\(\int (c-a^2 c x^2)^{3/2} \sqrt {\text {arccosh}(a x)} \, dx\) [82]

Optimal result

Integrand size = 24, antiderivative size = 351 \[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\text {arccosh}(a x)} \, dx=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\text {arccosh}(a x)}-\frac {c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}} \] Output:

3/8*c*x*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(1/2)+1/4*x*(-a^2*c*x^2+c)^(3/2) 
*arccosh(a*x)^(1/2)-1/4*c*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(3/2)/a/(a*x-1 
)^(1/2)/(a*x+1)^(1/2)-1/256*c*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erf(2*arccosh( 
a*x)^(1/2))/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+1/32*c*2^(1/2)*Pi^(1/2)*(-a^2*c* 
x^2+c)^(1/2)*erf(2^(1/2)*arccosh(a*x)^(1/2))/a/(a*x-1)^(1/2)/(a*x+1)^(1/2) 
+1/256*c*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erfi(2*arccosh(a*x)^(1/2))/a/(a*x-1 
)^(1/2)/(a*x+1)^(1/2)-1/32*c*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erfi(2^ 
(1/2)*arccosh(a*x)^(1/2))/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.44 \[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\text {arccosh}(a x)} \, dx=-\frac {c \sqrt {c-a^2 c x^2} \left (-\sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},-4 \text {arccosh}(a x)\right )+8 \sqrt {2} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},-2 \text {arccosh}(a x)\right )+\sqrt {\text {arccosh}(a x)} \left (32 \text {arccosh}(a x)^{3/2}+8 \sqrt {2} \Gamma \left (\frac {3}{2},2 \text {arccosh}(a x)\right )-\Gamma \left (\frac {3}{2},4 \text {arccosh}(a x)\right )\right )\right )}{128 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\text {arccosh}(a x)}} \] Input:

Integrate[(c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]],x]
 

Output:

-1/128*(c*Sqrt[c - a^2*c*x^2]*(-(Sqrt[-ArcCosh[a*x]]*Gamma[3/2, -4*ArcCosh 
[a*x]]) + 8*Sqrt[2]*Sqrt[-ArcCosh[a*x]]*Gamma[3/2, -2*ArcCosh[a*x]] + Sqrt 
[ArcCosh[a*x]]*(32*ArcCosh[a*x]^(3/2) + 8*Sqrt[2]*Gamma[3/2, 2*ArcCosh[a*x 
]] - Gamma[3/2, 4*ArcCosh[a*x]])))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) 
*Sqrt[ArcCosh[a*x]])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.97 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {6312, 25, 6310, 6302, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634, 6308, 6327, 6367, 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.

\(\displaystyle \int \sqrt {\text {arccosh}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx\)

\(\Big \downarrow \) 6312

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6310

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

\(\Big \downarrow \) 6302

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3789

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

\(\displaystyle \frac {3}{4} c \left (\frac {i \sqrt {c-a^2 c x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-i \int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\right )-\frac {a c \sqrt {c-a^2 c x^2} \int \frac {x (1-a x) (a x+1)}{\sqrt {\text {arccosh}(a x)}}dx}{8 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \sqrt {\text {arccosh}(a x)} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {3}{4} c \left (-\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {i \sqrt {c-a^2 c x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\right )-\frac {a c \sqrt {c-a^2 c x^2} \int \frac {x (1-a x) (a x+1)}{\sqrt {\text {arccosh}(a x)}}dx}{8 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \sqrt {\text {arccosh}(a x)} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 6308

\(\displaystyle -\frac {a c \sqrt {c-a^2 c x^2} \int \frac {x (1-a x) (a x+1)}{\sqrt {\text {arccosh}(a x)}}dx}{8 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{4} c \left (\frac {i \sqrt {c-a^2 c x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{3 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \sqrt {\text {arccosh}(a x)} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 6327

\(\displaystyle -\frac {a c \sqrt {c-a^2 c x^2} \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\text {arccosh}(a x)}}dx}{8 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{4} c \left (\frac {i \sqrt {c-a^2 c x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{3 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \sqrt {\text {arccosh}(a x)} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 6367

\(\displaystyle \frac {c \sqrt {c-a^2 c x^2} \int \frac {a x \left (\frac {a x-1}{a x+1}\right )^{3/2} (a x+1)^3}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{4} c \left (\frac {i \sqrt {c-a^2 c x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{3 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \sqrt {\text {arccosh}(a x)} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c \sqrt {c-a^2 c x^2} \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{4} c \left (\frac {i \sqrt {c-a^2 c x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{3 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \sqrt {\text {arccosh}(a x)} \left (c-a^2 c x^2\right )^{3/2}\)

Input:

Int[(c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]],x]
 

Output:

(x*(c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]])/4 + (c*Sqrt[c - a^2*c*x^2]*(- 
1/32*(Sqrt[Pi]*Erf[2*Sqrt[ArcCosh[a*x]]]) + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[A 
rcCosh[a*x]]])/8 + (Sqrt[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/32 - (Sqrt[Pi/2]* 
Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/8))/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + 
 (3*c*((x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/2 - (Sqrt[c - a^2*c*x^2] 
*ArcCosh[a*x]^(3/2))/(3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + ((I/8)*Sqrt[c - 
a^2*c*x^2]*((-1/2*I)*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + (I/2)*Sq 
rt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]]))/(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x 
])))/4
 

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 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

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

rule 3789

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
/2   Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2   Int[(c + d*x)^m*E 
^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
 

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 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
 

rule 6310

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCosh[c*x])^n/2), x] + (-Simp[( 
1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[(a + b*ArcC 
osh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sq 
rt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[x*(a + b*ArcCosh[c*x])^ 
(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n 
, 0]
 

rule 6312

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

Maple [F]

\[\int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\operatorname {arccosh}\left (a x \right )}d x\]

Input:

int((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(1/2),x)
 

Output:

int((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(1/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\text {arccosh}(a x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(1/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\text {arccosh}(a x)} \, dx=\int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {\operatorname {acosh}{\left (a x \right )}}\, dx \] Input:

integrate((-a**2*c*x**2+c)**(3/2)*acosh(a*x)**(1/2),x)
 

Output:

Integral((-c*(a*x - 1)*(a*x + 1))**(3/2)*sqrt(acosh(a*x)), x)
 

Maxima [F]

\[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\text {arccosh}(a x)} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {\operatorname {arcosh}\left (a x\right )} \,d x } \] Input:

integrate((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(1/2),x, algorithm="maxima")
 

Output:

integrate((-a^2*c*x^2 + c)^(3/2)*sqrt(arccosh(a*x)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\text {arccosh}(a x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(1/2),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
 

Mupad [F(-1)]

Timed out. \[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\text {arccosh}(a x)} \, dx=\int \sqrt {\mathrm {acosh}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \] Input:

int(acosh(a*x)^(1/2)*(c - a^2*c*x^2)^(3/2),x)
 

Output:

int(acosh(a*x)^(1/2)*(c - a^2*c*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\text {arccosh}(a x)} \, dx=\sqrt {c}\, c \left (-\left (\int \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{2}d x \right ) a^{2}+\int \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acosh} \left (a x \right )}d x \right ) \] Input:

int((-a^2*c*x^2+c)^(3/2)*acosh(a*x)^(1/2),x)
 

Output:

sqrt(c)*c*( - int(sqrt( - a**2*x**2 + 1)*sqrt(acosh(a*x))*x**2,x)*a**2 + i 
nt(sqrt( - a**2*x**2 + 1)*sqrt(acosh(a*x)),x))