\(\int \frac {1}{a+b \arccos (1+d x^2)} \, dx\) [77]

Optimal result

Integrand size = 14, antiderivative size = 99 \[ \int \frac {1}{a+b \arccos \left (1+d x^2\right )} \, dx=\frac {x \cos \left (\frac {a}{2 b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos \left (1+d x^2\right )}{2 b}\right )}{\sqrt {2} b \sqrt {-d x^2}}+\frac {x \sin \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (1+d x^2\right )}{2 b}\right )}{\sqrt {2} b \sqrt {-d x^2}} \]

[Out]

1/2*x*Ci(1/2*(a+b*arccos(d*x^2+1))/b)*cos(1/2*a/b)/b*2^(1/2)/(-d*x^2)^(1/2)+1/2*x*Si(1/2*(a+b*arccos(d*x^2+1))
/b)*sin(1/2*a/b)/b*2^(1/2)/(-d*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4901} \[ \int \frac {1}{a+b \arccos \left (1+d x^2\right )} \, dx=\frac {x \cos \left (\frac {a}{2 b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos \left (d x^2+1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {-d x^2}}+\frac {x \sin \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (d x^2+1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {-d x^2}} \]

[In]

Int[(a + b*ArcCos[1 + d*x^2])^(-1),x]

[Out]

(x*Cos[a/(2*b)]*CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[-(d*x^2)]) + (x*Sin[a/(2*b)]*Sin
Integral[(a + b*ArcCos[1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[-(d*x^2)])

Rule 4901

Int[((a_.) + ArcCos[1 + (d_.)*(x_)^2]*(b_.))^(-1), x_Symbol] :> Simp[x*Cos[a/(2*b)]*(CosIntegral[(a + b*ArcCos
[1 + d*x^2])/(2*b)]/(Sqrt[2]*b*Sqrt[(-d)*x^2])), x] + Simp[x*Sin[a/(2*b)]*(SinIntegral[(a + b*ArcCos[1 + d*x^2
])/(2*b)]/(Sqrt[2]*b*Sqrt[(-d)*x^2])), x] /; FreeQ[{a, b, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {x \cos \left (\frac {a}{2 b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos \left (1+d x^2\right )}{2 b}\right )}{\sqrt {2} b \sqrt {-d x^2}}+\frac {x \sin \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (1+d x^2\right )}{2 b}\right )}{\sqrt {2} b \sqrt {-d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.86 \[ \int \frac {1}{a+b \arccos \left (1+d x^2\right )} \, dx=-\frac {\sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right ) \left (\cos \left (\frac {a}{2 b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos \left (1+d x^2\right )}{2 b}\right )+\sin \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (1+d x^2\right )}{2 b}\right )\right )}{b d x} \]

[In]

Integrate[(a + b*ArcCos[1 + d*x^2])^(-1),x]

[Out]

-((Sin[ArcCos[1 + d*x^2]/2]*(Cos[a/(2*b)]*CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)] + Sin[a/(2*b)]*SinInteg
ral[(a + b*ArcCos[1 + d*x^2])/(2*b)]))/(b*d*x))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(1/(b*arccos(d*x^2 + 1) + a), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: sign: argument cannot be imaginary; found sqrt((-_SAGE_VAR_d*_SAGE
_VAR_x^2)-2)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{a+b \arccos \left (1+d x^2\right )} \, dx=\int \frac {1}{a+b\,\mathrm {acos}\left (d\,x^2+1\right )} \,d x \]

[In]

int(1/(a + b*acos(d*x^2 + 1)),x)

[Out]

int(1/(a + b*acos(d*x^2 + 1)), x)