Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\text {Int}\left (\frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2},x\right ) \] Output:
Defer(Int)((-c^2*x^2+1)^(1/2)/x/(a+b*arccosh(c*x))^2,x)
Not integrable
Time = 9.14 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx \] Input:
Integrate[Sqrt[1 - c^2*x^2]/(x*(a + b*ArcCosh[c*x])^2),x]
Output:
Integrate[Sqrt[1 - c^2*x^2]/(x*(a + b*ArcCosh[c*x])^2), x]
Not integrable
Time = 1.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
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.
\(\displaystyle \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx\) |
\(\Big \downarrow \) 6357 |
\(\displaystyle \frac {\sqrt {1-c x} \int \frac {1}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}+\frac {c \sqrt {1-c x} \int \frac {1}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c x (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 6296 |
\(\displaystyle \frac {\sqrt {1-c x} \int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {1}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c x (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {1-c x} \int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {1}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c x (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sqrt {1-c x} \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {1}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c x (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \sqrt {1-c x} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {1}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c x (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))+\cosh \left (\frac {a}{b}\right ) \int -\frac {i \sinh \left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))\right )}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {1}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c x (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-i \cosh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))\right )}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {1}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c x (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-i \cosh \left (\frac {a}{b}\right ) \int -\frac {i \sin \left (\frac {i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))\right )}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {1}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c x (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))\right )}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {1}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c x (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle \frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )\right )}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {1}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c x (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle \frac {\sqrt {1-c x} \int \frac {1}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}+\frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )\right )}{b^2 \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c x (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 6303 |
\(\displaystyle \frac {\sqrt {1-c x} \int \frac {1}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}+\frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )\right )}{b^2 \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c x (a+b \text {arccosh}(c x))}\) |
Input:
Int[Sqrt[1 - c^2*x^2]/(x*(a + b*ArcCosh[c*x])^2),x]
Output:
$Aborted
Not integrable
Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
\[\int \frac {\sqrt {-c^{2} x^{2}+1}}{x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}d x\]
Input:
int((-c^2*x^2+1)^(1/2)/x/(a+b*arccosh(c*x))^2,x)
Output:
int((-c^2*x^2+1)^(1/2)/x/(a+b*arccosh(c*x))^2,x)
Not integrable
Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x} \,d x } \] Input:
integrate((-c^2*x^2+1)^(1/2)/x/(a+b*arccosh(c*x))^2,x, algorithm="fricas")
Output:
integral(sqrt(-c^2*x^2 + 1)/(b^2*x*arccosh(c*x)^2 + 2*a*b*x*arccosh(c*x) + a^2*x), x)
Not integrable
Time = 2.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \] Input:
integrate((-c**2*x**2+1)**(1/2)/x/(a+b*acosh(c*x))**2,x)
Output:
Integral(sqrt(-(c*x - 1)*(c*x + 1))/(x*(a + b*acosh(c*x))**2), x)
Not integrable
Time = 0.53 (sec) , antiderivative size = 442, normalized size of antiderivative = 15.79 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x} \,d x } \] Input:
integrate((-c^2*x^2+1)^(1/2)/x/(a+b*arccosh(c*x))^2,x, algorithm="maxima")
Output:
-((c^2*x^2 - 1)*(c*x + 1)*sqrt(c*x - 1) + (c^3*x^3 - c*x)*sqrt(c*x + 1))*s qrt(-c*x + 1)/(a*b*c^3*x^3 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x^2 - a*b *c*x + (b^2*c^3*x^3 + sqrt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x^2 - b^2*c*x)*l og(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) + integrate(((c^3*x^3 + 2*c*x)*(c*x + 1)^(3/2)*(c*x - 1) + (2*c^4*x^4 + c^2*x^2 - 1)*(c*x + 1)*sqrt(c*x - 1) + (c^5*x^5 - c^3*x^3)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^5*x^6 + (c*x + 1)*(c*x - 1)*a*b*c^3*x^4 - 2*a*b*c^3*x^4 + a*b*c*x^2 + 2*(a*b*c^4*x^5 - a* b*c^2*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^6 + (c*x + 1)*(c*x - 1 )*b^2*c^3*x^4 - 2*b^2*c^3*x^4 + b^2*c*x^2 + 2*(b^2*c^4*x^5 - b^2*c^2*x^3)* sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)
Exception generated. \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*x^2+1)^(1/2)/x/(a+b*arccosh(c*x))^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Not integrable
Time = 3.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {1-c^2\,x^2}}{x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \] Input:
int((1 - c^2*x^2)^(1/2)/(x*(a + b*acosh(c*x))^2),x)
Output:
int((1 - c^2*x^2)^(1/2)/(x*(a + b*acosh(c*x))^2), x)
Not integrable
Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {-c^{2} x^{2}+1}}{\mathit {acosh} \left (c x \right )^{2} b^{2} x +2 \mathit {acosh} \left (c x \right ) a b x +a^{2} x}d x \] Input:
int((-c^2*x^2+1)^(1/2)/x/(a+b*acosh(c*x))^2,x)
Output:
int(sqrt( - c**2*x**2 + 1)/(acosh(c*x)**2*b**2*x + 2*acosh(c*x)*a*b*x + a* *2*x),x)