Optimal. Leaf size=68 \[ \frac {\sqrt {2} (A-B) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {2} \sqrt {a \cos (x)+a}}\right )}{\sqrt {a}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {a \cos (x)+a}}\right )}{\sqrt {a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2828, 2985, 2649, 206, 2773} \[ \frac {\sqrt {2} (A-B) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {2} \sqrt {a \cos (x)+a}}\right )}{\sqrt {a}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {a \cos (x)+a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 2649
Rule 2773
Rule 2828
Rule 2985
Rubi steps
\begin {align*} \int \frac {A+B \sec (x)}{\sqrt {a+a \cos (x)}} \, dx &=\int \frac {(B+A \cos (x)) \sec (x)}{\sqrt {a+a \cos (x)}} \, dx\\ &=\frac {B \int \sqrt {a+a \cos (x)} \sec (x) \, dx}{a}-(-A+B) \int \frac {1}{\sqrt {a+a \cos (x)}} \, dx\\ &=-\left ((2 (A-B)) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {a \sin (x)}{\sqrt {a+a \cos (x)}}\right )\right )-(2 B) \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,-\frac {a \sin (x)}{\sqrt {a+a \cos (x)}}\right )\\ &=\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {a+a \cos (x)}}\right )}{\sqrt {a}}+\frac {\sqrt {2} (A-B) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {2} \sqrt {a+a \cos (x)}}\right )}{\sqrt {a}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 52, normalized size = 0.76 \[ \frac {2 \cos \left (\frac {x}{2}\right ) \left ((A-B) \tanh ^{-1}\left (\sin \left (\frac {x}{2}\right )\right )+\sqrt {2} B \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {x}{2}\right )\right )\right )}{\sqrt {a (\cos (x)+1)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.61, size = 116, normalized size = 1.71 \[ -\frac {\sqrt {2} {\left (A - B\right )} \sqrt {a} \log \left (-\frac {\cos \relax (x)^{2} + \frac {2 \, \sqrt {2} \sqrt {a \cos \relax (x) + a} \sin \relax (x)}{\sqrt {a}} - 2 \, \cos \relax (x) - 3}{\cos \relax (x)^{2} + 2 \, \cos \relax (x) + 1}\right ) - B \sqrt {a} \log \left (\frac {a \cos \relax (x)^{3} - 7 \, a \cos \relax (x)^{2} - 4 \, \sqrt {a \cos \relax (x) + a} \sqrt {a} {\left (\cos \relax (x) - 2\right )} \sin \relax (x) + 8 \, a}{\cos \relax (x)^{3} + \cos \relax (x)^{2}}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.70, size = 133, normalized size = 1.96 \[ -\frac {\sqrt {2} {\left (A \sqrt {a} - B \sqrt {a}\right )} \log \left ({\left (\sqrt {a} \tan \left (\frac {1}{2} \, x\right ) - \sqrt {a \tan \left (\frac {1}{2} \, x\right )^{2} + a}\right )}^{2}\right )}{2 \, a} + \frac {B \log \left ({\left | {\left (\sqrt {a} \tan \left (\frac {1}{2} \, x\right ) - \sqrt {a \tan \left (\frac {1}{2} \, x\right )^{2} + a}\right )}^{2} - a {\left (2 \, \sqrt {2} + 3\right )} \right |}\right )}{\sqrt {a}} - \frac {B \log \left ({\left | {\left (\sqrt {a} \tan \left (\frac {1}{2} \, x\right ) - \sqrt {a \tan \left (\frac {1}{2} \, x\right )^{2} + a}\right )}^{2} + a {\left (2 \, \sqrt {2} - 3\right )} \right |}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.35, size = 192, normalized size = 2.82 \[ \frac {\cos \left (\frac {x}{2}\right ) \sqrt {a \left (\sin ^{2}\left (\frac {x}{2}\right )\right )}\, \left (\sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {x}{2}\right )\right )}+4 a}{\cos \left (\frac {x}{2}\right )}\right ) A -\sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {x}{2}\right )\right )}+4 a}{\cos \left (\frac {x}{2}\right )}\right ) B +B \ln \left (\frac {4 a \sqrt {2}\, \cos \left (\frac {x}{2}\right )+4 \sqrt {a}\, \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {x}{2}\right )\right )}+8 a}{2 \cos \left (\frac {x}{2}\right )+\sqrt {2}}\right )+B \ln \left (-\frac {4 \left (\sqrt {a}\, \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {x}{2}\right )\right )}-a \sqrt {2}\, \cos \left (\frac {x}{2}\right )+2 a \right )}{-2 \cos \left (\frac {x}{2}\right )+\sqrt {2}}\right )\right )}{\sqrt {a}\, \sin \left (\frac {x}{2}\right ) \sqrt {\left (\cos ^{2}\left (\frac {x}{2}\right )\right ) a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.05, size = 58, normalized size = 0.85 \[ \frac {{\left (\sqrt {2} \log \left (\cos \left (\frac {1}{2} \, x\right )^{2} + \sin \left (\frac {1}{2} \, x\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, x\right ) + 1\right ) - \sqrt {2} \log \left (\cos \left (\frac {1}{2} \, x\right )^{2} + \sin \left (\frac {1}{2} \, x\right )^{2} - 2 \, \sin \left (\frac {1}{2} \, x\right ) + 1\right )\right )} A}{2 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {A+\frac {B}{\cos \relax (x)}}{\sqrt {a+a\,\cos \relax (x)}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {A + B \sec {\relax (x )}}{\sqrt {a \left (\cos {\relax (x )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________