Integrand size = 37, antiderivative size = 611 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=-\frac {a \sqrt {d} g^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b^2 f}+\frac {a \sqrt {d} g^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b^2 f}-\frac {2 \sqrt {2} \sqrt {-a^2+b^2} \sqrt {d} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^2 f \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {2} \sqrt {-a^2+b^2} \sqrt {d} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^2 f \sqrt {g \cos (e+f x)}}+\frac {a \sqrt {d} g^{3/2} \log \left (\sqrt {d}-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b^2 f}-\frac {a \sqrt {d} g^{3/2} \log \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b^2 f}+\frac {g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b f}-\frac {d g^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{2 b f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \]
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Time = 0.67 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {2980, 2917, 2654, 303, 1176, 631, 210, 1179, 642, 2648, 2653, 2720, 2987, 2986, 1232} \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=-\frac {2 \sqrt {2} \sqrt {d} g^2 \sqrt {b^2-a^2} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{b^2 f \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {2} \sqrt {d} g^2 \sqrt {b^2-a^2} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{b^2 f \sqrt {g \cos (e+f x)}}-\frac {a \sqrt {d} g^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b^2 f}+\frac {a \sqrt {d} g^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} b^2 f}+\frac {a \sqrt {d} g^{3/2} \log \left (-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)+\sqrt {d}\right )}{2 \sqrt {2} b^2 f}-\frac {a \sqrt {d} g^{3/2} \log \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)+\sqrt {d}\right )}{2 \sqrt {2} b^2 f}-\frac {d g^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 b f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}+\frac {g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{b f} \]
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Rule 210
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1232
Rule 2648
Rule 2653
Rule 2654
Rule 2720
Rule 2917
Rule 2980
Rule 2986
Rule 2987
Rubi steps \begin{align*} \text {integral}& = \frac {g^2 \int \frac {\sqrt {d \sin (e+f x)} (a-b \sin (e+f x))}{\sqrt {g \cos (e+f x)}} \, dx}{b^2}-\frac {\left (\left (a^2-b^2\right ) g^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx}{b^2} \\ & = \frac {\left (a g^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}} \, dx}{b^2}-\frac {g^2 \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)}} \, dx}{b d}-\frac {\left (\left (a^2-b^2\right ) g^2 \sqrt {\cos (e+f x)}\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \sin (e+f x))} \, dx}{b^2 \sqrt {g \cos (e+f x)}} \\ & = \frac {g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b f}-\frac {\left (d g^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \, dx}{2 b}+\frac {\left (2 a d g^3\right ) \text {Subst}\left (\int \frac {x^2}{d^2+g^2 x^4} \, dx,x,\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{b^2 f}-\frac {\left (2 \sqrt {2} \left (a^2-b^2\right ) \left (1-\frac {b}{\sqrt {-a^2+b^2}}\right ) d g^2 \sqrt {\cos (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (b-\sqrt {-a^2+b^2}\right ) d+a x^2\right ) \sqrt {1-\frac {x^4}{d^2}}} \, dx,x,\frac {\sqrt {d \sin (e+f x)}}{\sqrt {1+\cos (e+f x)}}\right )}{b^2 f \sqrt {g \cos (e+f x)}}-\frac {\left (2 \sqrt {2} \left (a^2-b^2\right ) \left (1+\frac {b}{\sqrt {-a^2+b^2}}\right ) d g^2 \sqrt {\cos (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (b+\sqrt {-a^2+b^2}\right ) d+a x^2\right ) \sqrt {1-\frac {x^4}{d^2}}} \, dx,x,\frac {\sqrt {d \sin (e+f x)}}{\sqrt {1+\cos (e+f x)}}\right )}{b^2 f \sqrt {g \cos (e+f x)}} \\ & = -\frac {2 \sqrt {2} \sqrt {-a^2+b^2} \sqrt {d} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^2 f \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {2} \sqrt {-a^2+b^2} \sqrt {d} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^2 f \sqrt {g \cos (e+f x)}}+\frac {g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b f}-\frac {\left (a d g^2\right ) \text {Subst}\left (\int \frac {d-g x^2}{d^2+g^2 x^4} \, dx,x,\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{b^2 f}+\frac {\left (a d g^2\right ) \text {Subst}\left (\int \frac {d+g x^2}{d^2+g^2 x^4} \, dx,x,\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{b^2 f}-\frac {\left (d g^2 \sqrt {\sin (2 e+2 f x)}\right ) \int \frac {1}{\sqrt {\sin (2 e+2 f x)}} \, dx}{2 b \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \\ & = -\frac {2 \sqrt {2} \sqrt {-a^2+b^2} \sqrt {d} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^2 f \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {2} \sqrt {-a^2+b^2} \sqrt {d} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^2 f \sqrt {g \cos (e+f x)}}+\frac {g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b f}-\frac {d g^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{2 b f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}+\frac {(a d g) \text {Subst}\left (\int \frac {1}{\frac {d}{g}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt {g}}+x^2} \, dx,x,\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{2 b^2 f}+\frac {(a d g) \text {Subst}\left (\int \frac {1}{\frac {d}{g}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt {g}}+x^2} \, dx,x,\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{2 b^2 f}+\frac {\left (a \sqrt {d} g^{3/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt {g}}+2 x}{-\frac {d}{g}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt {g}}-x^2} \, dx,x,\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{2 \sqrt {2} b^2 f}+\frac {\left (a \sqrt {d} g^{3/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt {g}}-2 x}{-\frac {d}{g}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt {g}}-x^2} \, dx,x,\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{2 \sqrt {2} b^2 f} \\ & = -\frac {2 \sqrt {2} \sqrt {-a^2+b^2} \sqrt {d} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^2 f \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {2} \sqrt {-a^2+b^2} \sqrt {d} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^2 f \sqrt {g \cos (e+f x)}}+\frac {a \sqrt {d} g^{3/2} \log \left (\sqrt {d}-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b^2 f}-\frac {a \sqrt {d} g^{3/2} \log \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b^2 f}+\frac {g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b f}-\frac {d g^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{2 b f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}+\frac {\left (a \sqrt {d} g^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b^2 f}-\frac {\left (a \sqrt {d} g^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b^2 f} \\ & = -\frac {a \sqrt {d} g^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b^2 f}+\frac {a \sqrt {d} g^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b^2 f}-\frac {2 \sqrt {2} \sqrt {-a^2+b^2} \sqrt {d} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^2 f \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {2} \sqrt {-a^2+b^2} \sqrt {d} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^2 f \sqrt {g \cos (e+f x)}}+\frac {a \sqrt {d} g^{3/2} \log \left (\sqrt {d}-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b^2 f}-\frac {a \sqrt {d} g^{3/2} \log \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b^2 f}+\frac {g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b f}-\frac {d g^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{2 b f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 19.69 (sec) , antiderivative size = 601, normalized size of antiderivative = 0.98 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=-\frac {(g \cos (e+f x))^{5/2} (d \sin (e+f x))^{3/2} \left (a+b \sqrt {\sin ^2(e+f x)}\right ) \left (\frac {2 a \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{4},1,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )}{a^2-b^2}+\frac {\left (2 a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},1,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )}{-a^2 b+b^3}+\frac {5 \left (-5 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \left (a^2+b^2 \left (-2+\cos ^2(e+f x)\right )\right )+\left (-4 b^2 \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},2,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+3 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {7}{4},1,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) \sin ^2(e+f x) \left (a^2-b^2 \sin ^2(e+f x)\right )\right )}{b \left (-5 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+\left (4 b^2 \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},2,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+3 \left (-a^2+b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {7}{4},1,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) \cos ^2(e+f x)\right ) \sin ^2(e+f x)^{3/4} \left (a^2-b^2 \sin ^2(e+f x)\right )}\right )}{5 d f g \sin ^2(e+f x)^{3/4} (a+b \sin (e+f x))} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2844 vs. \(2 (517 ) = 1034\).
Time = 3.07 (sec) , antiderivative size = 2845, normalized size of antiderivative = 4.66
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Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {d \sin {\left (e + f x \right )}} \left (g \cos {\left (e + f x \right )}\right )^{\frac {3}{2}}}{a + b \sin {\left (e + f x \right )}}\, dx \]
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\[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {d \sin \left (f x + e\right )}}{b \sin \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {d \sin \left (f x + e\right )}}{b \sin \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {d\,\sin \left (e+f\,x\right )}}{a+b\,\sin \left (e+f\,x\right )} \,d x \]
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