Optimal. Leaf size=53 \[ \frac {4 \sqrt {a \cos (c+d x)+a}}{d^2}+\frac {2 x \tan \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3319, 3296, 2638} \[ \frac {4 \sqrt {a \cos (c+d x)+a}}{d^2}+\frac {2 x \tan \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2638
Rule 3296
Rule 3319
Rubi steps
\begin {align*} \int x \sqrt {a+a \cos (c+d x)} \, dx &=\left (\sqrt {a+a \cos (c+d x)} \csc \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int x \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx\\ &=\frac {2 x \sqrt {a+a \cos (c+d x)} \tan \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {\left (2 \sqrt {a+a \cos (c+d x)} \csc \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int \sin \left (\frac {c}{2}+\frac {d x}{2}\right ) \, dx}{d}\\ &=\frac {4 \sqrt {a+a \cos (c+d x)}}{d^2}+\frac {2 x \sqrt {a+a \cos (c+d x)} \tan \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 34, normalized size = 0.64 \[ \frac {2 \left (d x \tan \left (\frac {1}{2} (c+d x)\right )+2\right ) \sqrt {a (\cos (c+d x)+1)}}{d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.44, size = 57, normalized size = 1.08 \[ 2 \, \sqrt {2} {\left (\frac {x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} + \frac {2 \, \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d^{2}}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.06, size = 80, normalized size = 1.51 \[ -\frac {i \sqrt {2}\, \sqrt {a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2} {\mathrm e}^{-i \left (d x +c \right )}}\, \left (d x \,{\mathrm e}^{i \left (d x +c \right )}+2 i {\mathrm e}^{i \left (d x +c \right )}-d x +2 i\right )}{\left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.49, size = 61, normalized size = 1.15 \[ -\frac {2 \, {\left (\sqrt {2} \sqrt {a} c \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - {\left (\sqrt {2} {\left (d x + c\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}\right )}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.21, size = 46, normalized size = 0.87 \[ \frac {2\,\sqrt {a\,\left (\cos \left (c+d\,x\right )+1\right )}\,\left (2\,\cos \left (c+d\,x\right )+d\,x\,\sin \left (c+d\,x\right )+2\right )}{d^2\,\left (\cos \left (c+d\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________