Integrand size = 37, antiderivative size = 926 \[ \int \frac {\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx=\frac {a^2 d^{5/2} \sqrt {g} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b^3 f}+\frac {d^{5/2} \sqrt {g} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{4 \sqrt {2} b f}-\frac {a^2 d^{5/2} \sqrt {g} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b^3 f}-\frac {d^{5/2} \sqrt {g} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{4 \sqrt {2} b f}-\frac {a^2 d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b^3 f}-\frac {d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{8 \sqrt {2} b f}+\frac {a^2 d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b^3 f}+\frac {d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{8 \sqrt {2} b f}-\frac {2 \sqrt {2} a^3 d^3 \sqrt {g} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b^3 \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a^3 d^3 \sqrt {g} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b^3 \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \sin (e+f x)}}-\frac {d^2 (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f g}-\frac {a d^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{b^2 f \sqrt {\sin (2 e+2 f x)}} \]
[Out]
Time = 1.19 (sec) , antiderivative size = 926, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {2988, 2648, 2655, 303, 1176, 631, 210, 1179, 642, 2652, 2719, 2985, 2984, 504, 1232} \[ \int \frac {\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx=-\frac {2 \sqrt {2} a^3 \sqrt {g} \operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right ) \sqrt {\sin (e+f x)} d^3}{b^3 \sqrt {b-a} \sqrt {a+b} f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a^3 \sqrt {g} \operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right ) \sqrt {\sin (e+f x)} d^3}{b^3 \sqrt {b-a} \sqrt {a+b} f \sqrt {d \sin (e+f x)}}+\frac {\sqrt {g} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right ) d^{5/2}}{4 \sqrt {2} b f}+\frac {a^2 \sqrt {g} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right ) d^{5/2}}{\sqrt {2} b^3 f}-\frac {\sqrt {g} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right ) d^{5/2}}{4 \sqrt {2} b f}-\frac {a^2 \sqrt {g} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right ) d^{5/2}}{\sqrt {2} b^3 f}-\frac {\sqrt {g} \log \left (\sqrt {g} \cot (e+f x)+\sqrt {g}-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right ) d^{5/2}}{8 \sqrt {2} b f}-\frac {a^2 \sqrt {g} \log \left (\sqrt {g} \cot (e+f x)+\sqrt {g}-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right ) d^{5/2}}{2 \sqrt {2} b^3 f}+\frac {\sqrt {g} \log \left (\sqrt {g} \cot (e+f x)+\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right ) d^{5/2}}{8 \sqrt {2} b f}+\frac {a^2 \sqrt {g} \log \left (\sqrt {g} \cot (e+f x)+\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right ) d^{5/2}}{2 \sqrt {2} b^3 f}-\frac {(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)} d^2}{2 b f g}-\frac {a \sqrt {g \cos (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} d^2}{b^2 f \sqrt {\sin (2 e+2 f x)}} \]
[In]
[Out]
Rule 210
Rule 303
Rule 504
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1232
Rule 2648
Rule 2652
Rule 2655
Rule 2719
Rule 2984
Rule 2985
Rule 2988
Rubi steps \begin{align*} \text {integral}& = \frac {d \int \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2} \, dx}{b}-\frac {(a d) \int \frac {\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx}{b} \\ & = -\frac {d^2 (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f g}-\frac {\left (a d^2\right ) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{b^2}+\frac {\left (a^2 d^2\right ) \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx}{b^2}+\frac {d^3 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx}{4 b} \\ & = -\frac {d^2 (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f g}+\frac {\left (a^2 d^3\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx}{b^3}-\frac {\left (a^3 d^3\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx}{b^3}-\frac {\left (d^4 g\right ) \text {Subst}\left (\int \frac {x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 b f}-\frac {\left (a d^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{b^2 \sqrt {\sin (2 e+2 f x)}} \\ & = -\frac {d^2 (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f g}-\frac {a d^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{b^2 f \sqrt {\sin (2 e+2 f x)}}+\frac {\left (d^3 g\right ) \text {Subst}\left (\int \frac {g-d x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{4 b f}-\frac {\left (d^3 g\right ) \text {Subst}\left (\int \frac {g+d x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{4 b f}-\frac {\left (2 a^2 d^4 g\right ) \text {Subst}\left (\int \frac {x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{b^3 f}-\frac {\left (a^3 d^3 \sqrt {\sin (e+f x)}\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))} \, dx}{b^3 \sqrt {d \sin (e+f x)}} \\ & = -\frac {d^2 (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f g}-\frac {a d^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{b^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {\left (d^{5/2} \sqrt {g}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {g}}{\sqrt {d}}+2 x}{-\frac {g}{d}-\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{8 \sqrt {2} b f}-\frac {\left (d^{5/2} \sqrt {g}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {g}}{\sqrt {d}}-2 x}{-\frac {g}{d}+\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{8 \sqrt {2} b f}-\frac {\left (d^2 g\right ) \text {Subst}\left (\int \frac {1}{\frac {g}{d}-\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{8 b f}-\frac {\left (d^2 g\right ) \text {Subst}\left (\int \frac {1}{\frac {g}{d}+\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{8 b f}+\frac {\left (a^2 d^3 g\right ) \text {Subst}\left (\int \frac {g-d x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{b^3 f}-\frac {\left (a^2 d^3 g\right ) \text {Subst}\left (\int \frac {g+d x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{b^3 f}+\frac {\left (4 \sqrt {2} a^3 d^3 g \sqrt {\sin (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 \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{b^3 f \sqrt {d \sin (e+f x)}} \\ & = -\frac {d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{8 \sqrt {2} b f}+\frac {d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{8 \sqrt {2} b f}-\frac {d^2 (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f g}-\frac {a d^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{b^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {\left (a^2 d^{5/2} \sqrt {g}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {g}}{\sqrt {d}}+2 x}{-\frac {g}{d}-\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b^3 f}-\frac {\left (a^2 d^{5/2} \sqrt {g}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {g}}{\sqrt {d}}-2 x}{-\frac {g}{d}+\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b^3 f}-\frac {\left (d^{5/2} \sqrt {g}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{4 \sqrt {2} b f}+\frac {\left (d^{5/2} \sqrt {g}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{4 \sqrt {2} b f}-\frac {\left (a^2 d^2 g\right ) \text {Subst}\left (\int \frac {1}{\frac {g}{d}-\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 b^3 f}-\frac {\left (a^2 d^2 g\right ) \text {Subst}\left (\int \frac {1}{\frac {g}{d}+\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 b^3 f}+\frac {\left (2 \sqrt {2} a^3 d^3 g \sqrt {\sin (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 \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{b^3 \sqrt {-a+b} f \sqrt {d \sin (e+f x)}}-\frac {\left (2 \sqrt {2} a^3 d^3 g \sqrt {\sin (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 \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{b^3 \sqrt {-a+b} f \sqrt {d \sin (e+f x)}} \\ & = \frac {d^{5/2} \sqrt {g} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{4 \sqrt {2} b f}-\frac {d^{5/2} \sqrt {g} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{4 \sqrt {2} b f}-\frac {a^2 d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b^3 f}-\frac {d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{8 \sqrt {2} b f}+\frac {a^2 d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b^3 f}+\frac {d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{8 \sqrt {2} b f}-\frac {2 \sqrt {2} a^3 d^3 \sqrt {g} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b^3 \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a^3 d^3 \sqrt {g} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b^3 \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \sin (e+f x)}}-\frac {d^2 (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f g}-\frac {a d^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{b^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {\left (a^2 d^{5/2} \sqrt {g}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b^3 f}+\frac {\left (a^2 d^{5/2} \sqrt {g}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b^3 f} \\ & = \frac {a^2 d^{5/2} \sqrt {g} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b^3 f}+\frac {d^{5/2} \sqrt {g} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{4 \sqrt {2} b f}-\frac {a^2 d^{5/2} \sqrt {g} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b^3 f}-\frac {d^{5/2} \sqrt {g} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{4 \sqrt {2} b f}-\frac {a^2 d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b^3 f}-\frac {d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{8 \sqrt {2} b f}+\frac {a^2 d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b^3 f}+\frac {d^{5/2} \sqrt {g} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{8 \sqrt {2} b f}-\frac {2 \sqrt {2} a^3 d^3 \sqrt {g} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b^3 \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a^3 d^3 \sqrt {g} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b^3 \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \sin (e+f x)}}-\frac {d^2 (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f g}-\frac {a d^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{b^2 f \sqrt {\sin (2 e+2 f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 48.70 (sec) , antiderivative size = 1623, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx=-\frac {\sqrt {g \cos (e+f x)} \cot (e+f x) \csc (e+f x) (d \sin (e+f x))^{5/2}}{2 b f}+\frac {\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2} \left (-\frac {2 b \left (-b \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) \cos ^{\frac {3}{2}}(e+f x) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \sin ^{\frac {3}{2}}(e+f x)}{3 \left (a^2-b^2\right ) \left (1-\cos ^2(e+f x)\right )^{3/4} (a+b \sin (e+f x))}-\frac {\sqrt {\tan (e+f x)} \left (\frac {3 \sqrt {2} a^{3/2} \left (-2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )-\log \left (-a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}-\sqrt {a^2-b^2} \tan (e+f x)\right )+\log \left (a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+\sqrt {a^2-b^2} \tan (e+f x)\right )\right )}{\sqrt [4]{a^2-b^2}}-8 b \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(e+f x),\frac {\left (-a^2+b^2\right ) \tan ^2(e+f x)}{a^2}\right ) \tan ^{\frac {3}{2}}(e+f x)\right ) \left (b \tan (e+f x)+a \sqrt {1+\tan ^2(e+f x)}\right )}{12 a \cos ^{\frac {3}{2}}(e+f x) \sqrt {\sin (e+f x)} (a+b \sin (e+f x)) \left (1+\tan ^2(e+f x)\right )^{3/2}}+\frac {\cos (2 (e+f x)) \sqrt {\tan (e+f x)} \left (b \tan (e+f x)+a \sqrt {1+\tan ^2(e+f x)}\right ) \left (56 b \left (-3 a^2+b^2\right ) \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)+24 b \left (-a^2+b^2\right ) \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {7}{2}}(e+f x)+21 a^{3/2} \left (4 \sqrt {2} a^{3/2} \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )-4 \sqrt {2} a^{3/2} \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )-\frac {4 \sqrt {2} a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+\frac {2 \sqrt {2} b^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+\frac {4 \sqrt {2} a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}-\frac {2 \sqrt {2} b^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+2 \sqrt {2} a^{3/2} \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-2 \sqrt {2} a^{3/2} \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-\frac {2 \sqrt {2} a^2 \log \left (-a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}-\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {\sqrt {2} b^2 \log \left (-a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}-\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {2 \sqrt {2} a^2 \log \left (a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}-\frac {\sqrt {2} b^2 \log \left (a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {8 \sqrt {a} b \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {1+\tan ^2(e+f x)}}\right )\right )}{42 a b^2 \cos ^{\frac {3}{2}}(e+f x) \sqrt {\sin (e+f x)} (a+b \sin (e+f x)) \left (-1+\tan ^2(e+f x)\right ) \sqrt {1+\tan ^2(e+f x)}}\right )}{4 b f \sqrt {\cos (e+f x)} \sin ^{\frac {5}{2}}(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. 3599 vs. \(2 (745 ) = 1490\).
Time = 12.88 (sec) , antiderivative size = 3600, normalized size of antiderivative = 3.89
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Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx=\int { \frac {\sqrt {g \cos \left (f x + e\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{b \sin \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx=\int { \frac {\sqrt {g \cos \left (f x + e\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{b \sin \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {g\,\cos \left (e+f\,x\right )}\,{\left (d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{a+b\,\sin \left (e+f\,x\right )} \,d x \]
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