Optimal. Leaf size=249 \[ \frac {i \text {Li}_2\left (-e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right ) \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f^2 \sqrt {a \sin (e+f x)+a}}-\frac {i \text {Li}_2\left (e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right ) \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f^2 \sqrt {a \sin (e+f x)+a}}-\frac {1}{a f^2 \sqrt {a \sin (e+f x)+a}}-\frac {x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \tanh ^{-1}\left (e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right )}{a f \sqrt {a \sin (e+f x)+a}}-\frac {x \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{2 a f \sqrt {a \sin (e+f x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3319, 4185, 4183, 2279, 2391} \[ \frac {i \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \text {PolyLog}\left (2,-e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right )}{a f^2 \sqrt {a \sin (e+f x)+a}}-\frac {i \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \text {PolyLog}\left (2,e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right )}{a f^2 \sqrt {a \sin (e+f x)+a}}-\frac {1}{a f^2 \sqrt {a \sin (e+f x)+a}}-\frac {x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \tanh ^{-1}\left (e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right )}{a f \sqrt {a \sin (e+f x)+a}}-\frac {x \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{2 a f \sqrt {a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2279
Rule 2391
Rule 3319
Rule 4183
Rule 4185
Rubi steps
\begin {align*} \int \frac {x}{(a+a \sin (e+f x))^{3/2}} \, dx &=\frac {\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \int x \csc ^3\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{2 a \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {1}{a f^2 \sqrt {a+a \sin (e+f x)}}-\frac {x \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+a \sin (e+f x)}}+\frac {\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \int x \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{4 a \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {1}{a f^2 \sqrt {a+a \sin (e+f x)}}-\frac {x \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+a \sin (e+f x)}}-\frac {x \tanh ^{-1}\left (e^{\frac {1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+a \sin (e+f x)}}-\frac {\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \int \log \left (1-e^{i \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}\right ) \, dx}{2 a f \sqrt {a+a \sin (e+f x)}}+\frac {\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \int \log \left (1+e^{i \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}\right ) \, dx}{2 a f \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {1}{a f^2 \sqrt {a+a \sin (e+f x)}}-\frac {x \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+a \sin (e+f x)}}-\frac {x \tanh ^{-1}\left (e^{\frac {1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+a \sin (e+f x)}}+\frac {\left (i \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}\right )}{a f^2 \sqrt {a+a \sin (e+f x)}}-\frac {\left (i \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}\right )}{a f^2 \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {1}{a f^2 \sqrt {a+a \sin (e+f x)}}-\frac {x \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+a \sin (e+f x)}}-\frac {x \tanh ^{-1}\left (e^{\frac {1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+a \sin (e+f x)}}+\frac {i \text {Li}_2\left (-e^{\frac {1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f^2 \sqrt {a+a \sin (e+f x)}}-\frac {i \text {Li}_2\left (e^{\frac {1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f^2 \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.72, size = 308, normalized size = 1.24 \[ \frac {\frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (2 i \left (\text {Li}_2\left (-e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right )-\text {Li}_2\left (e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right )\right )+\frac {1}{2} (2 e+2 f x+\pi ) \left (\log \left (1-e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right )-\log \left (1+e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right )\right )-\pi \tanh ^{-1}\left (\frac {\tan \left (\frac {1}{4} (e+f x)\right )-1}{\sqrt {2}}\right )\right )}{\sqrt {2}}-(f x+2) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2+2 f x \sin \left (\frac {1}{2} (e+f x)\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )+\frac {e (\sin (e+f x)+1) \sin \left (\frac {1}{4} (2 e+2 f x-\pi )\right ) \sin ^{-1}\left (\csc \left (\frac {1}{4} (2 e+2 f x+\pi )\right )\right )}{\sqrt {\frac {\sin (e+f x)-1}{\sin (e+f x)+1}}}}{2 f^2 (a (\sin (e+f x)+1))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {a \sin \left (f x + e\right ) + a} x}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________