Optimal. Leaf size=1064 \[ -\frac {2 \sqrt {2} a^3 \Pi \left (-\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\sin ^{-1}\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)} d^3}{b (b-a)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a^3 \Pi \left (\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\sin ^{-1}\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)} d^3}{b (b-a)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {b \tan ^{-1}\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} \left (a^2-b^2\right ) f g^{3/2}}-\frac {a^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 ) d^{5/2}}{\sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {b \tan ^{-1}\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} \left (a^2-b^2\right ) f g^{3/2}}+\frac {a^2 \tan ^{-1}\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 \left (a^2-b^2\right ) f g^{3/2}}-\frac {b \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} \left (a^2-b^2\right ) f g^{3/2}}+\frac {a^2 \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 \left (a^2-b^2\right ) f g^{3/2}}+\frac {b \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} \left (a^2-b^2\right ) f g^{3/2}}-\frac {a^2 \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 \left (a^2-b^2\right ) f g^{3/2}}-\frac {2 b \sqrt {d \sin (e+f x)} d^2}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 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}{\left (a^2-b^2\right ) f g^2 \sqrt {\sin (2 e+2 f x)}}+\frac {2 a (d \sin (e+f x))^{3/2} d}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}} \]
[Out]
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Rubi [A] time = 1.62, antiderivative size = 1064, 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 = {2902, 2571, 2572, 2639, 2566, 2575, 297, 1162, 617, 204, 1165, 628, 2909, 2906, 2905, 490, 1218} \[ -\frac {2 \sqrt {2} a^3 \Pi \left (-\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\sin ^{-1}\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)} d^3}{b (b-a)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a^3 \Pi \left (\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\sin ^{-1}\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)} d^3}{b (b-a)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {b \tan ^{-1}\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} \left (a^2-b^2\right ) f g^{3/2}}-\frac {a^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 ) d^{5/2}}{\sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {b \tan ^{-1}\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} \left (a^2-b^2\right ) f g^{3/2}}+\frac {a^2 \tan ^{-1}\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 \left (a^2-b^2\right ) f g^{3/2}}-\frac {b \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} \left (a^2-b^2\right ) f g^{3/2}}+\frac {a^2 \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 \left (a^2-b^2\right ) f g^{3/2}}+\frac {b \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} \left (a^2-b^2\right ) f g^{3/2}}-\frac {a^2 \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 \left (a^2-b^2\right ) f g^{3/2}}-\frac {2 b \sqrt {d \sin (e+f x)} d^2}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 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}{\left (a^2-b^2\right ) f g^2 \sqrt {\sin (2 e+2 f x)}}+\frac {2 a (d \sin (e+f x))^{3/2} d}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 297
Rule 490
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1218
Rule 2566
Rule 2571
Rule 2572
Rule 2575
Rule 2639
Rule 2902
Rule 2905
Rule 2906
Rule 2909
Rubi steps
\begin {align*} \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx &=-\frac {(b d) \int \frac {(d \sin (e+f x))^{3/2}}{(g \cos (e+f x))^{3/2}} \, dx}{a^2-b^2}+\frac {\left (a d^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}} \, dx}{a^2-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}{\left (a^2-b^2\right ) g^2}\\ &=-\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {\left (2 a d^2\right ) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) g^2}-\frac {\left (a^2 d^3\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx}{b \left (a^2-b^2\right ) g^2}+\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 \left (a^2-b^2\right ) g^2}+\frac {\left (b d^3\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx}{\left (a^2-b^2\right ) g^2}\\ &=-\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {\left (2 a^2 d^4\right ) \operatorname {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 \left (a^2-b^2\right ) f g}-\frac {\left (2 b d^4\right ) \operatorname {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 )}{\left (a^2-b^2\right ) f g}+\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 \left (a^2-b^2\right ) g^2 \sqrt {d \sin (e+f x)}}-\frac {\left (2 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}{\left (a^2-b^2\right ) g^2 \sqrt {\sin (2 e+2 f x)}}\\ &=-\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 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)}}{\left (a^2-b^2\right ) f g^2 \sqrt {\sin (2 e+2 f x)}}-\frac {\left (a^2 d^3\right ) \operatorname {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 \left (a^2-b^2\right ) f g}+\frac {\left (a^2 d^3\right ) \operatorname {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 \left (a^2-b^2\right ) f g}+\frac {\left (b d^3\right ) \operatorname {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 )}{\left (a^2-b^2\right ) f g}-\frac {\left (b d^3\right ) \operatorname {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 )}{\left (a^2-b^2\right ) f g}-\frac {\left (4 \sqrt {2} a^3 d^3 \sqrt {\sin (e+f x)}\right ) \operatorname {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 \left (a^2-b^2\right ) f g \sqrt {d \sin (e+f x)}}\\ &=-\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 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)}}{\left (a^2-b^2\right ) f g^2 \sqrt {\sin (2 e+2 f x)}}+\frac {\left (a^2 d^{5/2}\right ) \operatorname {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 \left (a^2-b^2\right ) f g^{3/2}}+\frac {\left (a^2 d^{5/2}\right ) \operatorname {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 \left (a^2-b^2\right ) f g^{3/2}}-\frac {\left (b d^{5/2}\right ) \operatorname {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} \left (a^2-b^2\right ) f g^{3/2}}-\frac {\left (b d^{5/2}\right ) \operatorname {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} \left (a^2-b^2\right ) f g^{3/2}}+\frac {\left (a^2 d^2\right ) \operatorname {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 \left (a^2-b^2\right ) f g}+\frac {\left (a^2 d^2\right ) \operatorname {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 \left (a^2-b^2\right ) f g}-\frac {\left (b d^2\right ) \operatorname {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 \left (a^2-b^2\right ) f g}-\frac {\left (b d^2\right ) \operatorname {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 \left (a^2-b^2\right ) f g}-\frac {\left (2 \sqrt {2} a^3 d^3 \sqrt {\sin (e+f x)}\right ) \operatorname {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 \sqrt {-a+b} \left (a^2-b^2\right ) f g \sqrt {d \sin (e+f x)}}+\frac {\left (2 \sqrt {2} a^3 d^3 \sqrt {\sin (e+f x)}\right ) \operatorname {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 \sqrt {-a+b} \left (a^2-b^2\right ) f g \sqrt {d \sin (e+f x)}}\\ &=\frac {a^2 d^{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 \left (a^2-b^2\right ) f g^{3/2}}-\frac {b d^{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} \left (a^2-b^2\right ) f g^{3/2}}-\frac {a^2 d^{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 \left (a^2-b^2\right ) f g^{3/2}}+\frac {b d^{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} \left (a^2-b^2\right ) f g^{3/2}}-\frac {2 \sqrt {2} a^3 d^3 \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 (-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a^3 d^3 \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 (-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}-\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 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)}}{\left (a^2-b^2\right ) f g^2 \sqrt {\sin (2 e+2 f x)}}+\frac {\left (a^2 d^{5/2}\right ) \operatorname {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 \left (a^2-b^2\right ) f g^{3/2}}-\frac {\left (a^2 d^{5/2}\right ) \operatorname {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 \left (a^2-b^2\right ) f g^{3/2}}-\frac {\left (b d^{5/2}\right ) \operatorname {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} \left (a^2-b^2\right ) f g^{3/2}}+\frac {\left (b d^{5/2}\right ) \operatorname {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} \left (a^2-b^2\right ) f g^{3/2}}\\ &=-\frac {a^2 d^{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 \left (a^2-b^2\right ) f g^{3/2}}+\frac {b d^{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} \left (a^2-b^2\right ) f g^{3/2}}+\frac {a^2 d^{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 \left (a^2-b^2\right ) f g^{3/2}}-\frac {b d^{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} \left (a^2-b^2\right ) f g^{3/2}}+\frac {a^2 d^{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 \left (a^2-b^2\right ) f g^{3/2}}-\frac {b d^{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} \left (a^2-b^2\right ) f g^{3/2}}-\frac {a^2 d^{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 \left (a^2-b^2\right ) f g^{3/2}}+\frac {b d^{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} \left (a^2-b^2\right ) f g^{3/2}}-\frac {2 \sqrt {2} a^3 d^3 \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 (-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a^3 d^3 \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 (-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}-\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 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)}}{\left (a^2-b^2\right ) f g^2 \sqrt {\sin (2 e+2 f x)}}\\ \end {align*}
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Mathematica [C] time = 80.66, size = 1290, normalized size = 1.21 \[ \frac {2 \cot (e+f x) \csc (e+f x) (d \sin (e+f x))^{5/2} (a \sin (e+f x)-b)}{\left (a^2-b^2\right ) f (g \cos (e+f x))^{3/2}}-\frac {\cos ^{\frac {3}{2}}(e+f x) (d \sin (e+f x))^{5/2} \left (-\frac {2 \left (3 a^2-b^2\right ) \left (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)}{b^2-a^2}\right )-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)}{b^2-a^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 {\cos (2 (e+f x)) \sqrt {\tan (e+f x)} \left (\sqrt {\tan ^2(e+f x)+1} a+b \tan (e+f x)\right ) \left (24 b \left (b^2-a^2\right ) F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};-\tan ^2(e+f x),\left (\frac {b^2}{a^2}-1\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {7}{2}}(e+f x)+56 b \left (b^2-3 a^2\right ) F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};-\tan ^2(e+f x),\left (\frac {b^2}{a^2}-1\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)+21 a^{3/2} \left (-\frac {4 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right ) a^2}{\sqrt [4]{a^2-b^2}}+\frac {4 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}+1\right ) a^2}{\sqrt [4]{a^2-b^2}}-\frac {2 \sqrt {2} \log \left (-a+\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)} \sqrt {a}-\sqrt {a^2-b^2} \tan (e+f x)\right ) a^2}{\sqrt [4]{a^2-b^2}}+\frac {2 \sqrt {2} \log \left (a+\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)} \sqrt {a}+\sqrt {a^2-b^2} \tan (e+f x)\right ) a^2}{\sqrt [4]{a^2-b^2}}+4 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) a^{3/2}-4 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right ) a^{3/2}+2 \sqrt {2} \log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right ) a^{3/2}-2 \sqrt {2} \log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right ) a^{3/2}+\frac {8 b \tan ^{\frac {3}{2}}(e+f x) \sqrt {a}}{\sqrt {\tan ^2(e+f x)+1}}+\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 {2 \sqrt {2} b^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}+1\right )}{\sqrt [4]{a^2-b^2}}+\frac {\sqrt {2} b^2 \log \left (-a+\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)} \sqrt {a}-\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 [4]{a^2-b^2} \sqrt {\tan (e+f x)} \sqrt {a}+\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}\right )\right )}{84 a b \cos ^{\frac {3}{2}}(e+f x) (a+b \sin (e+f x)) \left (\tan ^2(e+f x)-1\right ) \sqrt {\tan ^2(e+f x)+1} \sqrt {\sin (e+f x)}}\right )}{(a-b) (a+b) f (g \cos (e+f x))^{3/2} \sin ^{\frac {5}{2}}(e+f x)} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {\left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.89, size = 4619, normalized size = 4.34 \[ \text {output too large to display} \]
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 \[ \int \frac {\left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}}\,{d x} \]
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 \[ \int \frac {{\left (d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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