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

Optimal. Leaf size=120 \[ \frac {b \left (2 a^2-b^2\right ) \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 d \left (a^2-b^2\right )^{3/2}}-\frac {b^2 \cot \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )}+\frac {x^2}{2 a^2} \]

[Out]

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*co
t(d*x^2+c)/a/(a^2-b^2)/d/(a+b*csc(d*x^2+c))

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Rubi [A]  time = 0.24, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4205, 3785, 3919, 3831, 2660, 618, 206} \[ \frac {b \left (2 a^2-b^2\right ) \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 d \left (a^2-b^2\right )^{3/2}}-\frac {b^2 \cot \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )}+\frac {x^2}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Csc[c + d*x^2])^2,x]

[Out]

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(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 3785

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] + Dist[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] && IntegerQ[2*n]

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(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 4205

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(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]

Rubi steps

\begin {align*} \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(a+b \csc (c+d x))^2} \, dx,x,x^2\right )\\ &=-\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 )}-\frac {\operatorname {Subst}\left (\int \frac {-a^2+b^2+a b \csc (c+d x)}{a+b \csc (c+d x)} \, dx,x,x^2\right )}{2 a \left (a^2-b^2\right )}\\ &=\frac {x^2}{2 a^2}-\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 )}-\frac {\left (b \left (2 a^2-b^2\right )\right ) \operatorname {Subst}\left (\int \frac {\csc (c+d x)}{a+b \csc (c+d x)} \, dx,x,x^2\right )}{2 a^2 \left (a^2-b^2\right )}\\ &=\frac {x^2}{2 a^2}-\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 )}-\frac {\left (2 a^2-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a \sin (c+d x)}{b}} \, dx,x,x^2\right )}{2 a^2 \left (a^2-b^2\right )}\\ &=\frac {x^2}{2 a^2}-\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 )}-\frac {\left (2 a^2-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}+x^2} \, dx,x,\tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{a^2 \left (a^2-b^2\right ) d}\\ &=\frac {x^2}{2 a^2}-\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 )}+\frac {\left (2 \left (2 a^2-b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (1-\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}+2 \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{a^2 \left (a^2-b^2\right ) d}\\ &=\frac {x^2}{2 a^2}+\frac {b \left (2 a^2-b^2\right ) \tanh ^{-1}\left (\frac {b \left (\frac {a}{b}+\tan \left (\frac {1}{2} \left (c+d x^2\right )\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 )}\\ \end {align*}

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Mathematica [A]  time = 0.68, size = 158, normalized size = 1.32 \[ \frac {\csc \left (c+d x^2\right ) \left (a \sin \left (c+d x^2\right )+b\right ) \left (-\frac {2 b \left (b^2-2 a^2\right ) \tan ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {b^2-a^2}}\right ) \left (a+b \csc \left (c+d x^2\right )\right )}{\left (b^2-a^2\right )^{3/2}}+\frac {a b^2 \cot \left (c+d x^2\right )}{(b-a) (a+b)}+\left (c+d x^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )\right )}{2 a^2 d \left (a+b \csc \left (c+d x^2\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.49, size = 536, normalized size = 4.47 \[ \left [\frac {2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \sin \left (d x^{2} + c\right ) + 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x^{2} + {\left (2 \, a^{2} b^{2} - b^{4} + {\left (2 \, a^{3} b - a b^{3}\right )} \sin \left (d x^{2} + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x^{2} + c\right )^{2} + 2 \, a b \sin \left (d x^{2} + c\right ) + a^{2} + b^{2} + 2 \, {\left (b \cos \left (d x^{2} + c\right ) \sin \left (d x^{2} + c\right ) + a \cos \left (d x^{2} + c\right )\right )} \sqrt {a^{2} - b^{2}}}{a^{2} \cos \left (d x^{2} + c\right )^{2} - 2 \, a b \sin \left (d x^{2} + c\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x^{2} + c\right )}{4 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sin \left (d x^{2} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )}}, \frac {{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \sin \left (d x^{2} + c\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x^{2} + {\left (2 \, a^{2} b^{2} - b^{4} + {\left (2 \, a^{3} b - a b^{3}\right )} \sin \left (d x^{2} + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \sin \left (d x^{2} + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (d x^{2} + c\right )}\right ) - {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x^{2} + c\right )}{2 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sin \left (d x^{2} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[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))*sqrt(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)]

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giac [A]  time = 0.54, size = 174, normalized size = 1.45 \[ -\frac {{\left (2 \, a^{2} b - b^{3}\right )} {\left (\pi \left \lfloor \frac {d x^{2} + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{4} d - a^{2} b^{2} d\right )} \sqrt {-a^{2} + b^{2}}} - \frac {a b \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) + b^{2}}{{\left (a^{3} d - a b^{2} d\right )} {\left (b \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) + b\right )}} + \frac {d x^{2} + c}{2 \, a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.77, size = 261, normalized size = 2.18 \[ -\frac {b \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{d \left (\left (\tan ^{2}\left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right ) b +2 a \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+b \right ) \left (a^{2}-b^{2}\right )}-\frac {b^{2}}{d a \left (\left (\tan ^{2}\left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right ) b +2 a \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+b \right ) \left (a^{2}-b^{2}\right )}-\frac {2 b \arctan \left (\frac {2 b \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{d \left (a^{2}-b^{2}\right ) \sqrt {-a^{2}+b^{2}}}+\frac {b^{3} \arctan \left (\frac {2 b \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{d \,a^{2} \left (a^{2}-b^{2}\right ) \sqrt {-a^{2}+b^{2}}}+\frac {\arctan \left (\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [B]  time = 5.67, size = 2755, normalized size = 22.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b/sin(c + d*x^2))^2,x)

[Out]

- 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*at
an(((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 - b^2)*((a + b)^3*(a - b)^3)^(1/2))/(2*(a^8 - a^2*b^6 + 3*a^4*b
^4 - 3*a^6*b^2))))/(2*(a^8 - a^2*b^6 + 3*a^4*b^4 - 3*a^6*b^2)))*1i)/(2*(a^8 - a^2*b^6 + 3*a^4*b^4 - 3*a^6*b^2)
) - (b*(2*a^2 - b^2)*((a + b)^3*(a - b)^3)^(1/2)*((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) - (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) + (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 - b^2)*((a + b)^3*(a - b)^3)^(1/2))/(2*(a^8 - a^2*b^6 + 3*a^4*b
^4 - 3*a^6*b^2))))/(2*(a^8 - a^2*b^6 + 3*a^4*b^4 - 3*a^6*b^2)))*1i)/(2*(a^8 - a^2*b^6 + 3*a^4*b^4 - 3*a^6*b^2)
))/((8*(b^5 - 2*a^2*b^3))/(a^6 + a^2*b^4 - 2*a^4*b^2) + (16*tan(c/2 + (d*x^2)/2)*(b^6 - 3*a^2*b^4 + 2*a^4*b^2)
)/(a^7 + a^3*b^4 - 2*a^5*b^2) + (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 - b^2)*((a + b)^3*(a - b)^3)^(1/2))
/(2*(a^8 - a^2*b^6 + 3*a^4*b^4 - 3*a^6*b^2))))/(2*(a^8 - a^2*b^6 + 3*a^4*b^4 - 3*a^6*b^2))))/(2*(a^8 - a^2*b^6
+ 3*a^4*b^4 - 3*a^6*b^2)) + (b*(2*a^2 - b^2)*((a + b)^3*(a - b)^3)^(1/2)*((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) - (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) + (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 - b^2)*((a + b)^3*(a - b)^3)^(1/2))/(2
*(a^8 - a^2*b^6 + 3*a^4*b^4 - 3*a^6*b^2))))/(2*(a^8 - a^2*b^6 + 3*a^4*b^4 - 3*a^6*b^2))))/(2*(a^8 - a^2*b^6 +
3*a^4*b^4 - 3*a^6*b^2))))*(2*a^2 - b^2)*((a + b)^3*(a - b)^3)^(1/2)*1i)/(d*(a^8 - a^2*b^6 + 3*a^4*b^4 - 3*a^6*
b^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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