Optimal. Leaf size=130 \[ -\frac{1}{2} d \sin \left (\frac{1}{4} (2 c-\pi )\right ) \text{CosIntegral}\left (\frac{d x}{2}\right ) \csc \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (c+d x)+a}-\frac{1}{2} d \sin \left (\frac{1}{4} (2 c+\pi )\right ) \text{Si}\left (\frac{d x}{2}\right ) \csc \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (c+d x)+a}-\frac{\sqrt{a \sin (c+d x)+a}}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15319, antiderivative size = 130, normalized size of antiderivative = 1., 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 = {3319, 3297, 3303, 3299, 3302} \[ -\frac{1}{2} d \sin \left (\frac{1}{4} (2 c-\pi )\right ) \text{CosIntegral}\left (\frac{d x}{2}\right ) \csc \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (c+d x)+a}-\frac{1}{2} d \sin \left (\frac{1}{4} (2 c+\pi )\right ) \text{Si}\left (\frac{d x}{2}\right ) \csc \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (c+d x)+a}-\frac{\sqrt{a \sin (c+d x)+a}}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3319
Rule 3297
Rule 3303
Rule 3299
Rule 3302
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*}
Mathematica [A] time = 0.300279, size = 117, normalized size = 0.9 \[ \frac{\sqrt{a (\sin (c+d x)+1)} \left (d x \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \text{CosIntegral}\left (\frac{d x}{2}\right )-d x \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \text{Si}\left (\frac{d x}{2}\right )-2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{2 x \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sqrt{a+a\sin \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \sin \left (d x + c\right ) + a}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \left (\sin{\left (c + d x \right )} + 1\right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \sin \left (d x + c\right ) + a}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]