Integrand size = 37, antiderivative size = 208 \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=-\frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{\sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}+\frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{\sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}} \]
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Time = 0.48 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {2985, 2984, 504, 1232} \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=\frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{f \sqrt {b-a} \sqrt {a+b} \sqrt {d \cos (e+f x)}}-\frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{f \sqrt {b-a} \sqrt {a+b} \sqrt {d \cos (e+f x)}} \]
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Rule 504
Rule 1232
Rule 2984
Rule 2985
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\cos (e+f x)} \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \cos (e+f x))} \, dx}{\sqrt {d \cos (e+f x)}} \\ & = \frac {\left (4 \sqrt {2} g \sqrt {\cos (e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{\left ((a+b) g^2+(a-b) x^4\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {1+\cos (e+f x)}}\right )}{f \sqrt {d \cos (e+f x)}} \\ & = \frac {\left (2 \sqrt {2} g \sqrt {\cos (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g-\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {1+\cos (e+f x)}}\right )}{\sqrt {-a+b} f \sqrt {d \cos (e+f x)}}-\frac {\left (2 \sqrt {2} g \sqrt {\cos (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g+\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {1+\cos (e+f x)}}\right )}{\sqrt {-a+b} f \sqrt {d \cos (e+f x)}} \\ & = -\frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{\sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}+\frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{\sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 2.89 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.80 \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=\frac {2 \left (b+a \sqrt {\sec ^2(e+f x)}\right ) \sqrt {g \sin (e+f x)} \left (\frac {-2 \arctan \left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\tan (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt {a} \sqrt {\tan (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+a \tan (e+f x)\right )-\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+a \tan (e+f x)\right )}{4 \sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2}}+\frac {b \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right ) \tan ^{\frac {3}{2}}(e+f x)}{3 \left (-a^2+b^2\right )}\right )}{f \sqrt {d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt {\sec ^2(e+f x)} \sqrt {\tan (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(463\) vs. \(2(164)=328\).
Time = 4.15 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.23
method | result | size |
default | \(-\frac {\sqrt {g \sin \left (f x +e \right )}\, \left (\sqrt {-a^{2}+b^{2}}\, \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-\sqrt {-a^{2}+b^{2}}\, \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-a \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )+\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b -a \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )+\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b \right ) \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) \sqrt {2}}{f \sqrt {\cos \left (f x +e \right ) d}\, \left (b +\sqrt {-a^{2}+b^{2}}-a \right ) \left (-b +\sqrt {-a^{2}+b^{2}}+a \right )}\) | \(464\) |
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Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=\int \frac {\sqrt {g \sin {\left (e + f x \right )}}}{\sqrt {d \cos {\left (e + f x \right )}} \left (a + b \cos {\left (e + f x \right )}\right )}\, dx \]
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\[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=\int { \frac {\sqrt {g \sin \left (f x + e\right )}}{{\left (b \cos \left (f x + e\right ) + a\right )} \sqrt {d \cos \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=\int { \frac {\sqrt {g \sin \left (f x + e\right )}}{{\left (b \cos \left (f x + e\right ) + a\right )} \sqrt {d \cos \left (f x + e\right )}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=\int \frac {\sqrt {g\,\sin \left (e+f\,x\right )}}{\sqrt {d\,\cos \left (e+f\,x\right )}\,\left (a+b\,\cos \left (e+f\,x\right )\right )} \,d x \]
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