3.4.36 \(\int \frac {x (1-c^2 x^2)^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx\) [336]

3.4.36.1 Optimal result

Integrand size = 26, antiderivative size = 448 \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}+\frac {5 \sqrt {1-c x} \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}-\frac {27 \sqrt {1-c x} \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}+\frac {25 \sqrt {1-c x} \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}-\frac {7 \sqrt {1-c x} \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {7 a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}-\frac {5 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}+\frac {27 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}-\frac {25 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}}+\frac {7 \sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x}} \]

output

-5/64*cosh(a/b)*Shi((a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^( 
1/2)+27/64*cosh(3*a/b)*Shi(3*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^2/ 
(c*x-1)^(1/2)-25/64*cosh(5*a/b)*Shi(5*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2) 
/b^2/c^2/(c*x-1)^(1/2)+7/64*cosh(7*a/b)*Shi(7*(a+b*arccosh(c*x))/b)*(-c*x+ 
1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)+5/64*Chi((a+b*arccosh(c*x))/b)*sinh(a/b)*(- 
c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)-27/64*Chi(3*(a+b*arccosh(c*x))/b)*sinh( 
3*a/b)*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)+25/64*Chi(5*(a+b*arccosh(c*x)) 
/b)*sinh(5*a/b)*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)-7/64*Chi(7*(a+b*arcco 
sh(c*x))/b)*sinh(7*a/b)*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)-x*(-c^2*x^2+1 
)^(5/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))
 

3.4.36.2 Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.97 \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (-64 b c x+192 b c^3 x^3-192 b c^5 x^5+64 b c^7 x^7-5 (a+b \text {arccosh}(c x)) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )+27 (a+b \text {arccosh}(c x)) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-25 a \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-25 b \text {arccosh}(c x) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )+7 a \text {Chi}\left (7 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {7 a}{b}\right )+7 b \text {arccosh}(c x) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {7 a}{b}\right )+5 a \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+5 b \text {arccosh}(c x) \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-27 a \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-27 b \text {arccosh}(c x) \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+25 a \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+25 b \text {arccosh}(c x) \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-7 a \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-7 b \text {arccosh}(c x) \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{64 b^2 c^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \]

input

Integrate[(x*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x])^2,x]
 

output

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-64*b*c*x + 192*b*c^3*x^3 - 192*b*c^5*x^5 + 
 64*b*c^7*x^7 - 5*(a + b*ArcCosh[c*x])*CoshIntegral[a/b + ArcCosh[c*x]]*Si 
nh[a/b] + 27*(a + b*ArcCosh[c*x])*CoshIntegral[3*(a/b + ArcCosh[c*x])]*Sin 
h[(3*a)/b] - 25*a*CoshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] - 25* 
b*ArcCosh[c*x]*CoshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] + 7*a*Co 
shIntegral[7*(a/b + ArcCosh[c*x])]*Sinh[(7*a)/b] + 7*b*ArcCosh[c*x]*CoshIn 
tegral[7*(a/b + ArcCosh[c*x])]*Sinh[(7*a)/b] + 5*a*Cosh[a/b]*SinhIntegral[ 
a/b + ArcCosh[c*x]] + 5*b*ArcCosh[c*x]*Cosh[a/b]*SinhIntegral[a/b + ArcCos 
h[c*x]] - 27*a*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] - 27*b*A 
rcCosh[c*x]*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 25*a*Cosh 
[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] + 25*b*ArcCosh[c*x]*Cosh[(5 
*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] - 7*a*Cosh[(7*a)/b]*SinhIntegr 
al[7*(a/b + ArcCosh[c*x])] - 7*b*ArcCosh[c*x]*Cosh[(7*a)/b]*SinhIntegral[7 
*(a/b + ArcCosh[c*x])]))/(64*b^2*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x] 
))
 

3.4.36.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.85 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6357, 6304, 6321, 25, 3042, 26, 3793, 2009, 6327, 6367, 25, 5971, 2009}

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.

\(\displaystyle \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx\)

\(\Big \downarrow \) 6357

\(\displaystyle \frac {7 c \sqrt {1-c x} \int \frac {x^2 (1-c x)^2 (c x+1)^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {\sqrt {1-c x} \int \frac {(1-c x)^2 (c x+1)^2}{a+b \text {arccosh}(c x)}dx}{b c \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 6304

\(\displaystyle -\frac {\sqrt {1-c x} \int \frac {\left (1-c^2 x^2\right )^2}{a+b \text {arccosh}(c x)}dx}{b c \sqrt {c x-1}}+\frac {7 c \sqrt {1-c x} \int \frac {x^2 (1-c x)^2 (c x+1)^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 6321

\(\displaystyle -\frac {\sqrt {1-c x} \int -\frac {\sinh ^5\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}+\frac {7 c \sqrt {1-c x} \int \frac {x^2 (1-c x)^2 (c x+1)^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {1-c x} \int \frac {\sinh ^5\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}+\frac {7 c \sqrt {1-c x} \int \frac {x^2 (1-c x)^2 (c x+1)^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {1-c x} \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^5}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}+\frac {7 c \sqrt {1-c x} \int \frac {x^2 (1-c x)^2 (c x+1)^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \sqrt {1-c x} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^5}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}+\frac {7 c \sqrt {1-c x} \int \frac {x^2 (1-c x)^2 (c x+1)^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {i \sqrt {1-c x} \int \left (\frac {i \sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}-\frac {5 i \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}+\frac {5 i \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}+\frac {7 c \sqrt {1-c x} \int \frac {x^2 (1-c x)^2 (c x+1)^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {7 c \sqrt {1-c x} \int \frac {x^2 (1-c x)^2 (c x+1)^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {i \sqrt {1-c x} \left (\frac {5}{8} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {5}{16} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} i \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{8} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {5}{16} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{16} i \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 6327

\(\displaystyle \frac {7 c \sqrt {1-c x} \int \frac {x^2 \left (1-c^2 x^2\right )^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {i \sqrt {1-c x} \left (\frac {5}{8} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {5}{16} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} i \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{8} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {5}{16} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{16} i \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 6367

\(\displaystyle \frac {7 \sqrt {1-c x} \int -\frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^5\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}-\frac {i \sqrt {1-c x} \left (\frac {5}{8} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {5}{16} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} i \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{8} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {5}{16} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{16} i \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {7 \sqrt {1-c x} \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^5\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}-\frac {i \sqrt {1-c x} \left (\frac {5}{8} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {5}{16} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} i \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{8} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {5}{16} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{16} i \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {7 \sqrt {1-c x} \int \left (\frac {\sinh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 (a+b \text {arccosh}(c x))}-\frac {3 \sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 (a+b \text {arccosh}(c x))}+\frac {\sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 (a+b \text {arccosh}(c x))}+\frac {5 \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}-\frac {i \sqrt {1-c x} \left (\frac {5}{8} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {5}{16} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} i \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{8} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {5}{16} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{16} i \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {i \sqrt {1-c x} \left (\frac {5}{8} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {5}{16} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} i \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{8} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {5}{16} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{16} i \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}+\frac {7 \sqrt {1-c x} \left (-\frac {5}{64} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{64} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {3}{64} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{64} \sinh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )+\frac {5}{64} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{64} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{64} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{64} \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

input

Int[(x*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x])^2,x]
 

output

-((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(b*c*(a + b*ArcCosh 
[c*x]))) - (I*Sqrt[1 - c*x]*(((5*I)/8)*CoshIntegral[(a + b*ArcCosh[c*x])/b 
]*Sinh[a/b] - ((5*I)/16)*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3* 
a)/b] + (I/16)*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b]*Sinh[(5*a)/b] - (( 
5*I)/8)*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b] + ((5*I)/16)*Cosh[( 
3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b] - (I/16)*Cosh[(5*a)/b]*Si 
nhIntegral[(5*(a + b*ArcCosh[c*x]))/b]))/(b^2*c^2*Sqrt[-1 + c*x]) + (7*Sqr 
t[1 - c*x]*((-5*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/64 - (Cosh 
Integral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/64 + (3*CoshIntegral[( 
5*(a + b*ArcCosh[c*x]))/b]*Sinh[(5*a)/b])/64 - (CoshIntegral[(7*(a + b*Arc 
Cosh[c*x]))/b]*Sinh[(7*a)/b])/64 + (5*Cosh[a/b]*SinhIntegral[(a + b*ArcCos 
h[c*x])/b])/64 + (Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/ 
64 - (3*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/64 + (Cosh 
[(7*a)/b]*SinhIntegral[(7*(a + b*ArcCosh[c*x]))/b])/64))/(b^2*c^2*Sqrt[-1 
+ c*x])
 

3.4.36.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6304

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( 
(d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*A 
rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + 
 d1*e2, 0] && IntegerQ[p]
 

rule 6321

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] 
  Subst[Int[x^n*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] 
/; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0]
 

rule 6327

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( 
e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 
*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 
, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
 

rule 6357

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c* 
x]*(d + e*x^2)^p]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Simp[ 
f*(m/(b*c*(n + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[(f 
*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^( 
n + 1), x], x] - Simp[c*((m + 2*p + 1)/(b*f*(n + 1)))*Simp[(d + e*x^2)^p/(( 
1 + c*x)^p*(-1 + c*x)^p)]   Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x 
)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, 
f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 
2*p + 1, 0] && IGtQ[m, -3]
 

rule 6367

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x 
)^p*(-1 + c*x)^p)]   Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p 
 + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && Eq 
Q[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

3.4.36.4 Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 759, normalized size of antiderivative = 1.69

method	result	size
default	\(\text {Expression too large to display}\)	\(759\)

input

int(x*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)
 

output

-1/128*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(38 
4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^5*x^5-384*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b* 
c^3*x^3+128*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c*x-128*(c*x-1)^(1/2)*(c*x+1)^(1 
/2)*b*c^7*x^7+384*b*c^6*x^6+5*a*Ei(1,-arccosh(c*x)-a/b)*exp(-(-b*arccosh(c 
*x)+a)/b)+25*a*Ei(1,-5*arccosh(c*x)-5*a/b)*exp(-(-b*arccosh(c*x)+5*a)/b)-2 
7*a*Ei(1,-3*arccosh(c*x)-3*a/b)*exp(-(-b*arccosh(c*x)+3*a)/b)-7*a*Ei(1,-7* 
arccosh(c*x)-7*a/b)*exp(-(-b*arccosh(c*x)+7*a)/b)+7*Ei(1,7*arccosh(c*x)+7* 
a/b)*exp((b*arccosh(c*x)+7*a)/b)*a-25*Ei(1,5*arccosh(c*x)+5*a/b)*exp((b*ar 
ccosh(c*x)+5*a)/b)*a+27*Ei(1,3*arccosh(c*x)+3*a/b)*exp((b*arccosh(c*x)+3*a 
)/b)*a-5*Ei(1,arccosh(c*x)+a/b)*exp((a+b*arccosh(c*x))/b)*a+5*arccosh(c*x) 
*b*Ei(1,-arccosh(c*x)-a/b)*exp(-(-b*arccosh(c*x)+a)/b)+25*arccosh(c*x)*b*E 
i(1,-5*arccosh(c*x)-5*a/b)*exp(-(-b*arccosh(c*x)+5*a)/b)+7*Ei(1,7*arccosh( 
c*x)+7*a/b)*exp((b*arccosh(c*x)+7*a)/b)*b*arccosh(c*x)-27*arccosh(c*x)*b*E 
i(1,-3*arccosh(c*x)-3*a/b)*exp(-(-b*arccosh(c*x)+3*a)/b)-7*arccosh(c*x)*b* 
Ei(1,-7*arccosh(c*x)-7*a/b)*exp(-(-b*arccosh(c*x)+7*a)/b)-25*Ei(1,5*arccos 
h(c*x)+5*a/b)*exp((b*arccosh(c*x)+5*a)/b)*b*arccosh(c*x)+27*Ei(1,3*arccosh 
(c*x)+3*a/b)*exp((b*arccosh(c*x)+3*a)/b)*b*arccosh(c*x)-5*Ei(1,arccosh(c*x 
)+a/b)*exp((a+b*arccosh(c*x))/b)*b*arccosh(c*x)-384*b*c^4*x^4+128*b*c^2*x^ 
2-128*b*c^8*x^8)/(c*x+1)/c^2/(c*x-1)/b^2/(a+b*arccosh(c*x))
 

3.4.36.5 Fricas [F]

\[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")
 

output

integral((c^4*x^5 - 2*c^2*x^3 + x)*sqrt(-c^2*x^2 + 1)/(b^2*arccosh(c*x)^2 
+ 2*a*b*arccosh(c*x) + a^2), x)
 

3.4.36.6 Sympy [F(-1)]

Timed out. \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\text {Timed out} \]

input

integrate(x*(-c**2*x**2+1)**(5/2)/(a+b*acosh(c*x))**2,x)
 

output

Timed out
 

3.4.36.7 Maxima [F]

\[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")
 

output

-((c^6*x^7 - 3*c^4*x^5 + 3*c^2*x^3 - x)*(c*x + 1)*sqrt(c*x - 1) + (c^7*x^8 
 - 3*c^5*x^6 + 3*c^3*x^4 - c*x^2)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^3*x 
^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + sqrt(c 
*x + 1)*sqrt(c*x - 1)*b^2*c^2*x - b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c*x 
- 1))) + integrate((7*(c^7*x^7 - 2*c^5*x^5 + c^3*x^3)*(c*x + 1)^(3/2)*(c*x 
 - 1) + (14*c^8*x^8 - 37*c^6*x^6 + 33*c^4*x^4 - 11*c^2*x^2 + 1)*(c*x + 1)* 
sqrt(c*x - 1) + (7*c^9*x^9 - 23*c^7*x^7 + 27*c^5*x^5 - 13*c^3*x^3 + 2*c*x) 
*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^5*x^4 + (c*x + 1)*(c*x - 1)*a*b*c^3* 
x^2 - 2*a*b*c^3*x^2 + a*b*c + 2*(a*b*c^4*x^3 - a*b*c^2*x)*sqrt(c*x + 1)*sq 
rt(c*x - 1) + (b^2*c^5*x^4 + (c*x + 1)*(c*x - 1)*b^2*c^3*x^2 - 2*b^2*c^3*x 
^2 + b^2*c + 2*(b^2*c^4*x^3 - b^2*c^2*x)*sqrt(c*x + 1)*sqrt(c*x - 1))*log( 
c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)
 

3.4.36.8 Giac [F]

\[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="giac")
 

output

integrate((-c^2*x^2 + 1)^(5/2)*x/(b*arccosh(c*x) + a)^2, x)
 

3.4.36.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x\,{\left (1-c^2\,x^2\right )}^{5/2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

input

int((x*(1 - c^2*x^2)^(5/2))/(a + b*acosh(c*x))^2,x)
 

output

int((x*(1 - c^2*x^2)^(5/2))/(a + b*acosh(c*x))^2, x)