\(\int \frac {x}{(a+b \csc (c+d x^2))^2} \, dx\) [27]

Optimal result

Integrand size = 16, antiderivative size = 120 \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\frac {x^2}{2 a^2}+\frac {b \left (2 a^2-b^2\right ) \text {arctanh}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2} d}-\frac {b^2 \cot \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d x^2\right )\right )} \] Output:

1/2*x^2/a^2+b*(2*a^2-b^2)*arctanh((a+b*tan(1/2*d*x^2+1/2*c))/(a^2-b^2)^(1/ 
2))/a^2/(a^2-b^2)^(3/2)/d-1/2*b^2*cot(d*x^2+c)/a/(a^2-b^2)/d/(a+b*csc(d*x^ 
2+c))
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.32 \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\frac {\csc \left (c+d x^2\right ) \left (\frac {a b^2 \cot \left (c+d x^2\right )}{(-a+b) (a+b)}+\left (c+d x^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )-\frac {2 b \left (-2 a^2+b^2\right ) \arctan \left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {-a^2+b^2}}\right ) \left (a+b \csc \left (c+d x^2\right )\right )}{\left (-a^2+b^2\right )^{3/2}}\right ) \left (b+a \sin \left (c+d x^2\right )\right )}{2 a^2 d \left (a+b \csc \left (c+d x^2\right )\right )^2} \] Input:

Integrate[x/(a + b*Csc[c + d*x^2])^2,x]
 

Output:

(Csc[c + d*x^2]*((a*b^2*Cot[c + d*x^2])/((-a + b)*(a + b)) + (c + d*x^2)*( 
a + b*Csc[c + d*x^2]) - (2*b*(-2*a^2 + b^2)*ArcTan[(a + b*Tan[(c + d*x^2)/ 
2])/Sqrt[-a^2 + b^2]]*(a + b*Csc[c + d*x^2]))/(-a^2 + b^2)^(3/2))*(b + a*S 
in[c + d*x^2]))/(2*a^2*d*(a + b*Csc[c + d*x^2])^2)
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.28, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4693, 3042, 4272, 25, 3042, 4407, 3042, 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.

\(\displaystyle \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx\)

\(\Big \downarrow \) 4693

\(\displaystyle \frac {1}{2} \int \frac {1}{\left (a+b \csc \left (d x^2+c\right )\right )^2}dx^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \int \frac {1}{\left (a+b \csc \left (d x^2+c\right )\right )^2}dx^2\)

\(\Big \downarrow \) 4272

\(\displaystyle \frac {1}{2} \left (-\frac {\int -\frac {a^2-b \csc \left (d x^2+c\right ) a-b^2}{a+b \csc \left (d x^2+c\right )}dx^2}{a \left (a^2-b^2\right )}-\frac {b^2 \cot \left (c+d x^2\right )}{a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {a^2-b \csc \left (d x^2+c\right ) a-b^2}{a+b \csc \left (d x^2+c\right )}dx^2}{a \left (a^2-b^2\right )}-\frac {b^2 \cot \left (c+d x^2\right )}{a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {a^2-b \csc \left (d x^2+c\right ) a-b^2}{a+b \csc \left (d x^2+c\right )}dx^2}{a \left (a^2-b^2\right )}-\frac {b^2 \cot \left (c+d x^2\right )}{a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 4407

\(\displaystyle \frac {1}{2} \left (\frac {\frac {x^2 \left (a^2-b^2\right )}{a}-\frac {b \left (2 a^2-b^2\right ) \int \frac {\csc \left (d x^2+c\right )}{a+b \csc \left (d x^2+c\right )}dx^2}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot \left (c+d x^2\right )}{a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {\frac {x^2 \left (a^2-b^2\right )}{a}-\frac {b \left (2 a^2-b^2\right ) \int \frac {\csc \left (d x^2+c\right )}{a+b \csc \left (d x^2+c\right )}dx^2}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot \left (c+d x^2\right )}{a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 4318

\(\displaystyle \frac {1}{2} \left (\frac {\frac {x^2 \left (a^2-b^2\right )}{a}-\frac {\left (2 a^2-b^2\right ) \int \frac {1}{\frac {a \sin \left (d x^2+c\right )}{b}+1}dx^2}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot \left (c+d x^2\right )}{a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {\frac {x^2 \left (a^2-b^2\right )}{a}-\frac {\left (2 a^2-b^2\right ) \int \frac {1}{\frac {a \sin \left (d x^2+c\right )}{b}+1}dx^2}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot \left (c+d x^2\right )}{a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 3139

\(\displaystyle \frac {1}{2} \left (\frac {\frac {x^2 \left (a^2-b^2\right )}{a}-\frac {2 \left (2 a^2-b^2\right ) \int \frac {1}{x^4+\frac {2 a \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )}{b}+1}d\tan \left (\frac {1}{2} \left (d x^2+c\right )\right )}{a d}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot \left (c+d x^2\right )}{a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{2} \left (\frac {\frac {4 \left (2 a^2-b^2\right ) \int \frac {1}{-x^4-4 \left (1-\frac {a^2}{b^2}\right )}d\left (\frac {2 a}{b}+2 \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )\right )}{a d}+\frac {x^2 \left (a^2-b^2\right )}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot \left (c+d x^2\right )}{a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {\frac {2 b \left (2 a^2-b^2\right ) \text {arctanh}\left (\frac {b \left (\frac {2 a}{b}+2 \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}+\frac {x^2 \left (a^2-b^2\right )}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot \left (c+d x^2\right )}{a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )}\right )\)

Input:

Int[x/(a + b*Csc[c + d*x^2])^2,x]
 

Output:

((((a^2 - b^2)*x^2)/a + (2*b*(2*a^2 - b^2)*ArcTanh[(b*((2*a)/b + 2*Tan[(c 
+ d*x^2)/2]))/(2*Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d))/(a*(a^2 - b^2)) 
 - (b^2*Cot[c + d*x^2])/(a*(a^2 - b^2)*d*(a + b*Csc[c + d*x^2])))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 219

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])
 

rule 1083

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3139

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]
 

rule 4272

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[ 
c + d*x]*((a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Sim 
p[1/(a*(n + 1)*(a^2 - b^2))   Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^2 - 
b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x 
], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ 
erQ[2*n]
 

rule 4318

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]
 

rule 4407

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
 (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a   Int[Csc[e + f* 
x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c 
- a*d, 0]
 

rule 4693

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^ 
p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 
 1)/n], 0] && IntegerQ[p]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.49

method	result	size
derivativedivides	\(\frac {-\frac {2 b \left (\frac {\frac {\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right ) a^{2}}{2 a^{2}-2 b^{2}}+\frac {b a}{2 a^{2}-2 b^{2}}}{\frac {\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )^{2} b}{2}+a \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+\frac {b}{2}}+\frac {2 \left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {2 b \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (2 a^{2}-2 b^{2}\right ) \sqrt {-a^{2}+b^{2}}}\right )}{a^{2}}+\frac {2 \arctan \left (\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{2 d}\)	\(179\)
default	\(\frac {-\frac {2 b \left (\frac {\frac {\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right ) a^{2}}{2 a^{2}-2 b^{2}}+\frac {b a}{2 a^{2}-2 b^{2}}}{\frac {\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )^{2} b}{2}+a \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+\frac {b}{2}}+\frac {2 \left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {2 b \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (2 a^{2}-2 b^{2}\right ) \sqrt {-a^{2}+b^{2}}}\right )}{a^{2}}+\frac {2 \arctan \left (\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{2 d}\)	\(179\)
risch	\(\frac {x^{2}}{2 a^{2}}-\frac {i b^{2} \left (i a +b \,{\mathrm e}^{i \left (d \,x^{2}+c \right )}\right )}{a^{2} \left (-a^{2}+b^{2}\right ) d \left (2 b \,{\mathrm e}^{i \left (d \,x^{2}+c \right )}-i a \,{\mathrm e}^{2 i \left (d \,x^{2}+c \right )}+i a \right )}+\frac {b \ln \left ({\mathrm e}^{i \left (d \,x^{2}+c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d \,x^{2}+c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{a \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}-\frac {b \ln \left ({\mathrm e}^{i \left (d \,x^{2}+c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d \,x^{2}+c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{a \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}\)	\(420\)

Input:

int(x/(a+b*csc(d*x^2+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/2/d*(-2/a^2*b*((1/2*a^2/(a^2-b^2)*tan(1/2*d*x^2+1/2*c)+1/2*b*a/(a^2-b^2) 
)/(1/2*tan(1/2*d*x^2+1/2*c)^2*b+a*tan(1/2*d*x^2+1/2*c)+1/2*b)+2*(2*a^2-b^2 
)/(2*a^2-2*b^2)/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*tan(1/2*d*x^2+1/2*c)+2*a) 
/(-a^2+b^2)^(1/2)))+2/a^2*arctan(tan(1/2*d*x^2+1/2*c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (111) = 222\).

Time = 0.10 (sec) , antiderivative size = 536, normalized size of antiderivative = 4.47 \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\left [\frac {2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \sin \left (d x^{2} + c\right ) + 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x^{2} + {\left (2 \, a^{2} b^{2} - b^{4} + {\left (2 \, a^{3} b - a b^{3}\right )} \sin \left (d x^{2} + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x^{2} + c\right )^{2} + 2 \, a b \sin \left (d x^{2} + c\right ) + a^{2} + b^{2} + 2 \, {\left (b \cos \left (d x^{2} + c\right ) \sin \left (d x^{2} + c\right ) + a \cos \left (d x^{2} + c\right )\right )} \sqrt {a^{2} - b^{2}}}{a^{2} \cos \left (d x^{2} + c\right )^{2} - 2 \, a b \sin \left (d x^{2} + c\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x^{2} + c\right )}{4 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sin \left (d x^{2} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )}}, \frac {{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \sin \left (d x^{2} + c\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x^{2} + {\left (2 \, a^{2} b^{2} - b^{4} + {\left (2 \, a^{3} b - a b^{3}\right )} \sin \left (d x^{2} + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \sin \left (d x^{2} + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (d x^{2} + c\right )}\right ) - {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x^{2} + c\right )}{2 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sin \left (d x^{2} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )}}\right ] \] Input:

integrate(x/(a+b*csc(d*x^2+c))^2,x, algorithm="fricas")
 

Output:

[1/4*(2*(a^5 - 2*a^3*b^2 + a*b^4)*d*x^2*sin(d*x^2 + c) + 2*(a^4*b - 2*a^2* 
b^3 + b^5)*d*x^2 + (2*a^2*b^2 - b^4 + (2*a^3*b - a*b^3)*sin(d*x^2 + c))*sq 
rt(a^2 - b^2)*log(((a^2 - 2*b^2)*cos(d*x^2 + c)^2 + 2*a*b*sin(d*x^2 + c) + 
 a^2 + b^2 + 2*(b*cos(d*x^2 + c)*sin(d*x^2 + c) + a*cos(d*x^2 + c))*sqrt(a 
^2 - b^2))/(a^2*cos(d*x^2 + c)^2 - 2*a*b*sin(d*x^2 + c) - a^2 - b^2)) - 2* 
(a^3*b^2 - a*b^4)*cos(d*x^2 + c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*sin(d*x^2 
 + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d), 1/2*((a^5 - 2*a^3*b^2 + a*b^4)*d 
*x^2*sin(d*x^2 + c) + (a^4*b - 2*a^2*b^3 + b^5)*d*x^2 + (2*a^2*b^2 - b^4 + 
 (2*a^3*b - a*b^3)*sin(d*x^2 + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^ 
2)*(b*sin(d*x^2 + c) + a)/((a^2 - b^2)*cos(d*x^2 + c))) - (a^3*b^2 - a*b^4 
)*cos(d*x^2 + c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*sin(d*x^2 + c) + (a^6*b - 
 2*a^4*b^3 + a^2*b^5)*d)]
 

Sympy [F]

\[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x}{\left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}}\, dx \] Input:

integrate(x/(a+b*csc(d*x**2+c))**2,x)
 

Output:

Integral(x/(a + b*csc(c + d*x**2))**2, x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x/(a+b*csc(d*x^2+c))^2,x, algorithm="maxima")
 

Output:

Timed out
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.45 \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=-\frac {{\left (2 \, a^{2} b - b^{3}\right )} {\left (\pi \left \lfloor \frac {d x^{2} + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{4} d - a^{2} b^{2} d\right )} \sqrt {-a^{2} + b^{2}}} - \frac {a b \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) + b^{2}}{{\left (a^{3} d - a b^{2} d\right )} {\left (b \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) + b\right )}} + \frac {d x^{2} + c}{2 \, a^{2} d} \] Input:

integrate(x/(a+b*csc(d*x^2+c))^2,x, algorithm="giac")
 

Output:

-(2*a^2*b - b^3)*(pi*floor(1/2*(d*x^2 + c)/pi + 1/2)*sgn(b) + arctan((b*ta 
n(1/2*d*x^2 + 1/2*c) + a)/sqrt(-a^2 + b^2)))/((a^4*d - a^2*b^2*d)*sqrt(-a^ 
2 + b^2)) - (a*b*tan(1/2*d*x^2 + 1/2*c) + b^2)/((a^3*d - a*b^2*d)*(b*tan(1 
/2*d*x^2 + 1/2*c)^2 + 2*a*tan(1/2*d*x^2 + 1/2*c) + b)) + 1/2*(d*x^2 + c)/( 
a^2*d)
 

Mupad [B] (verification not implemented)

Time = 20.60 (sec) , antiderivative size = 2755, normalized size of antiderivative = 22.96 \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\text {Too large to display} \] Input:

int(x/(a + b/sin(c + d*x^2))^2,x)
 

Output:

- atan((8*a^3*b^3*tan(c/2 + (d*x^2)/2))/((8*a^3*b^9)/(a^6 + a^2*b^4 - 2*a^ 
4*b^2) - (24*a^5*b^7)/(a^6 + a^2*b^4 - 2*a^4*b^2) + (16*a^7*b^5)/(a^6 + a^ 
2*b^4 - 2*a^4*b^2) + (8*a^9*b^3)/(a^6 + a^2*b^4 - 2*a^4*b^2) - (8*a^11*b)/ 
(a^6 + a^2*b^4 - 2*a^4*b^2)) - (8*a*b^5*tan(c/2 + (d*x^2)/2))/((8*a^3*b^9) 
/(a^6 + a^2*b^4 - 2*a^4*b^2) - (24*a^5*b^7)/(a^6 + a^2*b^4 - 2*a^4*b^2) + 
(16*a^7*b^5)/(a^6 + a^2*b^4 - 2*a^4*b^2) + (8*a^9*b^3)/(a^6 + a^2*b^4 - 2* 
a^4*b^2) - (8*a^11*b)/(a^6 + a^2*b^4 - 2*a^4*b^2)) + (8*a^5*b*tan(c/2 + (d 
*x^2)/2))/((8*a^3*b^9)/(a^6 + a^2*b^4 - 2*a^4*b^2) - (24*a^5*b^7)/(a^6 + a 
^2*b^4 - 2*a^4*b^2) + (16*a^7*b^5)/(a^6 + a^2*b^4 - 2*a^4*b^2) + (8*a^9*b^ 
3)/(a^6 + a^2*b^4 - 2*a^4*b^2) - (8*a^11*b)/(a^6 + a^2*b^4 - 2*a^4*b^2)))/ 
(a^2*d) - (b^2/(a*(a^2 - b^2)) + (b*tan(c/2 + (d*x^2)/2))/(a^2 - b^2))/(d* 
(b + b*tan(c/2 + (d*x^2)/2)^2 + 2*a*tan(c/2 + (d*x^2)/2))) - (b*atan(((b*( 
2*a^2 - b^2)*((a + b)^3*(a - b)^3)^(1/2)*((8*tan(c/2 + (d*x^2)/2)*(2*a*b^7 
 - 2*a^7*b - 8*a^3*b^5 + 9*a^5*b^3))/(a^7 + a^3*b^4 - 2*a^5*b^2) - (4*(2*a 
*b^6 - 4*a^3*b^4 + 2*a^5*b^2))/(a^6 + a^2*b^4 - 2*a^4*b^2) + (b*(2*a^2 - b 
^2)*((a + b)^3*(a - b)^3)^(1/2)*((4*(4*a^8*b - 4*a^6*b^3))/(a^6 + a^2*b^4 
- 2*a^4*b^2) + (8*tan(c/2 + (d*x^2)/2)*(4*a^4*b^6 - 12*a^6*b^4 + 8*a^8*b^2 
))/(a^7 + a^3*b^4 - 2*a^5*b^2) - (b*((4*(8*a^5*b^6 - 16*a^7*b^4 + 8*a^9*b^ 
2))/(a^6 + a^2*b^4 - 2*a^4*b^2) + (8*tan(c/2 + (d*x^2)/2)*(12*a^11*b - 8*a 
^5*b^7 + 28*a^7*b^5 - 32*a^9*b^3))/(a^7 + a^3*b^4 - 2*a^5*b^2))*(2*a^2 ...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 377, normalized size of antiderivative = 3.14 \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\frac {4 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right ) b +a}{\sqrt {-a^{2}+b^{2}}}\right ) \sin \left (d \,x^{2}+c \right ) a^{3} b -2 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right ) b +a}{\sqrt {-a^{2}+b^{2}}}\right ) \sin \left (d \,x^{2}+c \right ) a \,b^{3}+4 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right ) b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2} b^{2}-2 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right ) b +a}{\sqrt {-a^{2}+b^{2}}}\right ) b^{4}-\cos \left (d \,x^{2}+c \right ) a^{3} b^{2}+\cos \left (d \,x^{2}+c \right ) a \,b^{4}+\sin \left (d \,x^{2}+c \right ) a^{5} d \,x^{2}-2 \sin \left (d \,x^{2}+c \right ) a^{3} b^{2} d \,x^{2}+\sin \left (d \,x^{2}+c \right ) a \,b^{4} d \,x^{2}+a^{4} b d \,x^{2}-2 a^{2} b^{3} d \,x^{2}+b^{5} d \,x^{2}}{2 a^{2} d \left (\sin \left (d \,x^{2}+c \right ) a^{5}-2 \sin \left (d \,x^{2}+c \right ) a^{3} b^{2}+\sin \left (d \,x^{2}+c \right ) a \,b^{4}+a^{4} b -2 a^{2} b^{3}+b^{5}\right )} \] Input:

int(x/(a+b*csc(d*x^2+c))^2,x)
 

Output:

(4*sqrt( - a**2 + b**2)*atan((tan((c + d*x**2)/2)*b + a)/sqrt( - a**2 + b* 
*2))*sin(c + d*x**2)*a**3*b - 2*sqrt( - a**2 + b**2)*atan((tan((c + d*x**2 
)/2)*b + a)/sqrt( - a**2 + b**2))*sin(c + d*x**2)*a*b**3 + 4*sqrt( - a**2 
+ b**2)*atan((tan((c + d*x**2)/2)*b + a)/sqrt( - a**2 + b**2))*a**2*b**2 - 
 2*sqrt( - a**2 + b**2)*atan((tan((c + d*x**2)/2)*b + a)/sqrt( - a**2 + b* 
*2))*b**4 - cos(c + d*x**2)*a**3*b**2 + cos(c + d*x**2)*a*b**4 + sin(c + d 
*x**2)*a**5*d*x**2 - 2*sin(c + d*x**2)*a**3*b**2*d*x**2 + sin(c + d*x**2)* 
a*b**4*d*x**2 + a**4*b*d*x**2 - 2*a**2*b**3*d*x**2 + b**5*d*x**2)/(2*a**2* 
d*(sin(c + d*x**2)*a**5 - 2*sin(c + d*x**2)*a**3*b**2 + sin(c + d*x**2)*a* 
b**4 + a**4*b - 2*a**2*b**3 + b**5))