Optimal. Leaf size=54 \[ \frac {2 \sqrt {-2+3 \cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |6\right ) \sqrt {\sec (c+d x)}}{d \sqrt {3-2 \sec (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {3943, 2740}
\begin {gather*} \frac {2 \sqrt {3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |6\right )}{d \sqrt {3-2 \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2740
Rule 3943
Rubi steps
\begin {align*} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {3-2 \sec (c+d x)}} \, dx &=\frac {\left (\sqrt {-2+3 \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {-2+3 \cos (c+d x)}} \, dx}{\sqrt {3-2 \sec (c+d x)}}\\ &=\frac {2 \sqrt {-2+3 \cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |6\right ) \sqrt {\sec (c+d x)}}{d \sqrt {3-2 \sec (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 54, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-2+3 \cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |6\right ) \sqrt {\sec (c+d x)}}{d \sqrt {3-2 \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 0.20, size = 138, normalized size = 2.56
method | result | size |
default | \(\frac {2 \EllipticF \left (\frac {\left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {5}}{5 d \left (3 \left (\cos ^{2}\left (d x +c \right )\right )-5 \cos \left (d x +c \right )+2\right )}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.78, size = 54, normalized size = 1.00 \begin {gather*} \frac {-i \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {4}{9}\right ) + i \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {4}{9}\right )}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\sec {\left (c + d x \right )}}}{\sqrt {3 - 2 \sec {\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}}{\sqrt {3-\frac {2}{\cos \left (c+d\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________