Optimal. Leaf size=65 \[ -\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3661, 402, 217, 206, 377, 203} \[ -\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 217
Rule 377
Rule 402
Rule 3661
Rubi steps
\begin {align*} \int \sqrt {a+b \cot ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\left (b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\cot (x)\right )\right )+(-a+b) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right )\\ &=-\left (b \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )\right )+(-a+b) \operatorname {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )\\ &=-\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.41, size = 167, normalized size = 2.57 \[ \frac {1}{2} i \left (\sqrt {a-b} \log \left (-\frac {4 i \left (\sqrt {a-b} \sqrt {a+b \cot ^2(x)}+a-i b \cot (x)\right )}{(a-b)^{3/2} (\cot (x)+i)}\right )-\sqrt {a-b} \log \left (\frac {4 i \left (\sqrt {a-b} \sqrt {a+b \cot ^2(x)}+a+i b \cot (x)\right )}{(a-b)^{3/2} (\cot (x)-i)}\right )+2 i \sqrt {b} \log \left (\sqrt {b} \sqrt {a+b \cot ^2(x)}+b \cot (x)\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.51, size = 515, normalized size = 7.92 \[ \left [\frac {1}{2} \, \sqrt {-a + b} \log \left (-{\left (a - b\right )} \cos \left (2 \, x\right ) + \sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + b\right ) + \frac {1}{2} \, \sqrt {b} \log \left (\frac {{\left (a - 2 \, b\right )} \cos \left (2 \, x\right ) + 2 \, \sqrt {b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - a - 2 \, b}{\cos \left (2 \, x\right ) - 1}\right ), -\sqrt {a - b} \arctan \left (-\frac {\sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{{\left (a - b\right )} \cos \left (2 \, x\right ) + a - b}\right ) + \frac {1}{2} \, \sqrt {b} \log \left (\frac {{\left (a - 2 \, b\right )} \cos \left (2 \, x\right ) + 2 \, \sqrt {b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - a - 2 \, b}{\cos \left (2 \, x\right ) - 1}\right ), \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{b \cos \left (2 \, x\right ) + b}\right ) + \frac {1}{2} \, \sqrt {-a + b} \log \left (-{\left (a - b\right )} \cos \left (2 \, x\right ) + \sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + b\right ), -\sqrt {a - b} \arctan \left (-\frac {\sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{{\left (a - b\right )} \cos \left (2 \, x\right ) + a - b}\right ) + \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{b \cos \left (2 \, x\right ) + b}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.64, size = 210, normalized size = 3.23 \[ -\frac {1}{2} \, {\left (\frac {2 \, \sqrt {-a + b} b \arctan \left (\frac {{\left (\sqrt {-a + b} \cos \relax (x) - \sqrt {-a \cos \relax (x)^{2} + b \cos \relax (x)^{2} + a}\right )}^{2} + a - 2 \, b}{2 \, \sqrt {a b - b^{2}}}\right )}{\sqrt {a b - b^{2}}} + \sqrt {-a + b} \log \left ({\left (\sqrt {-a + b} \cos \relax (x) - \sqrt {-a \cos \relax (x)^{2} + b \cos \relax (x)^{2} + a}\right )}^{2}\right )\right )} \mathrm {sgn}\left (\sin \relax (x)\right ) - \frac {{\left (2 \, \sqrt {-a + b} b \arctan \left (\frac {\sqrt {-a + b} \sqrt {b}}{\sqrt {a b - b^{2}}}\right ) - \sqrt {a b - b^{2}} \sqrt {-a + b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right )\right )} \mathrm {sgn}\left (\sin \relax (x)\right )}{2 \, \sqrt {a b - b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.23, size = 137, normalized size = 2.11 \[ -\sqrt {b}\, \ln \left (\cot \relax (x ) \sqrt {b}+\sqrt {a +b \left (\cot ^{2}\relax (x )\right )}\right )+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \cot \relax (x )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}\right )}{b \left (a -b \right )}-\frac {a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \cot \relax (x )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}\right )}{b^{2} \left (a -b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {b\,{\mathrm {cot}\relax (x)}^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \cot ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________