Integrand size = 14, antiderivative size = 98 \[ \int \frac {1}{a+b \arccos \left (-1+d x^2\right )} \, dx=\frac {x \operatorname {CosIntegral}\left (\frac {a+b \arccos \left (-1+d x^2\right )}{2 b}\right ) \sin \left (\frac {a}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (-1+d x^2\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}} \] Output:
1/2*x*Ci(1/2*(a+b*arccos(d*x^2-1))/b)*sin(1/2*a/b)*2^(1/2)/b/(d*x^2)^(1/2) -1/2*x*cos(1/2*a/b)*Si(1/2*(a+b*arccos(d*x^2-1))/b)*2^(1/2)/b/(d*x^2)^(1/2 )
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.87 \[ \int \frac {1}{a+b \arccos \left (-1+d x^2\right )} \, dx=\frac {\cos \left (\frac {1}{2} \arccos \left (-1+d x^2\right )\right ) \left (\operatorname {CosIntegral}\left (\frac {a+b \arccos \left (-1+d x^2\right )}{2 b}\right ) \sin \left (\frac {a}{2 b}\right )-\cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (-1+d x^2\right )}{2 b}\right )\right )}{b d x} \] Input:
Integrate[(a + b*ArcCos[-1 + d*x^2])^(-1),x]
Output:
(Cos[ArcCos[-1 + d*x^2]/2]*(CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]* Sin[a/(2*b)] - Cos[a/(2*b)]*SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]) )/(b*d*x)
Time = 0.22 (sec) , antiderivative size = 98, 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 = {5317}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{a+b \arccos \left (d x^2-1\right )} \, dx\) |
\(\Big \downarrow \) 5317 |
\(\displaystyle \frac {x \sin \left (\frac {a}{2 b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos \left (d x^2-1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (d x^2-1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}\) |
Input:
Int[(a + b*ArcCos[-1 + d*x^2])^(-1),x]
Output:
(x*CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]*Sin[a/(2*b)])/(Sqrt[2]*b* Sqrt[d*x^2]) - (x*Cos[a/(2*b)]*SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b )])/(Sqrt[2]*b*Sqrt[d*x^2])
Int[((a_.) + ArcCos[-1 + (d_.)*(x_)^2]*(b_.))^(-1), x_Symbol] :> Simp[x*Sin [a/(2*b)]*(CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]/(Sqrt[2]*b*Sqrt[d* x^2])), x] - Simp[x*Cos[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]
\[\int \frac {1}{a +b \arccos \left (d \,x^{2}-1\right )}d x\]
Input:
int(1/(a+b*arccos(d*x^2-1)),x)
Output:
int(1/(a+b*arccos(d*x^2-1)),x)
\[ \int \frac {1}{a+b \arccos \left (-1+d x^2\right )} \, dx=\int { \frac {1}{b \arccos \left (d x^{2} - 1\right ) + a} \,d x } \] Input:
integrate(1/(a+b*arccos(d*x^2-1)),x, algorithm="fricas")
Output:
integral(1/(b*arccos(d*x^2 - 1) + a), x)
\[ \int \frac {1}{a+b \arccos \left (-1+d x^2\right )} \, dx=\int \frac {1}{a + b \operatorname {acos}{\left (d x^{2} - 1 \right )}}\, dx \] Input:
integrate(1/(a+b*acos(d*x**2-1)),x)
Output:
Integral(1/(a + b*acos(d*x**2 - 1)), x)
\[ \int \frac {1}{a+b \arccos \left (-1+d x^2\right )} \, dx=\int { \frac {1}{b \arccos \left (d x^{2} - 1\right ) + a} \,d x } \] Input:
integrate(1/(a+b*arccos(d*x^2-1)),x, algorithm="maxima")
Output:
integrate(1/(b*arccos(d*x^2 - 1) + a), x)
\[ \int \frac {1}{a+b \arccos \left (-1+d x^2\right )} \, dx=\int { \frac {1}{b \arccos \left (d x^{2} - 1\right ) + a} \,d x } \] Input:
integrate(1/(a+b*arccos(d*x^2-1)),x, algorithm="giac")
Output:
integrate(1/(b*arccos(d*x^2 - 1) + a), x)
Timed out. \[ \int \frac {1}{a+b \arccos \left (-1+d x^2\right )} \, dx=\int \frac {1}{a+b\,\mathrm {acos}\left (d\,x^2-1\right )} \,d x \] Input:
int(1/(a + b*acos(d*x^2 - 1)),x)
Output:
int(1/(a + b*acos(d*x^2 - 1)), x)
\[ \int \frac {1}{a+b \arccos \left (-1+d x^2\right )} \, dx=\int \frac {1}{\mathit {acos} \left (d \,x^{2}-1\right ) b +a}d x \] Input:
int(1/(a+b*acos(d*x^2-1)),x)
Output:
int(1/(acos(d*x**2 - 1)*b + a),x)