Optimal. Leaf size=23 \[ \frac{2 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0121487, antiderivative size = 23, normalized size of antiderivative = 1., 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 = {2661} \[ \frac{2 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2661
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3+4 \cos (c+d x)}} \, dx &=\frac{2 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d}\\ \end{align*}
Mathematica [A] time = 0.0277494, size = 23, normalized size = 1. \[ \frac{2 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.046, size = 23, normalized size = 1. \begin{align*}{\frac{2\,\sqrt{7}}{7\,d}{\it InverseJacobiAM} \left ({\frac{dx}{2}}+{\frac{c}{2}},{\frac{2\,\sqrt{14}}{7}} \right ) } \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{1}{\sqrt{4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{4 \, \cos \left (d x + c\right ) + 3}}, x\right ) \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{1}{\sqrt{4 \cos{\left (c + d x \right )} + 3}}\, 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{1}{\sqrt{4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]