Optimal. Leaf size=101 \[ \sin \left (\frac {1}{4} (2 c+\pi )\right ) \text {Ci}\left (\frac {d x}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a}+\cos \left (\frac {1}{4} (2 c+\pi )\right ) \text {Si}\left (\frac {d x}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \]
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Rubi [A] time = 0.14, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3319, 3303, 3299, 3302} \[ \sin \left (\frac {1}{4} (2 c+\pi )\right ) \text {CosIntegral}\left (\frac {d x}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a}+\cos \left (\frac {1}{4} (2 c+\pi )\right ) \text {Si}\left (\frac {d x}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 3319
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \sin (c+d x)}}{x} \, dx &=\left (\csc \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{x} \, dx\\ &=\left (\cos \left (\frac {1}{4} (2 c+\pi )\right ) \csc \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sin \left (\frac {d x}{2}\right )}{x} \, dx+\left (\csc \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sin \left (\frac {1}{4} (2 c+\pi )\right ) \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\cos \left (\frac {d x}{2}\right )}{x} \, dx\\ &=\text {Ci}\left (\frac {d x}{2}\right ) \csc \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sin \left (\frac {1}{4} (2 c+\pi )\right ) \sqrt {a+a \sin (c+d x)}+\cos \left (\frac {1}{4} (2 c+\pi )\right ) \csc \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)} \text {Si}\left (\frac {d x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.18, size = 83, normalized size = 0.82 \[ \frac {\sqrt {a (\sin (c+d x)+1)} \left (\left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \text {Ci}\left (\frac {d x}{2}\right )+\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \text {Si}\left (\frac {d x}{2}\right )\right )}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 2.25, size = 383, normalized size = 3.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a +a \sin \left (d x +c \right )}}{x}\, dx \]
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 \[ \int \frac {\sqrt {a \sin \left (d x + c\right ) + a}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}}{x} \,d x \]
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 \[ \int \frac {\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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