\(\int \frac {1}{\sqrt {1+b x^2} \arccos (\sqrt {1+b x^2})} \, dx\) [118]

Optimal result

Integrand size = 26, antiderivative size = 31 \[ \int \frac {1}{\sqrt {1+b x^2} \arccos \left (\sqrt {1+b x^2}\right )} \, dx=-\frac {\sqrt {-b x^2} \log \left (\arccos \left (\sqrt {1+b x^2}\right )\right )}{b x} \]

[Out]

-ln(arccos((b*x^2+1)^(1/2)))*(-b*x^2)^(1/2)/b/x

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4919, 4736} \[ \int \frac {1}{\sqrt {1+b x^2} \arccos \left (\sqrt {1+b x^2}\right )} \, dx=-\frac {\sqrt {-b x^2} \log \left (\arccos \left (\sqrt {b x^2+1}\right )\right )}{b x} \]

[In]

Int[1/(Sqrt[1 + b*x^2]*ArcCos[Sqrt[1 + b*x^2]]),x]

[Out]

-((Sqrt[-(b*x^2)]*Log[ArcCos[Sqrt[1 + b*x^2]]])/(b*x))

Rule 4736

Int[1/(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Simp[(-(b*c)^(-1))*Simp[Sqr
t[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(Log[a + b*ArcCos[c*x]]/(b*c*Sqrt[d])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ
[c^2*d + e, 0]

Rule 4919

Int[ArcCos[Sqrt[1 + (b_.)*(x_)^2]]^(n_.)/Sqrt[1 + (b_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[(-b)*x^2]/(b*x), Subst
[Int[ArcCos[x]^n/Sqrt[1 - x^2], x], x, Sqrt[1 + b*x^2]], x] /; FreeQ[{b, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-b x^2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \arccos (x)} \, dx,x,\sqrt {1+b x^2}\right )}{b x} \\ & = -\frac {\sqrt {-b x^2} \log \left (\arccos \left (\sqrt {1+b x^2}\right )\right )}{b x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {1+b x^2} \arccos \left (\sqrt {1+b x^2}\right )} \, dx=\frac {x \log \left (\arccos \left (\sqrt {1+b x^2}\right )\right )}{\sqrt {-b x^2}} \]

[In]

Integrate[1/(Sqrt[1 + b*x^2]*ArcCos[Sqrt[1 + b*x^2]]),x]

[Out]

(x*Log[ArcCos[Sqrt[1 + b*x^2]]])/Sqrt[-(b*x^2)]

Maple [F]

\[\int \frac {1}{\arccos \left (\sqrt {b \,x^{2}+1}\right ) \sqrt {b \,x^{2}+1}}d x\]

[In]

int(1/arccos((b*x^2+1)^(1/2))/(b*x^2+1)^(1/2),x)

[Out]

int(1/arccos((b*x^2+1)^(1/2))/(b*x^2+1)^(1/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt {1+b x^2} \arccos \left (\sqrt {1+b x^2}\right )} \, dx=-\frac {\sqrt {-b x^{2}} \log \left (\arccos \left (\sqrt {b x^{2} + 1}\right )\right )}{b x} \]

[In]

integrate(1/arccos((b*x^2+1)^(1/2))/(b*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(-b*x^2)*log(arccos(sqrt(b*x^2 + 1)))/(b*x)

Sympy [F]

\[ \int \frac {1}{\sqrt {1+b x^2} \arccos \left (\sqrt {1+b x^2}\right )} \, dx=\int \frac {1}{\sqrt {b x^{2} + 1} \operatorname {acos}{\left (\sqrt {b x^{2} + 1} \right )}}\, dx \]

[In]

integrate(1/acos((b*x**2+1)**(1/2))/(b*x**2+1)**(1/2),x)

[Out]

Integral(1/(sqrt(b*x**2 + 1)*acos(sqrt(b*x**2 + 1))), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {1+b x^2} \arccos \left (\sqrt {1+b x^2}\right )} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/arccos((b*x^2+1)^(1/2))/(b*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: sign: argument cannot be imaginary; found sqrt(-_SAGE_VAR_b)

Giac [F]

\[ \int \frac {1}{\sqrt {1+b x^2} \arccos \left (\sqrt {1+b x^2}\right )} \, dx=\int { \frac {1}{\sqrt {b x^{2} + 1} \arccos \left (\sqrt {b x^{2} + 1}\right )} \,d x } \]

[In]

integrate(1/arccos((b*x^2+1)^(1/2))/(b*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^2 + 1)*arccos(sqrt(b*x^2 + 1))), x)

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {1+b x^2} \arccos \left (\sqrt {1+b x^2}\right )} \, dx=\frac {\ln \left (\mathrm {acos}\left (\sqrt {b\,x^2+1}\right )\right )\,\sqrt {x^2}}{\sqrt {-b}\,x} \]

[In]

int(1/(acos((b*x^2 + 1)^(1/2))*(b*x^2 + 1)^(1/2)),x)

[Out]

(log(acos((b*x^2 + 1)^(1/2)))*(x^2)^(1/2))/((-b)^(1/2)*x)