Optimal. Leaf size=936 \[ -\frac {\sqrt {d} g^{5/2} \tan ^{-1}\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 {\left (a^2-b^2\right ) \sqrt {d} g^{5/2} \tan ^{-1}\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 {\sqrt {d} g^{5/2} \tan ^{-1}\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 {\left (a^2-b^2\right ) \sqrt {d} g^{5/2} \tan ^{-1}\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 {\sqrt {d} g^{5/2} \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 {\left (a^2-b^2\right ) \sqrt {d} g^{5/2} \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 {\sqrt {d} g^{5/2} \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 {\left (a^2-b^2\right ) \sqrt {d} g^{5/2} \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 {2 \sqrt {2} a \sqrt {-a+b} \sqrt {a+b} d g^{5/2} \Pi \left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{b^3 f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a \sqrt {-a+b} \sqrt {a+b} d g^{5/2} \Pi \left (\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{b^3 f \sqrt {d \sin (e+f x)}}+\frac {g (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f}+\frac {a g^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]
________________________________________________________________________________________
Rubi [A]
time = 0.99, antiderivative size = 936, normalized size of antiderivative = 1.00, number of steps
used = 31, number of rules used = 17, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.460, Rules used = {2980, 2917,
2652, 2719, 2648, 2655, 303, 1176, 631, 210, 1179, 642, 2988, 2985, 2984, 504, 1232}
\begin {gather*} -\frac {\left (a^2-b^2\right ) \sqrt {d} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right ) g^{5/2}}{\sqrt {2} b^3 f}-\frac {\sqrt {d} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right ) g^{5/2}}{4 \sqrt {2} b f}+\frac {\left (a^2-b^2\right ) \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right ) g^{5/2}}{\sqrt {2} b^3 f}+\frac {\sqrt {d} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right ) g^{5/2}}{4 \sqrt {2} b f}+\frac {\left (a^2-b^2\right ) \sqrt {d} \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 ) g^{5/2}}{2 \sqrt {2} b^3 f}+\frac {\sqrt {d} \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 ) g^{5/2}}{8 \sqrt {2} b f}-\frac {\left (a^2-b^2\right ) \sqrt {d} \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 ) g^{5/2}}{2 \sqrt {2} b^3 f}-\frac {\sqrt {d} \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 ) g^{5/2}}{8 \sqrt {2} b f}-\frac {2 \sqrt {2} a \sqrt {b-a} \sqrt {a+b} d \Pi \left (-\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right )\right |-1\right ) \sqrt {\sin (e+f x)} g^{5/2}}{b^3 f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a \sqrt {b-a} \sqrt {a+b} d \Pi \left (\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right )\right |-1\right ) \sqrt {\sin (e+f x)} g^{5/2}}{b^3 f \sqrt {d \sin (e+f x)}}+\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)} g^2}{b^2 f \sqrt {\sin (2 e+2 f x)}}+\frac {(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)} g}{2 b f} \end {gather*}
Antiderivative was successfully verified.
[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 2917
Rule 2980
Rule 2984
Rule 2985
Rule 2988
Rubi steps
\begin {align*} \int \frac {(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx &=\frac {g^2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} (a-b \sin (e+f x)) \, dx}{b^2}-\frac {\left (\left (a^2-b^2\right ) g^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 {\left (a g^2\right ) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{b^2}-\frac {g^2 \int \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2} \, dx}{b d}-\frac {\left (\left (a^2-b^2\right ) d g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx}{b^3}+\frac {\left (a \left (a^2-b^2\right ) d g^2\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 {g (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f}-\frac {\left (d g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx}{4 b}+\frac {\left (2 \left (a^2-b^2\right ) d^2 g^3\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 \left (a^2-b^2\right ) d g^2 \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 {\left (a g^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 {g (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f}+\frac {a g^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 (\left (a^2-b^2\right ) d g^3\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 (\left (a^2-b^2\right ) d g^3\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 (d^2 g^3\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 (4 \sqrt {2} a \left (a^2-b^2\right ) d g^3 \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 {g (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f}+\frac {a g^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 (\left (a^2-b^2\right ) \sqrt {d} g^{5/2}\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 (\left (a^2-b^2\right ) \sqrt {d} g^{5/2}\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 (\left (a^2-b^2\right ) g^3\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 (\left (a^2-b^2\right ) g^3\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 (d g^3\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 g^3\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 \sqrt {2} a \left (a^2-b^2\right ) d g^3 \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 \left (a^2-b^2\right ) d g^3 \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 (a^2-b^2\right ) \sqrt {d} g^{5/2} \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 {\left (a^2-b^2\right ) \sqrt {d} g^{5/2} \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 {2 \sqrt {2} a \sqrt {-a+b} \sqrt {a+b} d g^{5/2} \Pi \left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{b^3 f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a \sqrt {-a+b} \sqrt {a+b} d g^{5/2} \Pi \left (\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{b^3 f \sqrt {d \sin (e+f x)}}+\frac {g (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f}+\frac {a g^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 (\sqrt {d} g^{5/2}\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 (\sqrt {d} g^{5/2}\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 (\left (a^2-b^2\right ) \sqrt {d} g^{5/2}\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 (\left (a^2-b^2\right ) \sqrt {d} g^{5/2}\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 {g^3 \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 {g^3 \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-b^2\right ) \sqrt {d} g^{5/2} \tan ^{-1}\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 {\left (a^2-b^2\right ) \sqrt {d} g^{5/2} \tan ^{-1}\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 {\sqrt {d} g^{5/2} \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 {\left (a^2-b^2\right ) \sqrt {d} g^{5/2} \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 {\sqrt {d} g^{5/2} \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 {\left (a^2-b^2\right ) \sqrt {d} g^{5/2} \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 {2 \sqrt {2} a \sqrt {-a+b} \sqrt {a+b} d g^{5/2} \Pi \left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{b^3 f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a \sqrt {-a+b} \sqrt {a+b} d g^{5/2} \Pi \left (\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{b^3 f \sqrt {d \sin (e+f x)}}+\frac {g (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f}+\frac {a g^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 (\sqrt {d} g^{5/2}\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 (\sqrt {d} g^{5/2}\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 {\sqrt {d} g^{5/2} \tan ^{-1}\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 {\left (a^2-b^2\right ) \sqrt {d} g^{5/2} \tan ^{-1}\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 {\sqrt {d} g^{5/2} \tan ^{-1}\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 {\left (a^2-b^2\right ) \sqrt {d} g^{5/2} \tan ^{-1}\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 {\sqrt {d} g^{5/2} \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 {\left (a^2-b^2\right ) \sqrt {d} g^{5/2} \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 {\sqrt {d} g^{5/2} \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 {\left (a^2-b^2\right ) \sqrt {d} g^{5/2} \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 {2 \sqrt {2} a \sqrt {-a+b} \sqrt {a+b} d g^{5/2} \Pi \left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{b^3 f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a \sqrt {-a+b} \sqrt {a+b} d g^{5/2} \Pi \left (\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{b^3 f \sqrt {d \sin (e+f x)}}+\frac {g (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}{2 b f}+\frac {a g^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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 55.35, size = 1615, normalized size = 1.73 \begin {gather*} \frac {(g \cos (e+f x))^{5/2} \sec (e+f x) \sqrt {d \sin (e+f x)}}{2 b f}-\frac {(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)} \left (\frac {2 b \left (-b F_1\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 F_1\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)}{\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 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )+2 \tan ^{-1}\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 F_1\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 ) F_1\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 ) F_1\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} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )-4 \sqrt {2} a^{3/2} \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )-\frac {4 \sqrt {2} a^2 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \cos ^{\frac {5}{2}}(e+f x) \sqrt {\sin (e+f x)}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 6310 vs. \(2 (757 ) = 1514\).
time = 0.60, size = 6311, normalized size = 6.74
method | result | size |
default | \(\text {Expression too large to display}\) | \(6311\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{5/2}\,\sqrt {d\,\sin \left (e+f\,x\right )}}{a+b\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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