\(\int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx\) [106]

Optimal result

Integrand size = 25, antiderivative size = 328 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {8 e^2 (c+d x)}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {4 e^2 (c+d x)^3}{b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{6 b^{5/2} d}-\frac {e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{2 b^{5/2} d}+\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{6 b^{5/2} d}+\frac {e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{2 b^{5/2} d} \] Output:

-2/3*e^2*(d*x+c-1)^(1/2)*(d*x+c)^2*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c) 
)^(3/2)+8/3*e^2*(d*x+c)/b^2/d/(a+b*arccosh(d*x+c))^(1/2)-4*e^2*(d*x+c)^3/b 
^2/d/(a+b*arccosh(d*x+c))^(1/2)-1/6*e^2*exp(a/b)*Pi^(1/2)*erf((a+b*arccosh 
(d*x+c))^(1/2)/b^(1/2))/b^(5/2)/d-1/2*e^2*exp(3*a/b)*3^(1/2)*Pi^(1/2)*erf( 
3^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(5/2)/d+1/6*e^2*Pi^(1/2)*erf 
i((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(5/2)/d/exp(a/b)+1/2*e^2*3^(1/2)*P 
i^(1/2)*erfi(3^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(5/2)/d/exp(3*a 
/b)
 

Mathematica [A] (verified)

Time = 1.95 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.19 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\frac {e^2 e^{-3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )} \left (2 e^{\frac {4 a}{b}+3 \text {arccosh}(c+d x)} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} (a+b \text {arccosh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )-6 \sqrt {3} b e^{3 \text {arccosh}(c+d x)} \left (-\frac {a+b \text {arccosh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )-2 b e^{\frac {2 a}{b}+3 \text {arccosh}(c+d x)} \left (-\frac {a+b \text {arccosh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )+e^{\frac {3 a}{b}} \left (-\left (\left (1+e^{2 \text {arccosh}(c+d x)}\right ) \left (a \left (6-4 e^{2 \text {arccosh}(c+d x)}+6 e^{4 \text {arccosh}(c+d x)}\right )+b \left (-1+6 \text {arccosh}(c+d x)-4 e^{2 \text {arccosh}(c+d x)} \text {arccosh}(c+d x)+e^{4 \text {arccosh}(c+d x)} (1+6 \text {arccosh}(c+d x))\right )\right )\right )+6 \sqrt {3} e^{3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} (a+b \text {arccosh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )\right )\right )}{12 b^2 d (a+b \text {arccosh}(c+d x))^{3/2}} \] Input:

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

Output:

(e^2*(2*E^((4*a)/b + 3*ArcCosh[c + d*x])*Sqrt[a/b + ArcCosh[c + d*x]]*(a + 
 b*ArcCosh[c + d*x])*Gamma[1/2, a/b + ArcCosh[c + d*x]] - 6*Sqrt[3]*b*E^(3 
*ArcCosh[c + d*x])*(-((a + b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2, (-3*(a 
 + b*ArcCosh[c + d*x]))/b] - 2*b*E^((2*a)/b + 3*ArcCosh[c + d*x])*(-((a + 
b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b)] + 
E^((3*a)/b)*(-((1 + E^(2*ArcCosh[c + d*x]))*(a*(6 - 4*E^(2*ArcCosh[c + d*x 
]) + 6*E^(4*ArcCosh[c + d*x])) + b*(-1 + 6*ArcCosh[c + d*x] - 4*E^(2*ArcCo 
sh[c + d*x])*ArcCosh[c + d*x] + E^(4*ArcCosh[c + d*x])*(1 + 6*ArcCosh[c + 
d*x])))) + 6*Sqrt[3]*E^(3*(a/b + ArcCosh[c + d*x]))*Sqrt[a/b + ArcCosh[c + 
 d*x]]*(a + b*ArcCosh[c + d*x])*Gamma[1/2, (3*(a + b*ArcCosh[c + d*x]))/b] 
)))/(12*b^2*d*E^(3*(a/b + ArcCosh[c + d*x]))*(a + b*ArcCosh[c + d*x])^(3/2 
))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 3.31 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.26, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {6411, 27, 6301, 6366, 6296, 25, 3042, 26, 3789, 2611, 2633, 2634, 6302, 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.

\(\displaystyle \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {e^2 (c+d x)^2}{(a+b \text {arccosh}(c+d x))^{5/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \int \frac {(c+d x)^2}{(a+b \text {arccosh}(c+d x))^{5/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 6301

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

\(\Big \downarrow \) 6366

\(\displaystyle \frac {e^2 \left (-\frac {4 \left (\frac {2 \int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 6296

\(\displaystyle \frac {e^2 \left (-\frac {4 \left (\frac {2 \int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {e^2 \left (-\frac {4 \left (-\frac {2 \int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e^2 \left (-\frac {4 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {2 \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {e^2 \left (-\frac {4 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 3789

\(\displaystyle \frac {e^2 \left (-\frac {4 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \left (\frac {1}{2} i \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))-\frac {1}{2} i \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )}{b^2}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {e^2 \left (-\frac {4 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-i \int e^{\frac {a+b \text {arccosh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}\right )}{b^2}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {e^2 \left (-\frac {4 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {e^2 \left (\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {4 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}\right )}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 6302

\(\displaystyle \frac {e^2 \left (\frac {2 \left (\frac {6 \int -\frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {4 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}\right )}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {e^2 \left (\frac {2 \left (-\frac {6 \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {4 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}\right )}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {e^2 \left (\frac {2 \left (-\frac {6 \int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {4 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}\right )}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^2 \left (-\frac {4 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}\right )}{3 b}+\frac {2 \left (\frac {6 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\)

Input:

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

Output:

(e^2*((-2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(3*b*(a + b*Ar 
cCosh[c + d*x])^(3/2)) - (4*((-2*(c + d*x))/(b*Sqrt[a + b*ArcCosh[c + d*x] 
]) + ((2*I)*((I/2)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x 
]]/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sq 
rt[b]])/E^(a/b)))/b^2))/(3*b) + (2*((-2*(c + d*x)^3)/(b*Sqrt[a + b*ArcCosh 
[c + d*x]]) + (6*(-1/8*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c 
+ d*x]]/Sqrt[b]]) - (Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + 
b*ArcCosh[c + d*x]])/Sqrt[b]])/8 + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCo 
sh[c + d*x]]/Sqrt[b]])/(8*E^(a/b)) + (Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqr 
t[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8*E^((3*a)/b))))/b^2))/b))/d
 

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 6296

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   S 
ubst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, 
b, c, n}, x]
 

rule 6301

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) 
)), x] + (-Simp[c*((m + 1)/(b*(n + 1)))   Int[x^(m + 1)*((a + b*ArcCosh[c*x 
])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[m/(b*c*(n + 1)) 
 Int[x^(m - 1)*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]) 
), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
 

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 6366

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 
_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a 
 + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x 
]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp 
[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]   Int[ 
(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 
1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
 

rule 6411

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

Maple [F]

\[\int \frac {\left (d e x +c e \right )^{2}}{\left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}}d x\]

Input:

int((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^(5/2),x)
 

Output:

int((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=e^{2} \left (\int \frac {c^{2}}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx\right ) \] Input:

integrate((d*e*x+c*e)**2/(a+b*acosh(d*x+c))**(5/2),x)
 

Output:

e**2*(Integral(c**2/(a**2*sqrt(a + b*acosh(c + d*x)) + 2*a*b*sqrt(a + b*ac 
osh(c + d*x))*acosh(c + d*x) + b**2*sqrt(a + b*acosh(c + d*x))*acosh(c + d 
*x)**2), x) + Integral(d**2*x**2/(a**2*sqrt(a + b*acosh(c + d*x)) + 2*a*b* 
sqrt(a + b*acosh(c + d*x))*acosh(c + d*x) + b**2*sqrt(a + b*acosh(c + d*x) 
)*acosh(c + d*x)**2), x) + Integral(2*c*d*x/(a**2*sqrt(a + b*acosh(c + d*x 
)) + 2*a*b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x) + b**2*sqrt(a + b*aco 
sh(c + d*x))*acosh(c + d*x)**2), x))
 

Maxima [F]

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

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

Output:

integrate((d*e*x + c*e)^2/(b*arccosh(d*x + c) + a)^(5/2), x)
 

Giac [F]

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

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

Output:

integrate((d*e*x + c*e)^2/(b*arccosh(d*x + c) + a)^(5/2), x)
 

Mupad [F(-1)]

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

int((c*e + d*e*x)^2/(a + b*acosh(c + d*x))^(5/2),x)
 

Output:

int((c*e + d*e*x)^2/(a + b*acosh(c + d*x))^(5/2), x)
 

Reduce [F]

\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\text {too large to display} \] Input:

int((d*e*x+c*e)^2/(a+b*acosh(d*x+c))^(5/2),x)
                                                                                    
                                                                                    
 

Output:

(e**2*(3*acosh(c + d*x)**2*int((sqrt(acosh(c + d*x)*b + a)*x**4)/(acosh(c 
+ d*x)**3*b**3*c**2 + 2*acosh(c + d*x)**3*b**3*c*d*x + acosh(c + d*x)**3*b 
**3*d**2*x**2 - acosh(c + d*x)**3*b**3 + 3*acosh(c + d*x)**2*a*b**2*c**2 + 
 6*acosh(c + d*x)**2*a*b**2*c*d*x + 3*acosh(c + d*x)**2*a*b**2*d**2*x**2 - 
 3*acosh(c + d*x)**2*a*b**2 + 3*acosh(c + d*x)*a**2*b*c**2 + 6*acosh(c + d 
*x)*a**2*b*c*d*x + 3*acosh(c + d*x)*a**2*b*d**2*x**2 - 3*acosh(c + d*x)*a* 
*2*b + a**3*c**2 + 2*a**3*c*d*x + a**3*d**2*x**2 - a**3),x)*b**3*d**5 + 12 
*acosh(c + d*x)**2*int((sqrt(acosh(c + d*x)*b + a)*x**3)/(acosh(c + d*x)** 
3*b**3*c**2 + 2*acosh(c + d*x)**3*b**3*c*d*x + acosh(c + d*x)**3*b**3*d**2 
*x**2 - acosh(c + d*x)**3*b**3 + 3*acosh(c + d*x)**2*a*b**2*c**2 + 6*acosh 
(c + d*x)**2*a*b**2*c*d*x + 3*acosh(c + d*x)**2*a*b**2*d**2*x**2 - 3*acosh 
(c + d*x)**2*a*b**2 + 3*acosh(c + d*x)*a**2*b*c**2 + 6*acosh(c + d*x)*a**2 
*b*c*d*x + 3*acosh(c + d*x)*a**2*b*d**2*x**2 - 3*acosh(c + d*x)*a**2*b + a 
**3*c**2 + 2*a**3*c*d*x + a**3*d**2*x**2 - a**3),x)*b**3*c*d**4 + 15*acosh 
(c + d*x)**2*int((sqrt(acosh(c + d*x)*b + a)*x**2)/(acosh(c + d*x)**3*b**3 
*c**2 + 2*acosh(c + d*x)**3*b**3*c*d*x + acosh(c + d*x)**3*b**3*d**2*x**2 
- acosh(c + d*x)**3*b**3 + 3*acosh(c + d*x)**2*a*b**2*c**2 + 6*acosh(c + d 
*x)**2*a*b**2*c*d*x + 3*acosh(c + d*x)**2*a*b**2*d**2*x**2 - 3*acosh(c + d 
*x)**2*a*b**2 + 3*acosh(c + d*x)*a**2*b*c**2 + 6*acosh(c + d*x)*a**2*b*c*d 
*x + 3*acosh(c + d*x)*a**2*b*d**2*x**2 - 3*acosh(c + d*x)*a**2*b + a**3...