Optimal. Leaf size=79 \[ -\frac {4 \cot (c+d x) \Pi \left (\frac {5}{3};\left .\text {ArcSin}\left (\frac {\sqrt {-2-3 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |5\right ) \sqrt {-1-\sec (c+d x)} \sqrt {1-\sec (c+d x)}}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2888}
\begin {gather*} -\frac {4 \cot (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} \Pi \left (\frac {5}{3};\left .\text {ArcSin}\left (\frac {\sqrt {-3 \cos (c+d x)-2}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |5\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2888
Rubi steps
\begin {align*} \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx &=-\frac {4 \cot (c+d x) \Pi \left (\frac {5}{3};\left .\sin ^{-1}\left (\frac {\sqrt {-2-3 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |5\right ) \sqrt {-1-\sec (c+d x)} \sqrt {1-\sec (c+d x)}}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.53, size = 156, normalized size = 1.97 \begin {gather*} \frac {4 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {-\frac {(2+3 \cos (c+d x))^2}{(1+\cos (c+d x))^2}} \left (F\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {1}{5}\right )-2 \Pi \left (-1;\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {1}{5}\right )\right )}{\sqrt {5} d \sqrt {-2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)} \sqrt {\frac {-2-3 \cos (c+d x)}{1+\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. \(163\) vs.
\(2(67)=134\).
time = 0.21, size = 164, normalized size = 2.08
method | result | size |
default | \(\frac {\sqrt {10}\, \sqrt {2}\, \left (\EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {\sqrt {5}}{5}\right )-2 \EllipticPi \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, -1, \frac {\sqrt {5}}{5}\right )\right ) \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {-2-3 \cos \left (d x +c \right )}\, \sqrt {-\cos \left (d x +c \right )}}{5 d \left (3 \left (\cos ^{2}\left (d x +c \right )\right )-\cos \left (d x +c \right )-2\right ) \cos \left (d x +c \right )}\) | \(164\) |
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 {\sqrt {- \cos {\left (c + d x \right )}}}{\sqrt {- 3 \cos {\left (c + d x \right )} - 2}}\, 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 = -0.01 \begin {gather*} \int \frac {\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]
________________________________________________________________________________________