Optimal. Leaf size=23 \[ \frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2653} \[ \frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2653
Rubi steps
\begin {align*} \int \sqrt {3+4 \cos (c+d x)} \, dx &=\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 23, normalized size = 1.00 \[ \frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {4 \, \cos \left (d x + c\right ) + 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {4 \, \cos \left (d x + c\right ) + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.42, size = 137, normalized size = 5.96 \[ \frac {2 \sqrt {\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )}{\sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {4 \, \cos \left (d x + c\right ) + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {4\,\cos \left (c+d\,x\right )+3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {4 \cos {\left (c + d x \right )} + 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________