3.27 \(\int \frac {\cos ^{-1}(a x)}{c+d x^2} \, dx\)

Optimal. Leaf size=521 \[ \frac {i \text {Li}_2\left (-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {Li}_2\left (\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \text {Li}_2\left (-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {Li}_2\left (\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]

[Out]

1/2*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)^(1/2)/d^(1/2)-1
/2*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)^(1/2)/d^(1/2)+1/
2*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)^(1/2)/d^(1/2)-1/2
*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)^(1/2)/d^(1/2)+1/2*
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)^(1/2)/d^(1/2)-1/2*I*pol
ylog(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)^(1/2)/d^(1/2)+1/2*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)^(1/2)/d^(1/2)-1/2*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)^(1/2)/d^(1/2)

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Rubi [A]  time = 0.81, antiderivative size = 521, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4668, 4742, 4522, 2190, 2279, 2391} \[ \frac {i \text {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {PolyLog}\left (2,\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \text {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {PolyLog}\left (2,\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]/(c + d*x^2),x]

[Out]

(ArcCos[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) - (Ar
cCos[a*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) + (ArcCo
s[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) - (ArcCos[a
*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) + ((I/2)*PolyL
og[2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d]))])/(Sqrt[-c]*Sqrt[d]) - ((I/2)*PolyLog[2,
(Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(Sqrt[-c]*Sqrt[d]) + ((I/2)*PolyLog[2, -((Sqrt
[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d]))])/(Sqrt[-c]*Sqrt[d]) - ((I/2)*PolyLog[2, (Sqrt[d]*E^(
I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(Sqrt[-c]*Sqrt[d])

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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 4522

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 4668

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 4742

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]

Rubi steps

\begin {align*} \int \frac {\cos ^{-1}(a x)}{c+d x^2} \, dx &=\int \left (\frac {\sqrt {-c} \cos ^{-1}(a x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \cos ^{-1}(a x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\cos ^{-1}(a x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\int \frac {\cos ^{-1}(a x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-c}}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x \sin (x)}{a \sqrt {-c}-\sqrt {d} \cos (x)} \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt {-c}}+\frac {\operatorname {Subst}\left (\int \frac {x \sin (x)}{a \sqrt {-c}+\sqrt {d} \cos (x)} \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt {-c}}\\ &=\frac {\operatorname {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,\cos ^{-1}(a x)\right )}{2 \sqrt {-c}}+\frac {\operatorname {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,\cos ^{-1}(a x)\right )}{2 \sqrt {-c}}+\frac {\operatorname {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,\cos ^{-1}(a x)\right )}{2 \sqrt {-c}}+\frac {\operatorname {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,\cos ^{-1}(a x)\right )}{2 \sqrt {-c}}\\ &=\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {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,\cos ^{-1}(a x)\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {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,\cos ^{-1}(a x)\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {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,\cos ^{-1}(a x)\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {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,\cos ^{-1}(a x)\right )}{2 \sqrt {-c} \sqrt {d}}\\ &=\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \operatorname {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 \cos ^{-1}(a x)}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \operatorname {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 \cos ^{-1}(a x)}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \operatorname {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 \cos ^{-1}(a x)}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \operatorname {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 \cos ^{-1}(a x)}\right )}{2 \sqrt {-c} \sqrt {d}}\\ &=\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \text {Li}_2\left (-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {Li}_2\left (\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \text {Li}_2\left (-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {Li}_2\left (\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}\\ \end {align*}

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Mathematica [A]  time = 1.23, size = 811, normalized size = 1.56 \[ \frac {4 \sin ^{-1}\left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \tan ^{-1}\left (\frac {\left (a \sqrt {c}-i \sqrt {d}\right ) \tan \left (\frac {1}{2} \cos ^{-1}(a x)\right )}{\sqrt {c a^2+d}}\right )-4 \sin ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {c} a}{\sqrt {d}}+1}}{\sqrt {2}}\right ) \tan ^{-1}\left (\frac {\left (\sqrt {c} a+i \sqrt {d}\right ) \tan \left (\frac {1}{2} \cos ^{-1}(a x)\right )}{\sqrt {c a^2+d}}\right )+i \cos ^{-1}(a x) \log \left (1-\frac {i \left (\sqrt {c a^2+d}-a \sqrt {c}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )+2 i \sin ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {c} a}{\sqrt {d}}+1}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {c a^2+d}-a \sqrt {c}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )-i \cos ^{-1}(a x) \log \left (\frac {i e^{i \cos ^{-1}(a x)} \left (\sqrt {c a^2+d}-a \sqrt {c}\right )}{\sqrt {d}}+1\right )-2 i \sin ^{-1}\left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (\frac {i e^{i \cos ^{-1}(a x)} \left (\sqrt {c a^2+d}-a \sqrt {c}\right )}{\sqrt {d}}+1\right )-i \cos ^{-1}(a x) \log \left (1-\frac {i \left (\sqrt {c} a+\sqrt {c a^2+d}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )+2 i \sin ^{-1}\left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {c} a+\sqrt {c a^2+d}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )+i \cos ^{-1}(a x) \log \left (\frac {i e^{i \cos ^{-1}(a x)} \left (\sqrt {c} a+\sqrt {c a^2+d}\right )}{\sqrt {d}}+1\right )-2 i \sin ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {c} a}{\sqrt {d}}+1}}{\sqrt {2}}\right ) \log \left (\frac {i e^{i \cos ^{-1}(a x)} \left (\sqrt {c} a+\sqrt {c a^2+d}\right )}{\sqrt {d}}+1\right )-\text {Li}_2\left (-\frac {i \left (\sqrt {c a^2+d}-a \sqrt {c}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )+\text {Li}_2\left (\frac {i \left (\sqrt {c a^2+d}-a \sqrt {c}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )+\text {Li}_2\left (-\frac {i \left (\sqrt {c} a+\sqrt {c a^2+d}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )-\text {Li}_2\left (\frac {i \left (\sqrt {c} a+\sqrt {c a^2+d}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )}{2 \sqrt {c} \sqrt {d}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCos[a*x]/(c + d*x^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]] - PolyL
og[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]))/Sq
rt[d]] - PolyLog[2, (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]])/(2*Sqrt[c]*Sqrt[d])

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arccos \left (a x\right )}{d x^{2} + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccos(a*x)/(d*x^2 + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccos(a*x)/(d*x^2 + c), x)

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maple [C]  time = 1.04, size = 216, normalized size = 0.41 \[ -\frac {i a \left (\munderset {\textit {\_R1} =\RootOf \left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} d +2 a^{2} c +d}\right )}{2}+\frac {i a \left (\munderset {\textit {\_R1} =\RootOf \left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} d +2 a^{2} c +d \right )}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*a*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))+1/2*I*a*sum(1/_R1/(_R1^2*d+2*a^2*c+d)*(I*arcc
os(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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccos(a*x)/(d*x^2 + c), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {acos}\left (a\,x\right )}{d\,x^2+c} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(a*x)/(c + d*x^2),x)

[Out]

int(acos(a*x)/(c + d*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acos(a*x)/(c + d*x**2), x)

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