Integrand size = 14, antiderivative size = 46 \[ \int \frac {1}{a+b \csc ^2(c+d x)} \, dx=\frac {x}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {b}}\right )}{a \sqrt {a+b} d} \] Output:
x/a-b^(1/2)*arctan((a+b)^(1/2)*tan(d*x+c)/b^(1/2))/a/(a+b)^(1/2)/d
Time = 2.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+b \csc ^2(c+d x)} \, dx=\frac {c+d x-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {b}}\right )}{\sqrt {a+b}}}{a d} \] Input:
Integrate[(a + b*Csc[c + d*x]^2)^(-1),x]
Output:
(c + d*x - (Sqrt[b]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[b]])/Sqrt[a + b ])/(a*d)
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3042, 4615, 3042, 3660, 218}
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.
\(\displaystyle \int \frac {1}{a+b \csc ^2(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{a+b \sec \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
\(\Big \downarrow \) 4615 |
\(\displaystyle \frac {x}{a}-\frac {b \int \frac {1}{a \sin ^2(c+d x)+b}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x}{a}-\frac {b \int \frac {1}{a \sin (c+d x)^2+b}dx}{a}\) |
\(\Big \downarrow \) 3660 |
\(\displaystyle \frac {x}{a}-\frac {b \int \frac {1}{(a+b) \tan ^2(c+d x)+b}d\tan (c+d x)}{a d}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {x}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {b}}\right )}{a d \sqrt {a+b}}\) |
Input:
Int[(a + b*Csc[c + d*x]^2)^(-1),x]
Output:
x/a - (Sqrt[b]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[b]])/(a*Sqrt[a + b]* d)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[1/(a + (a + b)*ff^2*x^ 2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> Simp[x/a, x ] - Simp[b/a Int[1/(b + a*Cos[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0]
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {-\frac {b \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {b \left (a +b \right )}}\right )}{a \sqrt {b \left (a +b \right )}}+\frac {\arctan \left (\tan \left (d x +c \right )\right )}{a}}{d}\) | \(48\) |
default | \(\frac {-\frac {b \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {b \left (a +b \right )}}\right )}{a \sqrt {b \left (a +b \right )}}+\frac {\arctan \left (\tan \left (d x +c \right )\right )}{a}}{d}\) | \(48\) |
risch | \(\frac {x}{a}+\frac {\sqrt {-b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-b \left (a +b \right )}+a +2 b}{a}\right )}{2 \left (a +b \right ) d a}-\frac {\sqrt {-b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-b \left (a +b \right )}-a -2 b}{a}\right )}{2 \left (a +b \right ) d a}\) | \(114\) |
Input:
int(1/(a+b*csc(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/d*(-1/a*b/(b*(a+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(b*(a+b))^(1/2))+1/a*a rctan(tan(d*x+c)))
Time = 0.10 (sec) , antiderivative size = 260, normalized size of antiderivative = 5.65 \[ \int \frac {1}{a+b \csc ^2(c+d x)} \, dx=\left [\frac {4 \, d x + \sqrt {-\frac {b}{a + b}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + 5 \, a b + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {-\frac {b}{a + b}} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{a^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + a b\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right )}{4 \, a d}, \frac {2 \, d x + \sqrt {\frac {b}{a + b}} \arctan \left (\frac {{\left ({\left (a + 2 \, b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {\frac {b}{a + b}}}{2 \, b \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right )}{2 \, a d}\right ] \] Input:
integrate(1/(a+b*csc(d*x+c)^2),x, algorithm="fricas")
Output:
[1/4*(4*d*x + sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(d*x + c)^4 - 2*(a^2 + 5*a*b + 4*b^2)*cos(d*x + c)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-b/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(a^2*cos(d*x + c)^4 - 2*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)))/(a*d), 1/2*(2*d*x + sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(d*x + c)^2 - a - b)*sqrt(b/(a + b))/(b*cos(d*x + c)*sin(d*x + c)) ))/(a*d)]
\[ \int \frac {1}{a+b \csc ^2(c+d x)} \, dx=\int \frac {1}{a + b \csc ^{2}{\left (c + d x \right )}}\, dx \] Input:
integrate(1/(a+b*csc(d*x+c)**2),x)
Output:
Integral(1/(a + b*csc(c + d*x)**2), x)
Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+b \csc ^2(c+d x)} \, dx=-\frac {\frac {b \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} a} - \frac {d x + c}{a}}{d} \] Input:
integrate(1/(a+b*csc(d*x+c)^2),x, algorithm="maxima")
Output:
-(b*arctan((a + b)*tan(d*x + c)/sqrt((a + b)*b))/(sqrt((a + b)*b)*a) - (d* x + c)/a)/d
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (38) = 76\).
Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.76 \[ \int \frac {1}{a+b \csc ^2(c+d x)} \, dx=-\frac {\frac {{\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a b + b^{2}}}\right )\right )} b}{\sqrt {a b + b^{2}} a} - \frac {d x + c}{a}}{d} \] Input:
integrate(1/(a+b*csc(d*x+c)^2),x, algorithm="giac")
Output:
-((pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a*b + b^2)))*b/(sqrt(a*b + b^2)*a) - (d*x + c)/a)/d
Time = 14.54 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.26 \[ \int \frac {1}{a+b \csc ^2(c+d x)} \, dx=\frac {\mathrm {atan}\left (\frac {2\,a\,b^2\,\mathrm {tan}\left (c+d\,x\right )}{2\,a^2\,b+2\,a\,b^2}+\frac {2\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )}{2\,a^2\,b+2\,a\,b^2}\right )}{a\,d}+\frac {\mathrm {atanh}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\sqrt {-b\,\left (a+b\right )}}{b}\right )\,\sqrt {-b\,\left (a+b\right )}}{d\,\left (a^2+b\,a\right )} \] Input:
int(1/(a + b/sin(c + d*x)^2),x)
Output:
atan((2*a*b^2*tan(c + d*x))/(2*a*b^2 + 2*a^2*b) + (2*a^2*b*tan(c + d*x))/( 2*a*b^2 + 2*a^2*b))/(a*d) + (atanh((tan(c + d*x)*(-b*(a + b))^(1/2))/b)*(- b*(a + b))^(1/2))/(d*(a*b + a^2))
Time = 0.20 (sec) , antiderivative size = 507, normalized size of antiderivative = 11.02 \[ \int \frac {1}{a+b \csc ^2(c+d x)} \, dx=\frac {2 \sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}}\right )-2 \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}}\right ) a -2 \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}}\right ) b +\sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+\sqrt {b}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+\sqrt {b}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+\sqrt {b}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+\sqrt {b}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+\sqrt {b}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+\sqrt {b}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +2 a b d x +2 b^{2} d x}{2 a b d \left (a +b \right )} \] Input:
int(1/(a+b*csc(d*x+c)^2),x)
Output:
(2*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan( (tan((c + d*x)/2)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b))) - 2* sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((tan((c + d*x)/2)*b)/(s qrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)))*a - 2*sqrt(b)*sqrt(2*sqrt(a )*sqrt(a + b) + 2*a + b)*atan((tan((c + d*x)/2)*b)/(sqrt(b)*sqrt(2*sqrt(a) *sqrt(a + b) + 2*a + b)))*b + sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*s qrt(a + b) - 2*a - b)*log( - sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + sqrt( b)*tan((c + d*x)/2)) - sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log(sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + sqrt(b)*tan((c + d*x)/2)) + sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log( - sqrt(2*s qrt(a)*sqrt(a + b) - 2*a - b) + sqrt(b)*tan((c + d*x)/2))*a + sqrt(b)*sqrt (2*sqrt(a)*sqrt(a + b) - 2*a - b)*log( - sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + sqrt(b)*tan((c + d*x)/2))*b - sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log(sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + sqrt(b)*tan((c + d*x) /2))*a - sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log(sqrt(2*sqrt(a)* sqrt(a + b) - 2*a - b) + sqrt(b)*tan((c + d*x)/2))*b + 2*a*b*d*x + 2*b**2* d*x)/(2*a*b*d*(a + b))