Integrand size = 14, antiderivative size = 98 \[ \int \frac {1}{a+b \text {arccosh}\left (-1+d x^2\right )} \, dx=-\frac {x \text {Chi}\left (\frac {a+b \text {arccosh}\left (-1+d x^2\right )}{2 b}\right ) \sinh \left (\frac {a}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}+\frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (-1+d x^2\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}} \] Output:
-1/2*x*Chi(1/2*(a+b*arccosh(d*x^2-1))/b)*sinh(1/2*a/b)*2^(1/2)/b/(d*x^2)^( 1/2)+1/2*x*cosh(1/2*a/b)*Shi(1/2*(a+b*arccosh(d*x^2-1))/b)*2^(1/2)/b/(d*x^ 2)^(1/2)
Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {1}{a+b \text {arccosh}\left (-1+d x^2\right )} \, dx=-\frac {\cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \left (\text {Chi}\left (\frac {a+b \text {arccosh}\left (-1+d x^2\right )}{2 b}\right ) \sinh \left (\frac {a}{2 b}\right )-\cosh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (-1+d x^2\right )}{2 b}\right )\right )}{b d x} \] Input:
Integrate[(a + b*ArcCosh[-1 + d*x^2])^(-1),x]
Output:
-((Cosh[ArcCosh[-1 + d*x^2]/2]*(CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/( 2*b)]*Sinh[a/(2*b)] - Cosh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[-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 = {6418}
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.
\(\displaystyle \int \frac {1}{a+b \text {arccosh}\left (d x^2-1\right )} \, dx\) |
\(\Big \downarrow \) 6418 |
\(\displaystyle \frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (d x^2-1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}-\frac {x \sinh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}\left (d x^2-1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}\) |
Input:
Int[(a + b*ArcCosh[-1 + d*x^2])^(-1),x]
Output:
-((x*CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]*Sinh[a/(2*b)])/(Sqrt[ 2]*b*Sqrt[d*x^2])) + (x*Cosh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[-1 + d*x ^2])/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2])
Int[((a_.) + ArcCosh[-1 + (d_.)*(x_)^2]*(b_.))^(-1), x_Symbol] :> Simp[(-x) *Sinh[a/(2*b)]*(CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[d*x^2])), x] + Simp[x*Cosh[a/(2*b)]*(SinhIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]/(Sqrt[2]*b*Sqrt[d*x^2])), x] /; FreeQ[{a, b, d}, x]
\[\int \frac {1}{a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )}d x\]
Input:
int(1/(a+b*arccosh(d*x^2-1)),x)
Output:
int(1/(a+b*arccosh(d*x^2-1)),x)
\[ \int \frac {1}{a+b \text {arccosh}\left (-1+d x^2\right )} \, dx=\int { \frac {1}{b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a} \,d x } \] Input:
integrate(1/(a+b*arccosh(d*x^2-1)),x, algorithm="fricas")
Output:
integral(1/(b*arccosh(d*x^2 - 1) + a), x)
\[ \int \frac {1}{a+b \text {arccosh}\left (-1+d x^2\right )} \, dx=\int \frac {1}{a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}}\, dx \] Input:
integrate(1/(a+b*acosh(d*x**2-1)),x)
Output:
Integral(1/(a + b*acosh(d*x**2 - 1)), x)
\[ \int \frac {1}{a+b \text {arccosh}\left (-1+d x^2\right )} \, dx=\int { \frac {1}{b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a} \,d x } \] Input:
integrate(1/(a+b*arccosh(d*x^2-1)),x, algorithm="maxima")
Output:
integrate(1/(b*arccosh(d*x^2 - 1) + a), x)
\[ \int \frac {1}{a+b \text {arccosh}\left (-1+d x^2\right )} \, dx=\int { \frac {1}{b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a} \,d x } \] Input:
integrate(1/(a+b*arccosh(d*x^2-1)),x, algorithm="giac")
Output:
integrate(1/(b*arccosh(d*x^2 - 1) + a), x)
Timed out. \[ \int \frac {1}{a+b \text {arccosh}\left (-1+d x^2\right )} \, dx=\int \frac {1}{a+b\,\mathrm {acosh}\left (d\,x^2-1\right )} \,d x \] Input:
int(1/(a + b*acosh(d*x^2 - 1)),x)
Output:
int(1/(a + b*acosh(d*x^2 - 1)), x)
\[ \int \frac {1}{a+b \text {arccosh}\left (-1+d x^2\right )} \, dx=\int \frac {1}{\mathit {acosh} \left (d \,x^{2}-1\right ) b +a}d x \] Input:
int(1/(a+b*acosh(d*x^2-1)),x)
Output:
int(1/(acosh(d*x**2 - 1)*b + a),x)