Integrand size = 37, antiderivative size = 209 \[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\frac {2 \sqrt {2} \sqrt {d} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{\sqrt {-a^2+b^2} f \sqrt {g \sin (e+f x)}}-\frac {2 \sqrt {2} \sqrt {d} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{\sqrt {-a^2+b^2} f \sqrt {g \sin (e+f x)}} \]
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Time = 0.54 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {2987, 2986, 1232} \[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\frac {2 \sqrt {2} \sqrt {d} \sqrt {\sin (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{f \sqrt {b^2-a^2} \sqrt {g \sin (e+f x)}}-\frac {2 \sqrt {2} \sqrt {d} \sqrt {\sin (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{f \sqrt {b^2-a^2} \sqrt {g \sin (e+f x)}} \]
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Rule 1232
Rule 2986
Rule 2987
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\sin (e+f x)} \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {\sin (e+f x)}} \, dx}{\sqrt {g \sin (e+f x)}} \\ & = -\frac {\left (2 \sqrt {2} \left (1-\frac {b}{\sqrt {-a^2+b^2}}\right ) d \sqrt {\sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (-\left (\left (-b+\sqrt {-a^2+b^2}\right ) d\right )+a x^2\right ) \sqrt {1-\frac {x^4}{d^2}}} \, dx,x,\frac {\sqrt {d \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{f \sqrt {g \sin (e+f x)}}-\frac {\left (2 \sqrt {2} \left (1+\frac {b}{\sqrt {-a^2+b^2}}\right ) d \sqrt {\sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (-\left (\left (-b-\sqrt {-a^2+b^2}\right ) d\right )+a x^2\right ) \sqrt {1-\frac {x^4}{d^2}}} \, dx,x,\frac {\sqrt {d \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{f \sqrt {g \sin (e+f x)}} \\ & = \frac {2 \sqrt {2} \sqrt {d} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{\sqrt {-a^2+b^2} f \sqrt {g \sin (e+f x)}}-\frac {2 \sqrt {2} \sqrt {d} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{\sqrt {-a^2+b^2} f \sqrt {g \sin (e+f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 17.92 (sec) , antiderivative size = 581, normalized size of antiderivative = 2.78 \[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\frac {2 \sqrt {d \cos (e+f x)} \left (b+a \sqrt {\sec ^2(e+f x)}\right ) \sqrt {\tan (e+f x)} \left (\frac {\sqrt {a} \left (-2 \arctan \left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\tan (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt {a} \sqrt {\tan (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+a \tan (e+f x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+a \tan (e+f x)\right )\right )}{4 \sqrt {2} \left (a^2-b^2\right )^{3/4}}+\frac {5 b \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right ) \sqrt {\tan (e+f x)}}{\sqrt {\sec ^2(e+f x)} \left (a^2-b^2+a^2 \tan ^2(e+f x)\right ) \left (-5 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right )+2 \left (2 a^2 \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right )+\left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right )\right ) \tan ^2(e+f x)\right )}\right )}{f (a+b \cos (e+f x)) \sqrt {\sec ^2(e+f x)} \sqrt {g \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(507\) vs. \(2(173)=346\).
Time = 4.77 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.43
method | result | size |
default | \(\frac {\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \left (2 \sqrt {-a^{2}+b^{2}}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+a \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b -\sqrt {-a^{2}+b^{2}}\, \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-a \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )+\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b -\sqrt {-a^{2}+b^{2}}\, \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {\cos \left (f x +e \right ) d}\, \left (\sec \left (f x +e \right )+1\right ) \sqrt {2}\, a}{f \sqrt {g \sin \left (f x +e \right )}\, \sqrt {-a^{2}+b^{2}}\, \left (-b +\sqrt {-a^{2}+b^{2}}+a \right ) \left (b +\sqrt {-a^{2}+b^{2}}-a \right )}\) | \(508\) |
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Timed out. \[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\int \frac {\sqrt {d \cos {\left (e + f x \right )}}}{\sqrt {g \sin {\left (e + f x \right )}} \left (a + b \cos {\left (e + f x \right )}\right )}\, dx \]
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\[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\int { \frac {\sqrt {d \cos \left (f x + e\right )}}{{\left (b \cos \left (f x + e\right ) + a\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\int { \frac {\sqrt {d \cos \left (f x + e\right )}}{{\left (b \cos \left (f x + e\right ) + a\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\int \frac {\sqrt {d\,\cos \left (e+f\,x\right )}}{\sqrt {g\,\sin \left (e+f\,x\right )}\,\left (a+b\,\cos \left (e+f\,x\right )\right )} \,d x \]
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