Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\text {Int}\left (\frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))^2},x\right ) \] Output:
Defer(Int)((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x))^2,x)
Not integrable
Time = 16.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))^2} \, dx \] Input:
Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcCosh[c*x])^2),x]
Output:
Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcCosh[c*x])^2), x]
Not integrable
Time = 2.57 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 11, 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.
| \(\displaystyle \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))^2} \, dx\) |
| \(\Big \downarrow \) 6357 |
| \(\displaystyle \frac {\sqrt {1-c x} \int \frac {(1-c x)^2 (c x+1)^2}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}+\frac {5 c \sqrt {1-c x} \int \frac {(1-c x)^2 (c x+1)^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c x (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle \frac {5 c \sqrt {1-c x} \int \frac {\left (1-c^2 x^2\right )^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {(1-c x)^2 (c x+1)^2}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c x (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 6321 |
| \(\displaystyle \frac {5 \sqrt {1-c x} \int -\frac {\sinh ^5\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {(1-c x)^2 (c x+1)^2}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c x (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5 \sqrt {1-c x} \int \frac {\sinh ^5\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {(1-c x)^2 (c x+1)^2}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c x (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 \sqrt {1-c x} \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^5}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {(1-c x)^2 (c x+1)^2}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c x (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {5 i \sqrt {1-c x} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^5}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {(1-c x)^2 (c x+1)^2}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c x (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {5 i \sqrt {1-c x} \int \left (\frac {i \sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}-\frac {5 i \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}+\frac {5 i \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int \frac {(1-c x)^2 (c x+1)^2}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c x (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-c x} \int \frac {(1-c x)^2 (c x+1)^2}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}+\frac {5 i \sqrt {1-c x} \left (\frac {5}{8} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {5}{16} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} i \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{8} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {5}{16} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{16} i \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c x (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {\sqrt {1-c x} \int \frac {\left (1-c^2 x^2\right )^2}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}+\frac {5 i \sqrt {1-c x} \left (\frac {5}{8} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {5}{16} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} i \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{8} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {5}{16} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{16} i \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c x (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 6375 |
| \(\displaystyle \frac {\sqrt {1-c x} \int \frac {\left (1-c^2 x^2\right )^2}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}+\frac {5 i \sqrt {1-c x} \left (\frac {5}{8} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {5}{16} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} i \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{8} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {5}{16} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{16} i \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c x (a+b \text {arccosh}(c x))}\) |
Input:
Int[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcCosh[c*x])^2),x]
Output:
$Aborted
Not integrable
Time = 0.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
\[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}d x\]
Input:
int((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x))^2,x)
Output:
int((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x))^2,x)
Not integrable
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x} \,d x } \] Input:
integrate((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x))^2,x, algorithm="fricas")
Output:
integral((c^4*x^4 - 2*c^2*x^2 + 1)*sqrt(-c^2*x^2 + 1)/(b^2*x*arccosh(c*x)^ 2 + 2*a*b*x*arccosh(c*x) + a^2*x), x)
Timed out. \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\text {Timed out} \] Input:
integrate((-c**2*x**2+1)**(5/2)/x/(a+b*acosh(c*x))**2,x)
Output:
Timed out
Not integrable
Time = 0.90 (sec) , antiderivative size = 524, normalized size of antiderivative = 18.71 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x} \,d x } \] Input:
integrate((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x))^2,x, algorithm="maxima")
Output:
-((c^6*x^6 - 3*c^4*x^4 + 3*c^2*x^2 - 1)*(c*x + 1)*sqrt(c*x - 1) + (c^7*x^7 - 3*c^5*x^5 + 3*c^3*x^3 - c*x)*sqrt(c*x + 1))*sqrt(-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)*log(c*x + sqrt(c*x + 1)*sqr t(c*x - 1))) + integrate(((5*c^7*x^7 - 8*c^5*x^5 + c^3*x^3 + 2*c*x)*(c*x + 1)^(3/2)*(c*x - 1) + (10*c^8*x^8 - 23*c^6*x^6 + 15*c^4*x^4 - c^2*x^2 - 1) *(c*x + 1)*sqrt(c*x - 1) + 5*(c^9*x^9 - 3*c^7*x^7 + 3*c^5*x^5 - c^3*x^3)*s qrt(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. \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*x^2+1)^(5/2)/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 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{5/2}}{x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \] Input:
int((1 - c^2*x^2)^(5/2)/(x*(a + b*acosh(c*x))^2),x)
Output:
int((1 - c^2*x^2)^(5/2)/(x*(a + b*acosh(c*x))^2), x)
Not integrable
Time = 0.26 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.71 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {-c^{2} x^{2}+1}}{\mathit {acosh} \left (c x \right )^{2} b^{2} x +2 \mathit {acosh} \left (c x \right ) a b x +a^{2} x}d x +\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, x^{3}}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}-2 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, x}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) c^{2} \] Input:
int((-c^2*x^2+1)^(5/2)/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) + int((sqrt( - c**2*x**2 + 1)*x**3)/(acosh(c*x)**2*b**2 + 2*acosh (c*x)*a*b + a**2),x)*c**4 - 2*int((sqrt( - c**2*x**2 + 1)*x)/(acosh(c*x)** 2*b**2 + 2*acosh(c*x)*a*b + a**2),x)*c**2