Integrand size = 24, antiderivative size = 40 \[ \int \csc (c+d x) \sqrt {-a-a \sin (c+d x)} \, dx=\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {-a-a \sin (c+d x)}}\right )}{d} \] Output:
2*a^(1/2)*arctan(a^(1/2)*cos(d*x+c)/(-a-a*sin(d*x+c))^(1/2))/d
Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.68 \[ \int \csc (c+d x) \sqrt {-a-a \sin (c+d x)} \, dx=\frac {\left (\coth ^{-1}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\text {arctanh}\left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sqrt {-a (1+\sin (c+d x))}}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \] Input:
Integrate[Csc[c + d*x]*Sqrt[-a - a*Sin[c + d*x]],x]
Output:
((ArcCoth[Sin[(c + d*x)/2]] - ArcTanh[Cos[(c + d*x)/2]])*Sqrt[-(a*(1 + Sin [c + d*x]))])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))
Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3042, 3252, 217}
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.
\(\displaystyle \int \csc (c+d x) \sqrt {a (-\sin (c+d x))-a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a (-\sin (c+d x))-a}}{\sin (c+d x)}dx\) |
\(\Big \downarrow \) 3252 |
\(\displaystyle \frac {2 a \int \frac {1}{-\frac {a^2 \cos ^2(c+d x)}{-\sin (c+d x) a-a}-a}d\left (-\frac {a \cos (c+d x)}{\sqrt {-\sin (c+d x) a-a}}\right )}{d}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a (-\sin (c+d x))-a}}\right )}{d}\) |
Input:
Int[Csc[c + d*x]*Sqrt[-a - a*Sin[c + d*x]],x]
Output:
(2*Sqrt[a]*ArcTan[(Sqrt[a]*Cos[c + d*x])/Sqrt[-a - a*Sin[c + d*x]]])/d
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Time = 0.34 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.72
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {a \left (\sin \left (d x +c \right )-1\right )}\, \sqrt {a}\, \arctan \left (\frac {\sqrt {a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right )}{\cos \left (d x +c \right ) \sqrt {-a \sin \left (d x +c \right )-a}\, d}\) | \(69\) |
Input:
int(csc(d*x+c)*(-a*sin(d*x+c)-a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2*(1+sin(d*x+c))*(a*(sin(d*x+c)-1))^(1/2)*a^(1/2)*arctan((a*(sin(d*x+c)-1 ))^(1/2)/a^(1/2))/cos(d*x+c)/(-a*sin(d*x+c)-a)^(1/2)/d
Time = 0.09 (sec) , antiderivative size = 221, normalized size of antiderivative = 5.52 \[ \int \csc (c+d x) \sqrt {-a-a \sin (c+d x)} \, dx=\left [\frac {\sqrt {-a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {-a \sin \left (d x + c\right ) - a} \sqrt {-a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right )}{2 \, d}, \frac {\sqrt {a} \arctan \left (\frac {\sqrt {-a \sin \left (d x + c\right ) - a} {\left (\sin \left (d x + c\right ) - 2\right )}}{2 \, \sqrt {a} \cos \left (d x + c\right )}\right )}{d}\right ] \] Input:
integrate(csc(d*x+c)*(-a-a*sin(d*x+c))^(1/2),x, algorithm="fricas")
Output:
[1/2*sqrt(-a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 + 4*(cos(d*x + c) ^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(-a*sin(d*x + c) - a)*sqrt(-a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c )^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1))/d, sqrt(a)*arctan(1/2*sqrt(-a*s in(d*x + c) - a)*(sin(d*x + c) - 2)/(sqrt(a)*cos(d*x + c)))/d]
\[ \int \csc (c+d x) \sqrt {-a-a \sin (c+d x)} \, dx=\int \sqrt {- a \left (\sin {\left (c + d x \right )} + 1\right )} \csc {\left (c + d x \right )}\, dx \] Input:
integrate(csc(d*x+c)*(-a-a*sin(d*x+c))**(1/2),x)
Output:
Integral(sqrt(-a*(sin(c + d*x) + 1))*csc(c + d*x), x)
\[ \int \csc (c+d x) \sqrt {-a-a \sin (c+d x)} \, dx=\int { \sqrt {-a \sin \left (d x + c\right ) - a} \csc \left (d x + c\right ) \,d x } \] Input:
integrate(csc(d*x+c)*(-a-a*sin(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-a*sin(d*x + c) - a)*csc(d*x + c), x)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (34) = 68\).
Time = 0.15 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.72 \[ \int \csc (c+d x) \sqrt {-a-a \sin (c+d x)} \, dx=-\frac {\sqrt {-a} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \] Input:
integrate(csc(d*x+c)*(-a-a*sin(d*x+c))^(1/2),x, algorithm="giac")
Output:
-sqrt(-a)*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c))/abs(2*sqr t(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c)))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2* c))/d
Timed out. \[ \int \csc (c+d x) \sqrt {-a-a \sin (c+d x)} \, dx=\int \frac {\sqrt {-a-a\,\sin \left (c+d\,x\right )}}{\sin \left (c+d\,x\right )} \,d x \] Input:
int((- a - a*sin(c + d*x))^(1/2)/sin(c + d*x),x)
Output:
int((- a - a*sin(c + d*x))^(1/2)/sin(c + d*x), x)
\[ \int \csc (c+d x) \sqrt {-a-a \sin (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \csc \left (d x +c \right )d x \right ) i \] Input:
int(csc(d*x+c)*(-a-a*sin(d*x+c))^(1/2),x)
Output:
sqrt(a)*int(sqrt(sin(c + d*x) + 1)*csc(c + d*x),x)*i