Optimal. Leaf size=17 \[ -\sqrt {2} \sin ^{-1}\left (\frac {\cos (x)}{1+\sin (x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2860, 222}
\begin {gather*} -\sqrt {2} \text {ArcSin}\left (\frac {\cos (x)}{\sin (x)+1}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 222
Rule 2860
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\sin (x)} \sqrt {1+\sin (x)}} \, dx &=-\left (\sqrt {2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\frac {\cos (x)}{1+\sin (x)}\right )\right )\\ &=-\sqrt {2} \sin ^{-1}\left (\frac {\cos (x)}{1+\sin (x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 1.34, size = 123, normalized size = 7.24 \begin {gather*} \frac {2 \left (F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{4}\right )}}\right )\right |-1\right )-\Pi \left (1-\sqrt {2};\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{4}\right )}}\right )\right |-1\right )-\Pi \left (1+\sqrt {2};\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{4}\right )}}\right )\right |-1\right )\right ) \sec ^2\left (\frac {x}{4}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \sqrt {\sin (x)}}{\sqrt {1-\cot ^2\left (\frac {x}{4}\right )} \sqrt {1+\sin (x)} \tan ^{\frac {3}{2}}\left (\frac {x}{4}\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs.
\(2(15)=30\).
time = 0.37, size = 52, normalized size = 3.06
method | result | size |
default | \(\frac {2 \sqrt {-\frac {-1+\cos \left (x \right )}{\sin \left (x \right )}}\, \left (-1+\cos \left (x \right )-\sin \left (x \right )\right ) \left (\sqrt {\sin }\left (x \right )\right ) \arctan \left (\sqrt {-\frac {-1+\cos \left (x \right )}{\sin \left (x \right )}}\right )}{\sqrt {1+\sin \left (x \right )}\, \left (-1+\cos \left (x \right )\right )}\) | \(52\) |
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 [A]
time = 0.33, size = 28, normalized size = 1.65 \begin {gather*} 2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\sin \left (x\right ) + 1} \sqrt {\sin \left (x\right )}}{\cos \left (x\right ) + \sin \left (x\right ) + 1}\right ) \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 {\sin {\left (x \right )} + 1} \sqrt {\sin {\left (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 = -0.06 \begin {gather*} \int \frac {1}{\sqrt {\sin \left (x\right )}\,\sqrt {\sin \left (x\right )+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________