Integrand size = 14, antiderivative size = 150 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2} \, dx=-\frac {\sqrt {d x^2} \sqrt {2+d x^2}}{2 b d x \left (a+b \text {arccosh}\left (1+d x^2\right )\right )}-\frac {x \text {Chi}\left (\frac {a+b \text {arccosh}\left (1+d x^2\right )}{2 b}\right ) \sinh \left (\frac {a}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}+\frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}} \] Output:
-1/2*(d*x^2)^(1/2)*(d*x^2+2)^(1/2)/b/d/x/(a+b*arccosh(d*x^2+1))-1/4*x*Chi( 1/2*(a+b*arccosh(d*x^2+1))/b)*sinh(1/2*a/b)*2^(1/2)/b^2/(d*x^2)^(1/2)+1/4* x*cosh(1/2*a/b)*Shi(1/2*(a+b*arccosh(d*x^2+1))/b)*2^(1/2)/b^2/(d*x^2)^(1/2 )
Time = 0.78 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2} \, dx=-\frac {\frac {2 b \sqrt {d x^2} \sqrt {2+d x^2}}{a d+b d \text {arccosh}\left (1+d x^2\right )}+x^2 \text {csch}\left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right ) \left (\text {Chi}\left (\frac {a+b \text {arccosh}\left (1+d x^2\right )}{2 b}\right ) \sinh \left (\frac {a}{2 b}\right )-\cosh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (1+d x^2\right )}{2 b}\right )\right )}{4 b^2 x} \] Input:
Integrate[(a + b*ArcCosh[1 + d*x^2])^(-2),x]
Output:
-1/4*((2*b*Sqrt[d*x^2]*Sqrt[2 + d*x^2])/(a*d + b*d*ArcCosh[1 + d*x^2]) + x ^2*Csch[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^2*x)
Time = 0.26 (sec) , antiderivative size = 150, 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 = {6423}
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.
\(\displaystyle \int \frac {1}{\left (a+b \text {arccosh}\left (d x^2+1\right )\right )^2} \, dx\) |
\(\Big \downarrow \) 6423 |
\(\displaystyle -\frac {x \sinh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}\left (d x^2+1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}+\frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (d x^2+1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}-\frac {\sqrt {d x^2} \sqrt {d x^2+2}}{2 b d x \left (a+b \text {arccosh}\left (d x^2+1\right )\right )}\) |
Input:
Int[(a + b*ArcCosh[1 + d*x^2])^(-2),x]
Output:
-1/2*(Sqrt[d*x^2]*Sqrt[2 + d*x^2])/(b*d*x*(a + b*ArcCosh[1 + d*x^2])) - (x *CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)]*Sinh[a/(2*b)])/(2*Sqrt[2]* b^2*Sqrt[d*x^2]) + (x*Cosh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[1 + d*x^2] )/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[d*x^2])
Int[((a_.) + ArcCosh[1 + (d_.)*(x_)^2]*(b_.))^(-2), x_Symbol] :> Simp[(-Sqr t[d*x^2])*(Sqrt[2 + d*x^2]/(2*b*d*x*(a + b*ArcCosh[1 + d*x^2]))), x] + (-Si mp[x*Sinh[a/(2*b)]*(CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)]/(2*Sqrt[ 2]*b^2*Sqrt[d*x^2])), x] + Simp[x*Cosh[a/(2*b)]*(SinhIntegral[(a + b*ArcCos h[1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2*Sqrt[d*x^2])), x]) /; FreeQ[{a, b, d}, x]
\[\int \frac {1}{{\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}+1\right )\right )}^{2}}d x\]
Input:
int(1/(a+b*arccosh(d*x^2+1))^2,x)
Output:
int(1/(a+b*arccosh(d*x^2+1))^2,x)
\[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} + 1\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(1/(a+b*arccosh(d*x^2+1))^2,x, algorithm="fricas")
Output:
integral(1/(b^2*arccosh(d*x^2 + 1)^2 + 2*a*b*arccosh(d*x^2 + 1) + a^2), x)
\[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \operatorname {acosh}{\left (d x^{2} + 1 \right )}\right )^{2}}\, dx \] Input:
integrate(1/(a+b*acosh(d*x**2+1))**2,x)
Output:
Integral((a + b*acosh(d*x**2 + 1))**(-2), x)
\[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} + 1\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(1/(a+b*arccosh(d*x^2+1))^2,x, algorithm="maxima")
Output:
-1/2*(d^2*x^4 + 3*d*x^2 + (d^(3/2)*x^3 + 2*sqrt(d)*x)*sqrt(d*x^2 + 2) + 2) /(a*b*d^2*x^3 + 2*a*b*d*x + (b^2*d^2*x^3 + 2*b^2*d*x + (b^2*d^(3/2)*x^2 + b^2*sqrt(d))*sqrt(d*x^2 + 2))*log(d*x^2 + sqrt(d*x^2 + 2)*sqrt(d)*x + 1) + (a*b*d^(3/2)*x^2 + a*b*sqrt(d))*sqrt(d*x^2 + 2)) + integrate(1/2*(d^3*x^6 + 3*d^2*x^4 + (d^2*x^4 + d*x^2 + 2)*(d*x^2 + 2) + (2*d^(5/2)*x^5 + 4*d^(3 /2)*x^3 + sqrt(d)*x)*sqrt(d*x^2 + 2) - 4)/(a*b*d^3*x^6 + 4*a*b*d^2*x^4 + 4 *a*b*d*x^2 + (a*b*d^2*x^4 + 2*a*b*d*x^2 + a*b)*(d*x^2 + 2) + (b^2*d^3*x^6 + 4*b^2*d^2*x^4 + 4*b^2*d*x^2 + (b^2*d^2*x^4 + 2*b^2*d*x^2 + b^2)*(d*x^2 + 2) + 2*(b^2*d^(5/2)*x^5 + 3*b^2*d^(3/2)*x^3 + 2*b^2*sqrt(d)*x)*sqrt(d*x^2 + 2))*log(d*x^2 + sqrt(d*x^2 + 2)*sqrt(d)*x + 1) + 2*(a*b*d^(5/2)*x^5 + 3 *a*b*d^(3/2)*x^3 + 2*a*b*sqrt(d)*x)*sqrt(d*x^2 + 2)), x)
\[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} + 1\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(1/(a+b*arccosh(d*x^2+1))^2,x, algorithm="giac")
Output:
integrate((b*arccosh(d*x^2 + 1) + a)^(-2), x)
Timed out. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (d\,x^2+1\right )\right )}^2} \,d x \] Input:
int(1/(a + b*acosh(d*x^2 + 1))^2,x)
Output:
int(1/(a + b*acosh(d*x^2 + 1))^2, x)
\[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2} \, dx=\int \frac {1}{\mathit {acosh} \left (d \,x^{2}+1\right )^{2} b^{2}+2 \mathit {acosh} \left (d \,x^{2}+1\right ) a b +a^{2}}d x \] Input:
int(1/(a+b*acosh(d*x^2+1))^2,x)
Output:
int(1/(acosh(d*x**2 + 1)**2*b**2 + 2*acosh(d*x**2 + 1)*a*b + a**2),x)