Optimal. Leaf size=982 \[ \frac {3 d^{3/2} g^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{4 \sqrt {2} b f}+\frac {\left (a^2-b^2\right ) d^{3/2} g^{3/2} \tan ^{-1}\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^3 f}-\frac {3 d^{3/2} g^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{4 \sqrt {2} b f}-\frac {\left (a^2-b^2\right ) d^{3/2} g^{3/2} \tan ^{-1}\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^3 f}+\frac {2 \sqrt {2} a \sqrt {-a^2+b^2} d^{3/2} g^2 \sqrt {\cos (e+f x)} \Pi \left (-\frac {a}{b-\sqrt {-a^2+b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b^3 f \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} a \sqrt {-a^2+b^2} d^{3/2} g^2 \sqrt {\cos (e+f x)} \Pi \left (-\frac {a}{b+\sqrt {-a^2+b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b^3 f \sqrt {g \cos (e+f x)}}-\frac {3 d^{3/2} 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 )}{8 \sqrt {2} b f}-\frac {\left (a^2-b^2\right ) d^{3/2} 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^3 f}+\frac {3 d^{3/2} 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 )}{8 \sqrt {2} b f}+\frac {\left (a^2-b^2\right ) d^{3/2} 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^3 f}-\frac {a d g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b^2 f}+\frac {g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{2 b f}+\frac {a d^2 g^2 F\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (2 e+2 f x)}}{2 b^2 f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.11, antiderivative size = 982, normalized size of antiderivative = 1.00, number of steps
used = 31, number of rules used = 16, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.432, Rules used = {2980,
2917, 2648, 2653, 2720, 2654, 303, 1176, 631, 210, 1179, 642, 2988, 2987, 2986, 1232}
\begin {gather*} \frac {2 \sqrt {2} a \sqrt {b^2-a^2} d^{3/2} \sqrt {\cos (e+f x)} \Pi \left (-\frac {a}{b-\sqrt {b^2-a^2}};\left .\text {ArcSin}\left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right )\right |-1\right ) g^2}{b^3 f \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} a \sqrt {b^2-a^2} d^{3/2} \sqrt {\cos (e+f x)} \Pi \left (-\frac {a}{b+\sqrt {b^2-a^2}};\left .\text {ArcSin}\left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right )\right |-1\right ) g^2}{b^3 f \sqrt {g \cos (e+f x)}}+\frac {a d^2 F\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {\sin (2 e+2 f x)} g^2}{2 b^2 f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}+\frac {\left (a^2-b^2\right ) d^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right ) g^{3/2}}{\sqrt {2} b^3 f}+\frac {3 d^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right ) g^{3/2}}{4 \sqrt {2} b f}-\frac {\left (a^2-b^2\right ) d^{3/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right ) g^{3/2}}{\sqrt {2} b^3 f}-\frac {3 d^{3/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right ) g^{3/2}}{4 \sqrt {2} b f}-\frac {\left (a^2-b^2\right ) d^{3/2} \log \left (\sqrt {d} \tan (e+f x)+\sqrt {d}-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right ) g^{3/2}}{2 \sqrt {2} b^3 f}-\frac {3 d^{3/2} \log \left (\sqrt {d} \tan (e+f x)+\sqrt {d}-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right ) g^{3/2}}{8 \sqrt {2} b f}+\frac {\left (a^2-b^2\right ) d^{3/2} \log \left (\sqrt {d} \tan (e+f x)+\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right ) g^{3/2}}{2 \sqrt {2} b^3 f}+\frac {3 d^{3/2} \log \left (\sqrt {d} \tan (e+f x)+\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right ) g^{3/2}}{8 \sqrt {2} b f}+\frac {\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2} g}{2 b f}-\frac {a d \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} g}{b^2 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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
Rule 2988
Rubi steps
\begin {align*} \int \frac {(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx &=\frac {g^2 \int \frac {(d \sin (e+f x))^{3/2} (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 {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx}{b^2}\\ &=\frac {\left (a g^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)}} \, dx}{b^2}-\frac {g^2 \int \frac {(d \sin (e+f x))^{5/2}}{\sqrt {g \cos (e+f x)}} \, dx}{b d}-\frac {\left (\left (a^2-b^2\right ) d g^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}} \, dx}{b^3}+\frac {\left (a \left (a^2-b^2\right ) d 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^3}\\ &=-\frac {a d g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b^2 f}+\frac {g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{2 b f}-\frac {\left (3 d g^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}} \, dx}{4 b}+\frac {\left (a d^2 g^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \, dx}{2 b^2}-\frac {\left (2 \left (a^2-b^2\right ) d^2 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^3 f}+\frac {\left (a \left (a^2-b^2\right ) d 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^3 \sqrt {g \cos (e+f x)}}\\ &=-\frac {a d g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b^2 f}+\frac {g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{2 b f}+\frac {\left (\left (a^2-b^2\right ) d^2 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^3 f}-\frac {\left (\left (a^2-b^2\right ) d^2 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^3 f}-\frac {\left (3 d^2 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 )}{2 b f}+\frac {\left (2 \sqrt {2} a \left (a^2-b^2\right ) \left (1-\frac {b}{\sqrt {-a^2+b^2}}\right ) d^2 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^3 f \sqrt {g \cos (e+f x)}}+\frac {\left (2 \sqrt {2} a \left (a^2-b^2\right ) \left (1+\frac {b}{\sqrt {-a^2+b^2}}\right ) d^2 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^3 f \sqrt {g \cos (e+f x)}}+\frac {\left (a d^2 g^2 \sqrt {\sin (2 e+2 f x)}\right ) \int \frac {1}{\sqrt {\sin (2 e+2 f x)}} \, dx}{2 b^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}\\ &=\frac {2 \sqrt {2} a \sqrt {-a^2+b^2} d^{3/2} g^2 \sqrt {\cos (e+f x)} \Pi \left (-\frac {a}{b-\sqrt {-a^2+b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b^3 f \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} a \sqrt {-a^2+b^2} d^{3/2} g^2 \sqrt {\cos (e+f x)} \Pi \left (-\frac {a}{b+\sqrt {-a^2+b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b^3 f \sqrt {g \cos (e+f x)}}-\frac {a d g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b^2 f}+\frac {g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{2 b f}+\frac {a d^2 g^2 F\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (2 e+2 f x)}}{2 b^2 f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {\left (\left (a^2-b^2\right ) d^2 g\right ) \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^3 f}-\frac {\left (\left (a^2-b^2\right ) d^2 g\right ) \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^3 f}-\frac {\left (\left (a^2-b^2\right ) d^{3/2} 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^3 f}-\frac {\left (\left (a^2-b^2\right ) d^{3/2} 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^3 f}+\frac {\left (3 d^2 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 )}{4 b f}-\frac {\left (3 d^2 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 )}{4 b f}\\ &=\frac {2 \sqrt {2} a \sqrt {-a^2+b^2} d^{3/2} g^2 \sqrt {\cos (e+f x)} \Pi \left (-\frac {a}{b-\sqrt {-a^2+b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b^3 f \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} a \sqrt {-a^2+b^2} d^{3/2} g^2 \sqrt {\cos (e+f x)} \Pi \left (-\frac {a}{b+\sqrt {-a^2+b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b^3 f \sqrt {g \cos (e+f x)}}-\frac {\left (a^2-b^2\right ) d^{3/2} 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^3 f}+\frac {\left (a^2-b^2\right ) d^{3/2} 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^3 f}-\frac {a d g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b^2 f}+\frac {g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{2 b f}+\frac {a d^2 g^2 F\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (2 e+2 f x)}}{2 b^2 f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {\left (3 d^2 g\right ) \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 )}{8 b f}-\frac {\left (3 d^2 g\right ) \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 )}{8 b f}-\frac {\left (3 d^{3/2} 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 )}{8 \sqrt {2} b f}-\frac {\left (3 d^{3/2} 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 )}{8 \sqrt {2} b f}-\frac {\left (\left (a^2-b^2\right ) d^{3/2} 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^3 f}+\frac {\left (\left (a^2-b^2\right ) d^{3/2} 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^3 f}\\ &=\frac {\left (a^2-b^2\right ) d^{3/2} g^{3/2} \tan ^{-1}\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^3 f}-\frac {\left (a^2-b^2\right ) d^{3/2} g^{3/2} \tan ^{-1}\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^3 f}+\frac {2 \sqrt {2} a \sqrt {-a^2+b^2} d^{3/2} g^2 \sqrt {\cos (e+f x)} \Pi \left (-\frac {a}{b-\sqrt {-a^2+b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b^3 f \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} a \sqrt {-a^2+b^2} d^{3/2} g^2 \sqrt {\cos (e+f x)} \Pi \left (-\frac {a}{b+\sqrt {-a^2+b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b^3 f \sqrt {g \cos (e+f x)}}-\frac {3 d^{3/2} 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 )}{8 \sqrt {2} b f}-\frac {\left (a^2-b^2\right ) d^{3/2} 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^3 f}+\frac {3 d^{3/2} 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 )}{8 \sqrt {2} b f}+\frac {\left (a^2-b^2\right ) d^{3/2} 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^3 f}-\frac {a d g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b^2 f}+\frac {g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{2 b f}+\frac {a d^2 g^2 F\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (2 e+2 f x)}}{2 b^2 f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {\left (3 d^{3/2} 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 )}{4 \sqrt {2} b f}+\frac {\left (3 d^{3/2} 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 )}{4 \sqrt {2} b f}\\ &=\frac {3 d^{3/2} g^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{4 \sqrt {2} b f}+\frac {\left (a^2-b^2\right ) d^{3/2} g^{3/2} \tan ^{-1}\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^3 f}-\frac {3 d^{3/2} g^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{4 \sqrt {2} b f}-\frac {\left (a^2-b^2\right ) d^{3/2} g^{3/2} \tan ^{-1}\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^3 f}+\frac {2 \sqrt {2} a \sqrt {-a^2+b^2} d^{3/2} g^2 \sqrt {\cos (e+f x)} \Pi \left (-\frac {a}{b-\sqrt {-a^2+b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b^3 f \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} a \sqrt {-a^2+b^2} d^{3/2} g^2 \sqrt {\cos (e+f x)} \Pi \left (-\frac {a}{b+\sqrt {-a^2+b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{b^3 f \sqrt {g \cos (e+f x)}}-\frac {3 d^{3/2} 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 )}{8 \sqrt {2} b f}-\frac {\left (a^2-b^2\right ) d^{3/2} 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^3 f}+\frac {3 d^{3/2} 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 )}{8 \sqrt {2} b f}+\frac {\left (a^2-b^2\right ) d^{3/2} 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^3 f}-\frac {a d g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b^2 f}+\frac {g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{2 b f}+\frac {a d^2 g^2 F\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (2 e+2 f x)}}{2 b^2 f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+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 = 57.72, size = 1898, normalized size = 1.93 \begin {gather*} \frac {(g \cos (e+f x))^{3/2} \sec (e+f x) (d \sin (e+f x))^{3/2}}{2 b f}-\frac {(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} \left (\frac {10 b \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \left (\frac {b F_1\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 ) \sqrt {1-\cos ^2(e+f x)}}{-5 \left (a^2-b^2\right ) F_1\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 F_1\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 ) F_1\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 )\right ) \cos ^2(e+f x)}+\frac {a F_1\left (\frac {1}{4};-\frac {1}{4},1;\frac {5}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )}{5 \left (a^2-b^2\right ) F_1\left (\frac {1}{4};-\frac {1}{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 F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+\left (-a^2+b^2\right ) F_1\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 )\right ) \cos ^2(e+f x)}\right ) \sin ^{\frac {5}{2}}(e+f x)}{\left (1-\cos ^2(e+f x)\right ) \left (a^2+b^2 \left (-1+\cos ^2(e+f x)\right )\right ) (a+b \sin (e+f x))}+\frac {2 a \sqrt {\sin (e+f x)} \left (\frac {\sqrt {a} \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 )}{4 \sqrt {2} \left (a^2-b^2\right )^{3/4}}-\frac {b F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};-\tan ^2(e+f x),\frac {\left (-a^2+b^2\right ) \tan ^2(e+f x)}{a^2}\right ) \tan ^{\frac {5}{2}}(e+f x)}{5 a^2}\right ) \left (b \tan (e+f x)+a \sqrt {1+\tan ^2(e+f x)}\right )}{\cos ^{\frac {5}{2}}(e+f x) (a+b \sin (e+f x)) \sqrt {\tan (e+f x)} \left (1+\tan ^2(e+f x)\right )^{3/2}}-\frac {a \cos (2 (e+f x)) \sqrt {\sin (e+f x)} \left (b \tan (e+f x)+a \sqrt {1+\tan ^2(e+f x)}\right ) \left (-20 \sqrt {2} a \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )+20 \sqrt {2} a \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )+\frac {10 \sqrt {2} \sqrt {a} \left (2 a^2-b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\left (a^2-b^2\right )^{3/4}}-\frac {10 \sqrt {2} \sqrt {a} \left (2 a^2-b^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\left (a^2-b^2\right )^{3/4}}+10 \sqrt {2} a \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-10 \sqrt {2} a \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-\frac {5 \sqrt {2} \sqrt {a} \left (2 a^2-b^2\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 )}{\left (a^2-b^2\right )^{3/4}}+\frac {5 \sqrt {2} \sqrt {a} \left (2 a^2-b^2\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 )}{\left (a^2-b^2\right )^{3/4}}+8 b F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {5}{2}}(e+f x)+\frac {40 b \sqrt {\tan (e+f x)}}{\sqrt {1+\tan ^2(e+f x)}}+\frac {200 a^4 b F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \sqrt {\tan (e+f x)}}{\sqrt {1+\tan ^2(e+f x)} \left (-5 a^2 F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right )+2 \left (2 \left (a^2-b^2\right ) F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right )+a^2 F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)\right ) \left (-b^2 \tan ^2(e+f x)+a^2 \left (1+\tan ^2(e+f x)\right )\right )}\right )}{10 b^2 \cos ^{\frac {5}{2}}(e+f x) (a+b \sin (e+f x)) \sqrt {\tan (e+f x)} \left (-1+\tan ^2(e+f x)\right ) \sqrt {1+\tan ^2(e+f x)}}\right )}{4 b f \cos ^{\frac {3}{2}}(e+f x) \sin ^{\frac {3}{2}}(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. 2546 vs. \(2 (806 ) = 1612\).
time = 0.60, size = 2547, normalized size = 2.59
method | result | size |
default | \(\text {Expression too large to display}\) | \(2547\) |
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 )}^{3/2}\,{\left (d\,\sin \left (e+f\,x\right )\right )}^{3/2}}{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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