Optimal. Leaf size=130 \[ -\frac {\sqrt {a+a \sin (c+d x)}}{x}-\frac {1}{2} d \text {Ci}\left (\frac {d x}{2}\right ) \csc \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sin \left (\frac {1}{4} (2 c-\pi )\right ) \sqrt {a+a \sin (c+d x)}-\frac {1}{2} d \csc \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sin \left (\frac {1}{4} (2 c+\pi )\right ) \sqrt {a+a \sin (c+d x)} \text {Si}\left (\frac {d x}{2}\right ) \]
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Rubi [A]
time = 0.10, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3400, 3378,
3384, 3380, 3383} \begin {gather*} -\frac {1}{2} d \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}-\frac {1}{2} d \sin \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}-\frac {\sqrt {a \sin (c+d x)+a}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3400
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \sin (c+d x)}}{x^2} \, 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^2} \, dx\\ &=-\frac {\sqrt {a+a \sin (c+d x)}}{x}+\frac {1}{2} \left (d \csc \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{x} \, dx\\ &=-\frac {\sqrt {a+a \sin (c+d x)}}{x}-\frac {1}{2} \left (d \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-\frac {1}{2} \left (d \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 {\sin \left (\frac {d x}{2}\right )}{x} \, dx\\ &=-\frac {\sqrt {a+a \sin (c+d x)}}{x}-\frac {1}{2} d \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)}-\frac {1}{2} d \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)} \text {Si}\left (\frac {d x}{2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 117, normalized size = 0.90 \begin {gather*} \frac {\sqrt {a (1+\sin (c+d x))} \left (d x \text {Ci}\left (\frac {d x}{2}\right ) \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right )-2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-d x \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \text {Si}\left (\frac {d x}{2}\right )\right )}{2 x \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a +a \sin \left (d x +c \right )}}{x^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 2.20, size = 1140, normalized size = 8.77 \begin {gather*} \text {Too large to display} \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 = -0.01 \begin {gather*} \int \frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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