Integrand size = 14, antiderivative size = 98 \[ \int \frac {1}{a+b \text {arccosh}\left (1+d x^2\right )} \, dx=\frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}\left (1+d x^2\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}-\frac {x \sinh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (1+d x^2\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}} \] Output:
1/2*x*cosh(1/2*a/b)*Chi(1/2*(a+b*arccosh(d*x^2+1))/b)*2^(1/2)/b/(d*x^2)^(1 /2)-1/2*x*sinh(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 = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.20 \[ \int \frac {1}{a+b \text {arccosh}\left (1+d x^2\right )} \, dx=\frac {x \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}\left (1+d x^2\right )}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (1+d x^2\right )}{2 b}\right )\right )}{b \sqrt {d x^2} \sqrt {\frac {d x^2}{2+d x^2}} \sqrt {2+d x^2}} \] Input:
Integrate[(a + b*ArcCosh[1 + d*x^2])^(-1),x]
Output:
(x*Sinh[ArcCosh[1 + d*x^2]/2]*(Cosh[a/(2*b)]*CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)] - Sinh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[1 + d*x^2])/ (2*b)]))/(b*Sqrt[d*x^2]*Sqrt[(d*x^2)/(2 + d*x^2)]*Sqrt[2 + d*x^2])
Time = 0.24 (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 = {6417}
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.
\(\displaystyle \int \frac {1}{a+b \text {arccosh}\left (d x^2+1\right )} \, dx\) |
\(\Big \downarrow \) 6417 |
\(\displaystyle \frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}\left (d x^2+1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}-\frac {x \sinh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (d x^2+1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}\) |
Input:
Int[(a + b*ArcCosh[1 + d*x^2])^(-1),x]
Output:
(x*Cosh[a/(2*b)]*CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)])/(Sqrt[2]* b*Sqrt[d*x^2]) - (x*Sinh[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*Cos h[a/(2*b)]*(CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)]/(Sqrt[2]*b*Sqrt[ d*x^2])), x] - Simp[x*Sinh[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)