\(\int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx\) [274]

Optimal result

Integrand size = 28, antiderivative size = 154 \[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{2 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3 \sqrt {-1+c x}} \] Output:

-x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arccosh(c*x)) 
-1/2*(-c*x+1)^(1/2)*Chi(4*(a+b*arccosh(c*x))/b)*sinh(4*a/b)/b^2/c^3/(c*x-1 
)^(1/2)+1/2*(-c*x+1)^(1/2)*cosh(4*a/b)*Shi(4*(a+b*arccosh(c*x))/b)/b^2/c^3 
/(c*x-1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-2 b c^2 x^2 \left (-1+c^2 x^2\right )-(a+b \text {arccosh}(c x)) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+(a+b \text {arccosh}(c x)) \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))} \] Input:

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

Output:

(Sqrt[1 - c^2*x^2]*(-2*b*c^2*x^2*(-1 + c^2*x^2) - (a + b*ArcCosh[c*x])*Cos 
hIntegral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] + (a + b*ArcCosh[c*x])*Cos 
h[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])]))/(2*b^2*c^3*Sqrt[-1 + c*x 
]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.29 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.70, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6357, 6302, 25, 5971, 27, 2009, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}

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.

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

\(\Big \downarrow \) 6357

\(\displaystyle \frac {4 c \sqrt {1-c x} \int \frac {x^3}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {2 \sqrt {1-c x} \int \frac {x}{a+b \text {arccosh}(c x)}dx}{b c \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 6302

\(\displaystyle \frac {4 \sqrt {1-c x} \int -\frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}-\frac {2 \sqrt {1-c x} \int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4 \sqrt {1-c x} \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}+\frac {2 \sqrt {1-c x} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {2 \sqrt {1-c x} \int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 (a+b \text {arccosh}(c x))}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}-\frac {4 \sqrt {1-c x} \int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 (a+b \text {arccosh}(c x))}+\frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{4 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {1-c x} \int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}-\frac {4 \sqrt {1-c x} \int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 (a+b \text {arccosh}(c x))}+\frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{4 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {1-c x} \int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}+\frac {4 \sqrt {1-c x} \left (-\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {1-c x} \int -\frac {i \sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}+\frac {4 \sqrt {1-c x} \left (-\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \sqrt {1-c x} \int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}+\frac {4 \sqrt {1-c x} \left (-\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 3784

\(\displaystyle -\frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))+\cosh \left (\frac {2 a}{b}\right ) \int -\frac {i \sinh \left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))\right )}{b^2 c^3 \sqrt {c x-1}}+\frac {4 \sqrt {1-c x} \left (-\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-i \cosh \left (\frac {2 a}{b}\right ) \int \frac {\sinh \left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))\right )}{b^2 c^3 \sqrt {c x-1}}+\frac {4 \sqrt {1-c x} \left (-\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-i \cosh \left (\frac {2 a}{b}\right ) \int -\frac {i \sin \left (\frac {2 i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))\right )}{b^2 c^3 \sqrt {c x-1}}+\frac {4 \sqrt {1-c x} \left (-\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))\right )}{b^2 c^3 \sqrt {c x-1}}+\frac {4 \sqrt {1-c x} \left (-\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 3779

\(\displaystyle -\frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {c x-1}}+\frac {4 \sqrt {1-c x} \left (-\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 3782

\(\displaystyle -\frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {c x-1}}+\frac {4 \sqrt {1-c x} \left (-\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

Input:

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

Output:

-((x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcCosh 
[c*x]))) - (I*Sqrt[1 - c*x]*(I*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b]*Si 
nh[(2*a)/b] - I*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b]))/( 
b^2*c^3*Sqrt[-1 + c*x]) + (4*Sqrt[1 - c*x]*(-1/4*(CoshIntegral[(2*(a + b*A 
rcCosh[c*x]))/b]*Sinh[(2*a)/b]) - (CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b 
]*Sinh[(4*a)/b])/8 + (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/ 
b])/4 + (Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/8))/(b^2* 
c^3*Sqrt[-1 + c*x])
 

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

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 6302

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(278\) vs. \(2(136)=272\).

Time = 0.23 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.81

method	result	size
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (4 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{4} x^{4}+4 b \,c^{5} x^{5}-4 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{2} x^{2}-4 b \,c^{3} x^{3}+\operatorname {arccosh}\left (c x \right ) b \,\operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}-\operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+a \,\operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}-\operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} a \right )}{4 \left (c x +1\right ) c^{3} \left (c x -1\right ) b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\)	\(279\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^{2} \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \] Input:

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

Output:

Integral(x**2*sqrt(-(c*x - 1)*(c*x + 1))/(a + b*acosh(c*x))**2, x)
 

Maxima [F]

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

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

Output:

-((c^2*x^4 - x^2)*(c*x + 1)*sqrt(c*x - 1) + (c^3*x^5 - c*x^3)*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(((4*c^3*x^4 - c*x^2)*(c*x + 
 1)^(3/2)*(c*x - 1) + 2*(4*c^4*x^5 - 4*c^2*x^3 + x)*(c*x + 1)*sqrt(c*x - 1 
) + (4*c^5*x^6 - 7*c^3*x^4 + 3*c*x^2)*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)*sqrt(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)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {-c^{2} x^{2}+1}\, x^{2}}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \] Input:

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

Output:

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