Integrand size = 13, antiderivative size = 62 \[ \int \frac {\csc ^3(x)}{a+b \csc (x)} \, dx=\frac {a \text {arctanh}(\cos (x))}{b^2}-\frac {2 a^2 \text {arctanh}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2}}-\frac {\cot (x)}{b} \]
a*arctanh(cos(x))/b^2-cot(x)/b-2*a^2*arctanh((a+b*tan(1/2*x))/(a^2-b^2)^(1 /2))/b^2/(a^2-b^2)^(1/2)
Time = 0.37 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.71 \[ \int \frac {\csc ^3(x)}{a+b \csc (x)} \, dx=\frac {\csc \left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right ) \left (2 a^2 \arctan \left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right ) \sin (x)+\sqrt {-a^2+b^2} \left (-b \cos (x)+a \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sin (x)\right )\right )}{2 b^2 \sqrt {-a^2+b^2}} \]
(Csc[x/2]*Sec[x/2]*(2*a^2*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]]*Sin[x] + Sqrt[-a^2 + b^2]*(-(b*Cos[x]) + a*(Log[Cos[x/2]] - Log[Sin[x/2]])*Sin[x ])))/(2*b^2*Sqrt[-a^2 + b^2])
Time = 0.49 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.23, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {3042, 4277, 3042, 4276, 3042, 4257, 4318, 3042, 3139, 1083, 219}
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.
\(\displaystyle \int \frac {\csc ^3(x)}{a+b \csc (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc (x)^3}{a+b \csc (x)}dx\) |
\(\Big \downarrow \) 4277 |
\(\displaystyle -\frac {a \int \frac {\csc ^2(x)}{a+b \csc (x)}dx}{b}-\frac {\cot (x)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {\csc (x)^2}{a+b \csc (x)}dx}{b}-\frac {\cot (x)}{b}\) |
\(\Big \downarrow \) 4276 |
\(\displaystyle -\frac {a \left (\frac {\int \csc (x)dx}{b}-\frac {a \int \frac {\csc (x)}{a+b \csc (x)}dx}{b}\right )}{b}-\frac {\cot (x)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \left (\frac {\int \csc (x)dx}{b}-\frac {a \int \frac {\csc (x)}{a+b \csc (x)}dx}{b}\right )}{b}-\frac {\cot (x)}{b}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -\frac {a \left (-\frac {a \int \frac {\csc (x)}{a+b \csc (x)}dx}{b}-\frac {\text {arctanh}(\cos (x))}{b}\right )}{b}-\frac {\cot (x)}{b}\) |
\(\Big \downarrow \) 4318 |
\(\displaystyle -\frac {a \left (-\frac {a \int \frac {1}{\frac {a \sin (x)}{b}+1}dx}{b^2}-\frac {\text {arctanh}(\cos (x))}{b}\right )}{b}-\frac {\cot (x)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \left (-\frac {a \int \frac {1}{\frac {a \sin (x)}{b}+1}dx}{b^2}-\frac {\text {arctanh}(\cos (x))}{b}\right )}{b}-\frac {\cot (x)}{b}\) |
\(\Big \downarrow \) 3139 |
\(\displaystyle -\frac {a \left (-\frac {2 a \int \frac {1}{\tan ^2\left (\frac {x}{2}\right )+\frac {2 a \tan \left (\frac {x}{2}\right )}{b}+1}d\tan \left (\frac {x}{2}\right )}{b^2}-\frac {\text {arctanh}(\cos (x))}{b}\right )}{b}-\frac {\cot (x)}{b}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle -\frac {a \left (\frac {4 a \int \frac {1}{-\left (\frac {2 a}{b}+2 \tan \left (\frac {x}{2}\right )\right )^2-4 \left (1-\frac {a^2}{b^2}\right )}d\left (\frac {2 a}{b}+2 \tan \left (\frac {x}{2}\right )\right )}{b^2}-\frac {\text {arctanh}(\cos (x))}{b}\right )}{b}-\frac {\cot (x)}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {a \left (\frac {2 a \text {arctanh}\left (\frac {b \left (\frac {2 a}{b}+2 \tan \left (\frac {x}{2}\right )\right )}{2 \sqrt {a^2-b^2}}\right )}{b \sqrt {a^2-b^2}}-\frac {\text {arctanh}(\cos (x))}{b}\right )}{b}-\frac {\cot (x)}{b}\) |
-((a*(-(ArcTanh[Cos[x]]/b) + (2*a*ArcTanh[(b*((2*a)/b + 2*Tan[x/2]))/(2*Sq rt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2])))/b) - Cot[x]/b
3.1.41.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]^2/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Sym bol] :> Simp[1/b Int[Csc[e + f*x], x], x] - Simp[a/b Int[Csc[e + f*x]/( a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]
Int[csc[(e_.) + (f_.)*(x_)]^3/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Sym bol] :> Simp[-Cot[e + f*x]/(b*f), x] - Simp[a/b Int[Csc[e + f*x]^2/(a + b *Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Time = 0.47 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {\tan \left (\frac {x}{2}\right )}{2 b}+\frac {2 a^{2} \arctan \left (\frac {2 b \tan \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{b^{2} \sqrt {-a^{2}+b^{2}}}-\frac {1}{2 b \tan \left (\frac {x}{2}\right )}-\frac {a \ln \left (\tan \left (\frac {x}{2}\right )\right )}{b^{2}}\) | \(77\) |
risch | \(-\frac {2 i}{b \left ({\mathrm e}^{2 i x}-1\right )}+\frac {a \ln \left ({\mathrm e}^{i x}+1\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{i x}-1\right )}{b^{2}}+\frac {i a^{2} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {-a^{2}+b^{2}}\, b -a^{2}+b^{2}\right )}{a \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, b^{2}}-\frac {i a^{2} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {-a^{2}+b^{2}}\, b +a^{2}-b^{2}\right )}{a \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, b^{2}}\) | \(176\) |
1/2*tan(1/2*x)/b+2*a^2/b^2/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*tan(1/2*x)+2*a )/(-a^2+b^2)^(1/2))-1/2/b/tan(1/2*x)-a/b^2*ln(tan(1/2*x))
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (56) = 112\).
Time = 0.30 (sec) , antiderivative size = 308, normalized size of antiderivative = 4.97 \[ \int \frac {\csc ^3(x)}{a+b \csc (x)} \, dx=\left [\frac {\sqrt {a^{2} - b^{2}} a^{2} \log \left (-\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (x\right )^{2} + 2 \, a b \sin \left (x\right ) + a^{2} + b^{2} - 2 \, {\left (b \cos \left (x\right ) \sin \left (x\right ) + a \cos \left (x\right )\right )} \sqrt {a^{2} - b^{2}}}{a^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) \sin \left (x\right ) + {\left (a^{3} - a b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \sin \left (x\right ) - {\left (a^{3} - a b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \sin \left (x\right ) - 2 \, {\left (a^{2} b - b^{3}\right )} \cos \left (x\right )}{2 \, {\left (a^{2} b^{2} - b^{4}\right )} \sin \left (x\right )}, -\frac {2 \, \sqrt {-a^{2} + b^{2}} a^{2} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \sin \left (x\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (x\right )}\right ) \sin \left (x\right ) - {\left (a^{3} - a b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \sin \left (x\right ) + {\left (a^{3} - a b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \sin \left (x\right ) + 2 \, {\left (a^{2} b - b^{3}\right )} \cos \left (x\right )}{2 \, {\left (a^{2} b^{2} - b^{4}\right )} \sin \left (x\right )}\right ] \]
[1/2*(sqrt(a^2 - b^2)*a^2*log(-((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^ 2 + b^2 - 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2))*sin(x) + (a^3 - a*b^2)*log(1/2*cos(x) + 1/2)*si n(x) - (a^3 - a*b^2)*log(-1/2*cos(x) + 1/2)*sin(x) - 2*(a^2*b - b^3)*cos(x ))/((a^2*b^2 - b^4)*sin(x)), -1/2*(2*sqrt(-a^2 + b^2)*a^2*arctan(-sqrt(-a^ 2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x)))*sin(x) - (a^3 - a*b^2)*log(1 /2*cos(x) + 1/2)*sin(x) + (a^3 - a*b^2)*log(-1/2*cos(x) + 1/2)*sin(x) + 2* (a^2*b - b^3)*cos(x))/((a^2*b^2 - b^4)*sin(x))]
\[ \int \frac {\csc ^3(x)}{a+b \csc (x)} \, dx=\int \frac {\csc ^{3}{\left (x \right )}}{a + b \csc {\left (x \right )}}\, dx \]
Exception generated. \[ \int \frac {\csc ^3(x)}{a+b \csc (x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.58 \[ \int \frac {\csc ^3(x)}{a+b \csc (x)} \, dx=\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, x\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )\right )} a^{2}}{\sqrt {-a^{2} + b^{2}} b^{2}} - \frac {a \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{b^{2}} + \frac {\tan \left (\frac {1}{2} \, x\right )}{2 \, b} + \frac {2 \, a \tan \left (\frac {1}{2} \, x\right ) - b}{2 \, b^{2} \tan \left (\frac {1}{2} \, x\right )} \]
2*(pi*floor(1/2*x/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*x) + a)/sqrt(-a^2 + b^2)))*a^2/(sqrt(-a^2 + b^2)*b^2) - a*log(abs(tan(1/2*x)))/b^2 + 1/2*tan( 1/2*x)/b + 1/2*(2*a*tan(1/2*x) - b)/(b^2*tan(1/2*x))
Time = 18.11 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.18 \[ \int \frac {\csc ^3(x)}{a+b \csc (x)} \, dx=-\frac {1}{b\,\mathrm {tan}\left (x\right )}-\frac {a\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{b^2}-\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {a^2-b^2}\,4{}\mathrm {i}-b^2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {a^2-b^2}\,1{}\mathrm {i}+a\,b\,\sqrt {a^2-b^2}\,2{}\mathrm {i}}{4\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^3+2\,a^2\,b-3\,\mathrm {tan}\left (\frac {x}{2}\right )\,a\,b^2-b^3}\right )\,2{}\mathrm {i}}{b^2\,\sqrt {a^2-b^2}} \]