Optimal. Leaf size=49 \[ -\frac {2 \sqrt {-\cos (c+d x)} F\left (\left .\text {ArcSin}\left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right )\right |5\right )}{d \sqrt {\cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 49, 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 = {2893, 2892}
\begin {gather*} -\frac {2 \sqrt {-\cos (c+d x)} F\left (\left .\text {ArcSin}\left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right )\right |5\right )}{d \sqrt {\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2892
Rule 2893
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx &=\frac {\sqrt {-\cos (c+d x)} \int \frac {1}{\sqrt {-2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx}{\sqrt {\cos (c+d x)}}\\ &=-\frac {2 \sqrt {-\cos (c+d x)} F\left (\left .\sin ^{-1}\left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right )\right |5\right )}{d \sqrt {\cos (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(153\) vs. \(2(49)=98\).
time = 1.59, size = 153, normalized size = 3.12 \begin {gather*} \frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) F\left (\text {ArcSin}\left (\sqrt {\frac {5}{2}} \sqrt {\frac {\cos (c+d x)}{-1+\cos (c+d x)}}\right )|\frac {4}{5}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{\sqrt {5} d \sqrt {-2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(131\) vs.
\(2(46)=92\).
time = 0.64, size = 132, normalized size = 2.69
method | result | size |
default | \(\frac {\sqrt {2}\, \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {-2-3 \cos \left (d x +c \right )}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \left (\sin ^{4}\left (d x +c \right )\right ) \sqrt {5}}{5 d \cos \left (d x +c \right )^{\frac {3}{2}} \left (2+3 \cos \left (d x +c \right )\right ) \left (-1+\cos \left (d x +c \right )\right )^{2}}\) | \(132\) |
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 [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- 3 \cos {\left (c + d x \right )} - 2} \sqrt {\cos {\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 {1}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {-3\,\cos \left (c+d\,x\right )-2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________