Optimal. Leaf size=156 \[ \frac {4 i \text {Li}_2\left (-i e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cos (c+d x)+a}}-\frac {4 i \text {Li}_2\left (i e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cos (c+d x)+a}}-\frac {4 i x \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \tan ^{-1}\left (e^{\frac {1}{2} i (c+d x)}\right )}{d \sqrt {a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.09, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3319, 4181, 2279, 2391} \[ \frac {4 i \text {Li}_2\left (-i e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cos (c+d x)+a}}-\frac {4 i \text {Li}_2\left (i e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cos (c+d x)+a}}-\frac {4 i x \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \tan ^{-1}\left (e^{\frac {1}{2} i (c+d x)}\right )}{d \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 3319
Rule 4181
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+a \cos (c+d x)}} \, dx &=\frac {\sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right ) \int x \csc \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{\sqrt {a+a \cos (c+d x)}}\\ &=-\frac {4 i x \tan ^{-1}\left (e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cos (c+d x)}}-\frac {\left (2 \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int \log \left (1-i e^{i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{d \sqrt {a+a \cos (c+d x)}}+\frac {\left (2 \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int \log \left (1+i e^{i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{d \sqrt {a+a \cos (c+d x)}}\\ &=-\frac {4 i x \tan ^{-1}\left (e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cos (c+d x)}}+\frac {\left (4 i \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}-\frac {\left (4 i \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}\\ &=-\frac {4 i x \tan ^{-1}\left (e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cos (c+d x)}}+\frac {4 i \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \text {Li}_2\left (-i e^{\frac {1}{2} i (c+d x)}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}-\frac {4 i \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \text {Li}_2\left (i e^{\frac {1}{2} i (c+d x)}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 89, normalized size = 0.57 \[ -\frac {4 i \cos \left (\frac {1}{2} (c+d x)\right ) \left (-\text {Li}_2\left (-i e^{\frac {1}{2} i (c+d x)}\right )+\text {Li}_2\left (i e^{\frac {1}{2} i (c+d x)}\right )+d x \tan ^{-1}\left (e^{\frac {1}{2} i (c+d x)}\right )\right )}{d^2 \sqrt {a (\cos (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\sqrt {a \cos \left (d x + c\right ) + a}}, x\right ) \]
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 \[ \int \frac {x}{\sqrt {a \cos \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {a +a \cos \left (d x +c \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{\sqrt {a+a\,\cos \left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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