\(\int \frac {\arccos (a x)}{(c+d x^2)^2} \, dx\) [28]

Optimal result

Integrand size = 14, antiderivative size = 727 \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^2} \, dx=-\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}+a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}} \]

[Out]

-1/4*arccos(a*x)*ln(1-(a*x+I*(-a^2*x^2+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)-I*(a^2*c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)+
1/4*arccos(a*x)*ln(1+(a*x+I*(-a^2*x^2+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)-I*(a^2*c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)-1
/4*arccos(a*x)*ln(1-(a*x+I*(-a^2*x^2+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)+I*(a^2*c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)+1/
4*arccos(a*x)*ln(1+(a*x+I*(-a^2*x^2+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)+I*(a^2*c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)-1/4
*I*polylog(2,-(a*x+I*(-a^2*x^2+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)-I*(a^2*c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)+1/4*I*po
lylog(2,(a*x+I*(-a^2*x^2+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)-I*(a^2*c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)-1/4*I*polylog(
2,-(a*x+I*(-a^2*x^2+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)+I*(a^2*c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)+1/4*I*polylog(2,(a*
x+I*(-a^2*x^2+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)+I*(a^2*c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)-1/4*arccos(a*x)/c/d^(1/2)
/((-c)^(1/2)-x*d^(1/2))+1/4*arccos(a*x)/c/d^(1/2)/((-c)^(1/2)+x*d^(1/2))-1/4*a*arctanh((-a^2*x*(-c)^(1/2)+d^(1
/2))/(a^2*c+d)^(1/2)/(-a^2*x^2+1)^(1/2))/c/d^(1/2)/(a^2*c+d)^(1/2)-1/4*a*arctanh((a^2*x*(-c)^(1/2)+d^(1/2))/(a
^2*c+d)^(1/2)/(-a^2*x^2+1)^(1/2))/c/d^(1/2)/(a^2*c+d)^(1/2)

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {4758, 4828, 739, 212, 4826, 4618, 2221, 2317, 2438} \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^2} \, dx=-\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {1-a^2 x^2} \sqrt {a^2 c+d}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \text {arctanh}\left (\frac {a^2 \sqrt {-c} x+\sqrt {d}}{\sqrt {1-a^2 x^2} \sqrt {a^2 c+d}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )} \]

[In]

Int[ArcCos[a*x]/(c + d*x^2)^2,x]

[Out]

-1/4*ArcCos[a*x]/(c*Sqrt[d]*(Sqrt[-c] - Sqrt[d]*x)) + ArcCos[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] + Sqrt[d]*x)) - (a*Ar
cTanh[(Sqrt[d] - a^2*Sqrt[-c]*x)/(Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2])])/(4*c*Sqrt[d]*Sqrt[a^2*c + d]) - (a*ArcT
anh[(Sqrt[d] + a^2*Sqrt[-c]*x)/(Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2])])/(4*c*Sqrt[d]*Sqrt[a^2*c + d]) - (ArcCos[a
*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCos[a*
x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) - (ArcCos[a*x
]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCos[a*x]
*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) - ((I/4)*PolyLo
g[2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d]))])/((-c)^(3/2)*Sqrt[d]) + ((I/4)*PolyLog[2
, (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/((-c)^(3/2)*Sqrt[d]) - ((I/4)*PolyLog[2, -((S
qrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d]))])/((-c)^(3/2)*Sqrt[d]) + ((I/4)*PolyLog[2, (Sqrt[d
]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/((-c)^(3/2)*Sqrt[d])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4618

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)])/(Cos[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :>
Simp[I*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a - Rt[-a^2 + b^2, 2] + I*
b*E^(I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a + Rt[-a^2 + b^2, 2] + I*b*E^(I*(c + d*x)))), x
]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

Rule 4758

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCos[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[n, 0])

Rule 4826

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Subst[Int[(a + b*x)^n*(Sin[x]/
(c*d + e*Cos[x])), x], x, ArcCos[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 4828

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*ArcCos[c*x])^n/(e*(m + 1))), x] + Dist[b*c*(n/(e*(m + 1))), Int[(d + e*x)^(m + 1)*((a + b*ArcCos[c*x])^(
n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d \arccos (a x)}{4 c \left (\sqrt {-c} \sqrt {d}-d x\right )^2}-\frac {d \arccos (a x)}{4 c \left (\sqrt {-c} \sqrt {d}+d x\right )^2}-\frac {d \arccos (a x)}{2 c \left (-c d-d^2 x^2\right )}\right ) \, dx \\ & = -\frac {d \int \frac {\arccos (a x)}{\left (\sqrt {-c} \sqrt {d}-d x\right )^2} \, dx}{4 c}-\frac {d \int \frac {\arccos (a x)}{\left (\sqrt {-c} \sqrt {d}+d x\right )^2} \, dx}{4 c}-\frac {d \int \frac {\arccos (a x)}{-c d-d^2 x^2} \, dx}{2 c} \\ & = -\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \int \frac {1}{\left (\sqrt {-c} \sqrt {d}-d x\right ) \sqrt {1-a^2 x^2}} \, dx}{4 c}+\frac {a \int \frac {1}{\left (\sqrt {-c} \sqrt {d}+d x\right ) \sqrt {1-a^2 x^2}} \, dx}{4 c}-\frac {d \int \left (-\frac {\sqrt {-c} \arccos (a x)}{2 c d \left (\sqrt {-c}-\sqrt {d} x\right )}-\frac {\sqrt {-c} \arccos (a x)}{2 c d \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx}{2 c} \\ & = -\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}+\frac {\int \frac {\arccos (a x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{4 (-c)^{3/2}}+\frac {\int \frac {\arccos (a x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{4 (-c)^{3/2}}+\frac {a \text {Subst}\left (\int \frac {1}{a^2 c d+d^2-x^2} \, dx,x,\frac {-d+a^2 \sqrt {-c} \sqrt {d} x}{\sqrt {1-a^2 x^2}}\right )}{4 c}-\frac {a \text {Subst}\left (\int \frac {1}{a^2 c d+d^2-x^2} \, dx,x,\frac {d+a^2 \sqrt {-c} \sqrt {d} x}{\sqrt {1-a^2 x^2}}\right )}{4 c} \\ & = -\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}+a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\text {Subst}\left (\int \frac {x \sin (x)}{a \sqrt {-c}-\sqrt {d} \cos (x)} \, dx,x,\arccos (a x)\right )}{4 (-c)^{3/2}}-\frac {\text {Subst}\left (\int \frac {x \sin (x)}{a \sqrt {-c}+\sqrt {d} \cos (x)} \, dx,x,\arccos (a x)\right )}{4 (-c)^{3/2}} \\ & = -\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}+a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\text {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}-i \sqrt {d} e^{i x}} \, dx,x,\arccos (a x)\right )}{4 (-c)^{3/2}}-\frac {\text {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}-i \sqrt {d} e^{i x}} \, dx,x,\arccos (a x)\right )}{4 (-c)^{3/2}}-\frac {\text {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}+i \sqrt {d} e^{i x}} \, dx,x,\arccos (a x)\right )}{4 (-c)^{3/2}}-\frac {\text {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}+i \sqrt {d} e^{i x}} \, dx,x,\arccos (a x)\right )}{4 (-c)^{3/2}} \\ & = -\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}+a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\text {Subst}\left (\int \log \left (1-\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right ) \, dx,x,\arccos (a x)\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\text {Subst}\left (\int \log \left (1+\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right ) \, dx,x,\arccos (a x)\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\text {Subst}\left (\int \log \left (1-\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right ) \, dx,x,\arccos (a x)\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\text {Subst}\left (\int \log \left (1+\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right ) \, dx,x,\arccos (a x)\right )}{4 (-c)^{3/2} \sqrt {d}} \\ & = -\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}+a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {d} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \arccos (a x)}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {d} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \arccos (a x)}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {d} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \arccos (a x)}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {d} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \arccos (a x)}\right )}{4 (-c)^{3/2} \sqrt {d}} \\ & = -\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}+a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.83 (sec) , antiderivative size = 1065, normalized size of antiderivative = 1.46 \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^2} \, dx=\frac {4 \arcsin \left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (a \sqrt {c}-i \sqrt {d}\right ) \tan \left (\frac {1}{2} \arccos (a x)\right )}{\sqrt {a^2 c+d}}\right )-4 \arcsin \left (\frac {\sqrt {1+\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (a \sqrt {c}+i \sqrt {d}\right ) \tan \left (\frac {1}{2} \arccos (a x)\right )}{\sqrt {a^2 c+d}}\right )+i \arccos (a x) \log \left (1-\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+2 i \arcsin \left (\frac {\sqrt {1+\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-i \arccos (a x) \log \left (1+\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-2 i \arcsin \left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-i \arccos (a x) \log \left (1-\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+2 i \arcsin \left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+i \arccos (a x) \log \left (1+\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-2 i \arcsin \left (\frac {\sqrt {1+\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+\sqrt {c} \left (\frac {\arccos (a x)}{-i \sqrt {c}+\sqrt {d} x}-\frac {a \log \left (\frac {2 d \left (\sqrt {d}-i a^2 \sqrt {c} x+\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}\right )}{a \sqrt {a^2 c+d} \left (-i \sqrt {c}+\sqrt {d} x\right )}\right )}{\sqrt {a^2 c+d}}\right )+\sqrt {c} \left (\frac {\arccos (a x)}{i \sqrt {c}+\sqrt {d} x}-\frac {a \log \left (-\frac {2 d \left (\sqrt {d}+i a^2 \sqrt {c} x+\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}\right )}{a \sqrt {a^2 c+d} \left (i \sqrt {c}+\sqrt {d} x\right )}\right )}{\sqrt {a^2 c+d}}\right )-\operatorname {PolyLog}\left (2,-\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-\operatorname {PolyLog}\left (2,\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )}{4 c^{3/2} \sqrt {d}} \]

[In]

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

[Out]

(4*ArcSin[Sqrt[1 - (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*ArcTan[((a*Sqrt[c] - I*Sqrt[d])*Tan[ArcCos[a*x]/2])/Sqrt[a^
2*c + d]] - 4*ArcSin[Sqrt[1 + (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*ArcTan[((a*Sqrt[c] + I*Sqrt[d])*Tan[ArcCos[a*x]/
2])/Sqrt[a^2*c + d]] + I*ArcCos[a*x]*Log[1 - (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] +
(2*I)*ArcSin[Sqrt[1 + (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 - (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos
[a*x]))/Sqrt[d]] - I*ArcCos[a*x]*Log[1 + (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - (2*
I)*ArcSin[Sqrt[1 - (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 + (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x
]))/Sqrt[d]] - I*ArcCos[a*x]*Log[1 - (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + (2*I)*ArcS
in[Sqrt[1 - (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 - (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d
]] + I*ArcCos[a*x]*Log[1 + (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - (2*I)*ArcSin[Sqrt[1
+ (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 + (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + Sqrt[
c]*(ArcCos[a*x]/((-I)*Sqrt[c] + Sqrt[d]*x) - (a*Log[(2*d*(Sqrt[d] - I*a^2*Sqrt[c]*x + Sqrt[a^2*c + d]*Sqrt[1 -
a^2*x^2]))/(a*Sqrt[a^2*c + d]*((-I)*Sqrt[c] + Sqrt[d]*x))])/Sqrt[a^2*c + d]) + Sqrt[c]*(ArcCos[a*x]/(I*Sqrt[c
] + Sqrt[d]*x) - (a*Log[(-2*d*(Sqrt[d] + I*a^2*Sqrt[c]*x + Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2]))/(a*Sqrt[a^2*c +
d]*(I*Sqrt[c] + Sqrt[d]*x))])/Sqrt[a^2*c + d]) - PolyLog[2, ((-I)*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCo
s[a*x]))/Sqrt[d]] + PolyLog[2, (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + PolyLog[2, ((
-I)*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - PolyLog[2, (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^
(I*ArcCos[a*x]))/Sqrt[d]])/(4*c^(3/2)*Sqrt[d])

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 5.75 (sec) , antiderivative size = 796, normalized size of antiderivative = 1.09

\[\frac {\frac {\arccos \left (a x \right ) a^{3} x}{2 c \left (a^{2} d \,x^{2}+c \,a^{2}\right )}-\frac {i \sqrt {\left (2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 a^{4} c^{2}-2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}\, a^{2} c +2 a^{2} c d -d \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}\right ) a^{2} \arctan \left (\frac {d \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )}{\sqrt {\left (2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) d}}\right )}{2 c \left (c \,a^{2}+d \right ) d^{3}}+\frac {i \sqrt {\left (2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 c \,a^{2}-2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) \arctan \left (\frac {d \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )}{\sqrt {\left (2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) d}}\right ) a^{2}}{2 c \,d^{3}}-\frac {i \sqrt {-\left (2 c \,a^{2}-2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 a^{4} c^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}\, a^{2} c +2 a^{2} c d +d \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}\right ) a^{2} \operatorname {arctanh}\left (\frac {d \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )}{\sqrt {\left (-2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}-d \right ) d}}\right )}{2 c \left (c \,a^{2}+d \right ) d^{3}}+\frac {i \sqrt {-\left (2 c \,a^{2}-2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) \operatorname {arctanh}\left (\frac {d \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )}{\sqrt {\left (-2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}-d \right ) d}}\right ) a^{2}}{2 c \,d^{3}}-\frac {i a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} d +2 c \,a^{2}+d}\right )}{4 c}+\frac {i a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} d +2 c \,a^{2}+d \right )}\right )}{4 c}}{a}\]

[In]

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

[Out]

1/a*(1/2*arccos(a*x)*a^3*x/c/(a^2*d*x^2+a^2*c)-1/2*I*((2*c*a^2+2*(c*a^2*(a^2*c+d))^(1/2)+d)*d)^(1/2)*(2*a^4*c^
2-2*(c*a^2*(a^2*c+d))^(1/2)*a^2*c+2*a^2*c*d-d*(c*a^2*(a^2*c+d))^(1/2))*a^2*arctan(d*(I*(-a^2*x^2+1)^(1/2)+a*x)
/((2*c*a^2+2*(c*a^2*(a^2*c+d))^(1/2)+d)*d)^(1/2))/c/(a^2*c+d)/d^3+1/2*I*((2*c*a^2+2*(c*a^2*(a^2*c+d))^(1/2)+d)
*d)^(1/2)*(2*c*a^2-2*(c*a^2*(a^2*c+d))^(1/2)+d)*arctan(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((2*c*a^2+2*(c*a^2*(a^2*c+
d))^(1/2)+d)*d)^(1/2))*a^2/c/d^3-1/2*I*(-(2*c*a^2-2*(c*a^2*(a^2*c+d))^(1/2)+d)*d)^(1/2)*(2*a^4*c^2+2*(c*a^2*(a
^2*c+d))^(1/2)*a^2*c+2*a^2*c*d+d*(c*a^2*(a^2*c+d))^(1/2))*a^2*arctanh(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((-2*c*a^2+
2*(c*a^2*(a^2*c+d))^(1/2)-d)*d)^(1/2))/c/(a^2*c+d)/d^3+1/2*I*(-(2*c*a^2-2*(c*a^2*(a^2*c+d))^(1/2)+d)*d)^(1/2)*
(2*c*a^2+2*(c*a^2*(a^2*c+d))^(1/2)+d)*arctanh(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((-2*c*a^2+2*(c*a^2*(a^2*c+d))^(1/2
)-d)*d)^(1/2))*a^2/c/d^3-1/4*I/c*a^2*sum(_R1/(_R1^2*d+2*a^2*c+d)*(I*arccos(a*x)*ln((_R1-a*x-I*(-a^2*x^2+1)^(1/
2))/_R1)+dilog((_R1-a*x-I*(-a^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(d*_Z^4+(4*a^2*c+2*d)*_Z^2+d))+1/4*I/c*a^2*sum(1
/_R1/(_R1^2*d+2*a^2*c+d)*(I*arccos(a*x)*ln((_R1-a*x-I*(-a^2*x^2+1)^(1/2))/_R1)+dilog((_R1-a*x-I*(-a^2*x^2+1)^(
1/2))/_R1)),_R1=RootOf(d*_Z^4+(4*a^2*c+2*d)*_Z^2+d)))

Fricas [F]

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

[In]

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

[Out]

integral(arccos(a*x)/(d^2*x^4 + 2*c*d*x^2 + c^2), x)

Sympy [F]

\[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^2} \, dx=\int \frac {\operatorname {acos}{\left (a x \right )}}{\left (c + d x^{2}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(arccos(a*x)/(d*x^2 + c)^2, x)

Giac [F]

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

[In]

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

[Out]

integrate(arccos(a*x)/(d*x^2 + c)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^2} \, dx=\int \frac {\mathrm {acos}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^2} \,d x \]

[In]

int(acos(a*x)/(c + d*x^2)^2,x)

[Out]

int(acos(a*x)/(c + d*x^2)^2, x)