Integrand size = 10, antiderivative size = 62 \[ \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right )}{\sqrt {a}}+\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a+a \csc (x)}}\right )}{\sqrt {a}} \]
-2*arctan(cot(x)*a^(1/2)/(a+a*csc(x))^(1/2))/a^(1/2)+arctan(1/2*cot(x)*a^( 1/2)*2^(1/2)/(a+a*csc(x))^(1/2))*2^(1/2)/a^(1/2)
Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx=\frac {\left (-2 \arctan \left (\sqrt {-1+\csc (x)}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {-1+\csc (x)}}{\sqrt {2}}\right )\right ) \cot (x)}{\sqrt {-1+\csc (x)} \sqrt {a (1+\csc (x))}} \]
((-2*ArcTan[Sqrt[-1 + Csc[x]]] + Sqrt[2]*ArcTan[Sqrt[-1 + Csc[x]]/Sqrt[2]] )*Cot[x])/(Sqrt[-1 + Csc[x]]*Sqrt[a*(1 + Csc[x])])
Time = 0.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3042, 4263, 3042, 4261, 216, 4282, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\sqrt {a \csc (x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {a \csc (x)+a}}dx\) |
\(\Big \downarrow \) 4263 |
\(\displaystyle \frac {\int \sqrt {\csc (x) a+a}dx}{a}-\int \frac {\csc (x)}{\sqrt {\csc (x) a+a}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {\csc (x) a+a}dx}{a}-\int \frac {\csc (x)}{\sqrt {\csc (x) a+a}}dx\) |
\(\Big \downarrow \) 4261 |
\(\displaystyle -2 \int \frac {1}{\frac {a^2 \cot ^2(x)}{\csc (x) a+a}+a}d\frac {a \cot (x)}{\sqrt {\csc (x) a+a}}-\int \frac {\csc (x)}{\sqrt {\csc (x) a+a}}dx\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\int \frac {\csc (x)}{\sqrt {\csc (x) a+a}}dx-\frac {2 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{\sqrt {a}}\) |
\(\Big \downarrow \) 4282 |
\(\displaystyle 2 \int \frac {1}{\frac {a^2 \cot ^2(x)}{\csc (x) a+a}+2 a}d\frac {a \cot (x)}{\sqrt {\csc (x) a+a}}-\frac {2 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{\sqrt {a}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a \csc (x)+a}}\right )}{\sqrt {a}}-\frac {2 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{\sqrt {a}}\) |
(-2*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]])/Sqrt[a] + (Sqrt[2]*ArcTan [(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a + a*Csc[x]])])/Sqrt[a]
3.1.16.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[1/a I nt[Sqrt[a + b*Csc[c + d*x]], x], x] - Simp[b/a Int[Csc[c + d*x]/Sqrt[a + b*Csc[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2/f Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(184\) vs. \(2(47)=94\).
Time = 0.53 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.98
method | result | size |
default | \(\frac {\left (\sqrt {2}\, \ln \left (\frac {\csc \left (x \right )-\cot \left (x \right )+\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1}{-\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+\csc \left (x \right )-\cot \left (x \right )+1}\right )+4 \sqrt {2}\, \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1\right )+4 \sqrt {2}\, \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}-1\right )+\sqrt {2}\, \ln \left (\frac {-\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+\csc \left (x \right )-\cot \left (x \right )+1}{\csc \left (x \right )-\cot \left (x \right )+\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1}\right )-8 \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\right )\right ) \left (\csc \left (x \right )-\cot \left (x \right )+1\right )}{4 \sqrt {a \left (\csc \left (x \right )+1\right )}\, \sqrt {\csc \left (x \right )-\cot \left (x \right )}}\) | \(185\) |
1/4*(2^(1/2)*ln((csc(x)-cot(x)+(csc(x)-cot(x))^(1/2)*2^(1/2)+1)/(-(csc(x)- cot(x))^(1/2)*2^(1/2)+csc(x)-cot(x)+1))+4*2^(1/2)*arctan((csc(x)-cot(x))^( 1/2)*2^(1/2)+1)+4*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)-1)+2^(1/2)* ln((-(csc(x)-cot(x))^(1/2)*2^(1/2)+csc(x)-cot(x)+1)/(csc(x)-cot(x)+(csc(x) -cot(x))^(1/2)*2^(1/2)+1))-8*arctan((csc(x)-cot(x))^(1/2)))/(a*(csc(x)+1)) ^(1/2)/(csc(x)-cot(x))^(1/2)*(csc(x)-cot(x)+1)
Time = 0.26 (sec) , antiderivative size = 219, normalized size of antiderivative = 3.53 \[ \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx=\left [\frac {\sqrt {2} a \sqrt {-\frac {1}{a}} \log \left (\frac {\sqrt {2} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} \sqrt {-\frac {1}{a}} \sin \left (x\right ) + \cos \left (x\right )}{\sin \left (x\right ) + 1}\right ) - \sqrt {-a} \log \left (\frac {2 \, a \cos \left (x\right )^{2} + 2 \, {\left (\cos \left (x\right )^{2} + {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} + a \cos \left (x\right ) - {\left (2 \, a \cos \left (x\right ) + a\right )} \sin \left (x\right ) - a}{\cos \left (x\right ) + \sin \left (x\right ) + 1}\right )}{a}, -\frac {2 \, {\left (\sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} \sin \left (x\right )}{\sqrt {a} {\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )}}\right ) - \sqrt {a} \arctan \left (-\frac {\sqrt {a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{a \cos \left (x\right ) + a \sin \left (x\right ) + a}\right )\right )}}{a}\right ] \]
[(sqrt(2)*a*sqrt(-1/a)*log((sqrt(2)*sqrt((a*sin(x) + a)/sin(x))*sqrt(-1/a) *sin(x) + cos(x))/(sin(x) + 1)) - sqrt(-a)*log((2*a*cos(x)^2 + 2*(cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x)) + a*cos(x ) - (2*a*cos(x) + a)*sin(x) - a)/(cos(x) + sin(x) + 1)))/a, -2*(sqrt(2)*sq rt(a)*arctan(sqrt(2)*sqrt((a*sin(x) + a)/sin(x))*sin(x)/(sqrt(a)*(cos(x) + sin(x) + 1))) - sqrt(a)*arctan(-sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos( x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a)))/a]
\[ \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx=\int \frac {1}{\sqrt {a \csc {\left (x \right )} + a}}\, dx \]
Time = 0.33 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.34 \[ \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx=\frac {\sqrt {2} {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}\right )\right )}}{\sqrt {a}} - \frac {2 \, \sqrt {2} \arctan \left (\sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}{\sqrt {a}} \]
sqrt(2)*(sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(sin(x)/(cos(x) + 1)) )) + sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(sin(x)/(cos(x) + 1))))) /sqrt(a) - 2*sqrt(2)*arctan(sqrt(sin(x)/(cos(x) + 1)))/sqrt(a)
Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (47) = 94\).
Time = 0.43 (sec) , antiderivative size = 205, normalized size of antiderivative = 3.31 \[ \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx=-\frac {4 \, \sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, x\right )}}{\sqrt {a}}\right ) - \frac {2 \, {\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} + 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right )}{a} - \frac {2 \, {\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} - 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right )}{a} - \frac {{\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) + \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right )}{a} + \frac {{\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) - \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right )}{a}}{2 \, a} \]
-1/2*(4*sqrt(2)*sqrt(a)*arctan(sqrt(a*tan(1/2*x))/sqrt(a)) - 2*(a*sqrt(abs (a)) + abs(a)^(3/2))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) + 2*sqrt(a*t an(1/2*x)))/sqrt(abs(a)))/a - 2*(a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(-1/ 2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) - 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a)))/a - (a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1/2*x) + sqrt(2)*sqrt(a*tan(1/2* x))*sqrt(abs(a)) + abs(a))/a + (a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1 /2*x) - sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))/a)/a
Timed out. \[ \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx=\int \frac {1}{\sqrt {a+\frac {a}{\sin \left (x\right )}}} \,d x \]