrt {\frac {\sin (e+f x)+\cos (e+f x)+1}{-\sin (e+f x)+\cos (e+f x)+1}} \sqrt {a+b \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (\sin (e) (-\cos (f x))-\cos (e) \sin (f x)+1)}}} \]
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Rule 226
Rule 2776
Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 \sqrt {2} \sqrt {c \cos (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {(a+b) x^4}{a-b}}} \, dx,x,\sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}}\right )}{c f \sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}} \sqrt {a+b \sin (e+f x)}} \\ & = \frac {\sqrt {2} \sqrt [4]{a-b} \sqrt {c \cos (e+f x)} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a+b} \sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}}}{\sqrt [4]{a-b}}\right ),\frac {1}{2}\right ) \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\cos (f x) \sin (e)-\cos (e) \sin (f x)) \left (1+\frac {\sqrt {a+b} (1+\cos (e+f x)+\sin (e+f x))}{\sqrt {a-b} (1+\cos (e+f x)-\sin (e+f x))}\right )^2}} \left (1+\frac {\sqrt {a+b} (1+\cos (e+f x)+\sin (e+f x))}{\sqrt {a-b} (1+\cos (e+f x)-\sin (e+f x))}\right )}{\sqrt [4]{a+b} c f \sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}} \sqrt {a+b \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\cos (f x) \sin (e)-\cos (e) \sin (f x))}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=-\frac {2 c \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\frac {2 (a+b \sin (e+f x))}{(a-b) (-1+\sin (e+f x))}\right ) (-1+\sin (e+f x)) \left (\frac {(a+b) (1+\sin (e+f x))}{(a-b) (-1+\sin (e+f x))}\right )^{3/4} \sqrt {a+b \sin (e+f x)}}{(a+b) f (c \cos (e+f x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(642\) vs. \(2(163)=326\).
Time = 10.40 (sec) , antiderivative size = 643, normalized size of antiderivative = 3.51
method | result | size |
default | \(-\frac {4 \left (a \sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )-a^{2} \sin \left (f x +e \right )+a b \sin \left (f x +e \right )-\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}\, a +\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}\, b -\cos \left (f x +e \right ) a^{2}+\cos \left (f x +e \right ) b^{2}+b \sqrt {-a^{2}+b^{2}}-a b +b^{2}\right ) \sqrt {\frac {\left (-a +b +\sqrt {-a^{2}+b^{2}}\right ) \left (\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )-b \sin \left (f x +e \right )+a \cos \left (f x +e \right )-a \right )}{\left (a -b +\sqrt {-a^{2}+b^{2}}\right ) \left (\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-a \cos \left (f x +e \right )+a \right )}}\, \sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \left (-\cos \left (f x +e \right )+1+\sin \left (f x +e \right )\right ) a}{\left (a -b +\sqrt {-a^{2}+b^{2}}\right ) \left (\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-a \cos \left (f x +e \right )+a \right )}}\, \sqrt {-\frac {\sqrt {-a^{2}+b^{2}}\, \left (\cos \left (f x +e \right )-1+\sin \left (f x +e \right )\right ) a}{\left (-a -b +\sqrt {-a^{2}+b^{2}}\right ) \left (\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-a \cos \left (f x +e \right )+a \right )}}\, F\left (\sqrt {-\frac {\left (a \csc \left (f x +e \right )-a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}+b \right ) \left (-a +b +\sqrt {-a^{2}+b^{2}}\right )}{\left (a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b \right ) \left (a -b +\sqrt {-a^{2}+b^{2}}\right )}}, \sqrt {\frac {\left (b +\sqrt {-a^{2}+b^{2}}+a \right ) \left (a -b +\sqrt {-a^{2}+b^{2}}\right )}{\left (-a -b +\sqrt {-a^{2}+b^{2}}\right ) \left (-a +b +\sqrt {-a^{2}+b^{2}}\right )}}\right )}{f \sqrt {c \cos \left (f x +e \right )}\, \sqrt {a +b \sin \left (f x +e \right )}\, \sqrt {-a^{2}+b^{2}}\, \left (-a +b +\sqrt {-a^{2}+b^{2}}\right )}\) | \(643\) |
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\[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {c \cos \left (f x + e\right )} \sqrt {b \sin \left (f x + e\right ) + a}} \,d x } \]
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\[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {c \cos {\left (e + f x \right )}} \sqrt {a + b \sin {\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {c \cos \left (f x + e\right )} \sqrt {b \sin \left (f x + e\right ) + a}} \,d x } \]
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\[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {c \cos \left (f x + e\right )} \sqrt {b \sin \left (f x + e\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {c\,\cos \left (e+f\,x\right )}\,\sqrt {a+b\,\sin \left (e+f\,x\right )}} \,d x \]
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