\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))}{x} \, dx\) [97]

Optimal result

Integrand size = 27, antiderivative size = 379 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {23 b c d^2 x \sqrt {d-c^2 d x^2}}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {11 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-23/15*b*c*d^2*x*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+11/45*b* 
c^3*d^2*x^3*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/25*b*c^5*d^ 
2*x^5*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+d^2*(-c^2*d*x^2+d)^ 
(1/2)*(a+b*arccosh(c*x))+1/3*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))+1/5 
*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))-2*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*a 
rccosh(c*x))*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*x-1)^(1/2)/(c*x+1) 
^(1/2)+I*b*d^2*(-c^2*d*x^2+d)^(1/2)*polylog(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1 
)^(1/2)))/(c*x-1)^(1/2)/(c*x+1)^(1/2)-I*b*d^2*(-c^2*d*x^2+d)^(1/2)*polylog 
(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 2.60 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x} \, dx=\frac {1}{15} a d^2 \sqrt {d-c^2 d x^2} \left (23-11 c^2 x^2+3 c^4 x^4\right )-\frac {b d^2 \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )}{18 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+a d^{5/2} \log (x)-a d^{5/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b d^2 \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+c x \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+i \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-i \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}-\frac {b d^2 \sqrt {d-c^2 d x^2} \left (25 \cosh (3 \text {arccosh}(c x))+9 (-50 c x+\cosh (5 \text {arccosh}(c x)))+15 \text {arccosh}(c x) \left (30 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-5 \sinh (3 \text {arccosh}(c x))-3 \sinh (5 \text {arccosh}(c x))\right )\right )}{3600 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \] Input:

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

Output:

(a*d^2*Sqrt[d - c^2*d*x^2]*(23 - 11*c^2*x^2 + 3*c^4*x^4))/15 - (b*d^2*Sqrt 
[d - c^2*d*x^2]*(9*c*x + 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCo 
sh[c*x] - Cosh[3*ArcCosh[c*x]]))/(18*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) 
 + a*d^(5/2)*Log[x] - a*d^(5/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b* 
d^2*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] 
+ c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + I*ArcCosh[c*x]*Log[1 - I/E 
^ArcCosh[c*x]] - I*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*PolyLog[2, ( 
-I)/E^ArcCosh[c*x]] - I*PolyLog[2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 
 + c*x)]*(1 + c*x)) - (b*d^2*Sqrt[d - c^2*d*x^2]*(25*Cosh[3*ArcCosh[c*x]] 
+ 9*(-50*c*x + Cosh[5*ArcCosh[c*x]]) + 15*ArcCosh[c*x]*(30*Sqrt[(-1 + c*x) 
/(1 + c*x)]*(1 + c*x) - 5*Sinh[3*ArcCosh[c*x]] - 3*Sinh[5*ArcCosh[c*x]]))) 
/(3600*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
 

Rubi [A] (verified)

Time = 1.62 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.86, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6345, 39, 210, 2009, 6345, 25, 39, 2009, 6341, 24, 6362, 3042, 4668, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6345

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

\(\Big \downarrow \) 39

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

\(\Big \downarrow \) 210

\(\displaystyle d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x}dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (c^4 x^4-2 c^2 x^2+1\right )dx}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right ) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6345

\(\displaystyle d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}dx+\frac {b c d \sqrt {d-c^2 d x^2} \int -((1-c x) (c x+1))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right ) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}dx-\frac {b c d \sqrt {d-c^2 d x^2} \int (1-c x) (c x+1)dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right ) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 39

\(\displaystyle d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}dx-\frac {b c d \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right ) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right ) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6341

\(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int 1dx}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right ) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 24

\(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right ) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6362

\(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{c x}d\text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right ) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 3042

\(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x)) \csc \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right ) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 4668

\(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (-i b \int \log \left (1-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+i b \int \log \left (1+i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right ) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 2715

\(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (-i b \int e^{-\text {arccosh}(c x)} \log \left (1-i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+i b \int e^{-\text {arccosh}(c x)} \log \left (1+i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right ) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 2838

\(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right ) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-1/5*(b*c*d^2*Sqrt[d - c^2*d*x^2]*(x - (2*c^2*x^3)/3 + (c^4*x^5)/5))/(Sqrt 
[-1 + c*x]*Sqrt[1 + c*x]) + ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/5 
 + d*(-1/3*(b*c*d*Sqrt[d - c^2*d*x^2]*(x - (c^2*x^3)/3))/(Sqrt[-1 + c*x]*S 
qrt[1 + c*x]) + ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/3 + d*(-((b*c 
*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + Sqrt[d - c^2*d*x 
^2]*(a + b*ArcCosh[c*x]) - (Sqrt[d - c^2*d*x^2]*(2*(a + b*ArcCosh[c*x])*Ar 
cTan[E^ArcCosh[c*x]] - I*b*PolyLog[2, (-I)*E^ArcCosh[c*x]] + I*b*PolyLog[2 
, I*E^ArcCosh[c*x]]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

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

rule 39

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( 
a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( 
IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
 

rule 210

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 
)^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
 

rule 2009

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

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 6341

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

rule 6345

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

rule 6362

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1 
_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/c^(m + 1))*Simp[ 
Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]   Subst 
[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, 
 e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && Inte 
gerQ[m]
 

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.64

method	result	size
default	\(\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} a}{5}+\frac {a d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}-a \,d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )+a \,d^{2} \sqrt {-c^{2} d \,x^{2}+d}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d^{2}}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {14 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}}{15 \left (c x -1\right ) \left (c x +1\right )}+\frac {34 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \operatorname {arccosh}\left (c x \right ) x^{2} c^{2}}{15 \left (c x -1\right ) \left (c x +1\right )}-\frac {23 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \operatorname {arccosh}\left (c x \right )}{15 \left (c x -1\right ) \left (c x +1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \operatorname {arccosh}\left (c x \right ) x^{6} c^{6}}{5 \left (c x -1\right ) \left (c x +1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d^{2}}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d^{2}}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {b \,c^{5} d^{2} x^{5} \sqrt {-d \left (c^{2} x^{2}-1\right )}}{25 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {11 b \,c^{3} d^{2} x^{3} \sqrt {-d \left (c^{2} x^{2}-1\right )}}{45 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {23 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c \,d^{2} x}{15 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d^{2}}{\sqrt {c x -1}\, \sqrt {c x +1}}\)	\(620\)
parts	\(\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} a}{5}+\frac {a d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}-a \,d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )+a \,d^{2} \sqrt {-c^{2} d \,x^{2}+d}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d^{2}}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {14 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}}{15 \left (c x -1\right ) \left (c x +1\right )}+\frac {34 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \operatorname {arccosh}\left (c x \right ) x^{2} c^{2}}{15 \left (c x -1\right ) \left (c x +1\right )}-\frac {23 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \operatorname {arccosh}\left (c x \right )}{15 \left (c x -1\right ) \left (c x +1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \operatorname {arccosh}\left (c x \right ) x^{6} c^{6}}{5 \left (c x -1\right ) \left (c x +1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d^{2}}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d^{2}}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {b \,c^{5} d^{2} x^{5} \sqrt {-d \left (c^{2} x^{2}-1\right )}}{25 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {11 b \,c^{3} d^{2} x^{3} \sqrt {-d \left (c^{2} x^{2}-1\right )}}{45 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {23 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c \,d^{2} x}{15 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d^{2}}{\sqrt {c x -1}\, \sqrt {c x +1}}\)	\(620\)

Input:

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

Output:

1/5*(-c^2*d*x^2+d)^(5/2)*a+1/3*a*d*(-c^2*d*x^2+d)^(3/2)-a*d^(5/2)*ln((2*d+ 
2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)+a*d^2*(-c^2*d*x^2+d)^(1/2)-I*b*(-d*(c^2 
*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c 
*x+1)^(1/2)))*d^2-14/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)/(c*x+1)*arcco 
sh(c*x)*x^4*c^4+34/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)/(c*x+1)*arccosh 
(c*x)*x^2*c^2-23/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)/(c*x+1)*arccosh(c 
*x)+1/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)/(c*x+1)*arccosh(c*x)*x^6*c^6+ 
I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*dilog(1+I*(c*x+(c*x 
-1)^(1/2)*(c*x+1)^(1/2)))*d^2-I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c* 
x+1)^(1/2)*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*d^2-1/25 
*b*c^5*d^2*x^5/(c*x+1)^(1/2)/(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)+11/45*b* 
c^3*d^2*x^3/(c*x+1)^(1/2)/(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)-23/15*b*(-d 
*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c*d^2*x+I*b*(-d*(c^2*x^2-1 
))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2 
)*(c*x+1)^(1/2)))*d^2
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/15*(15*d^(5/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) 
- 3*(-c^2*d*x^2 + d)^(5/2) - 5*(-c^2*d*x^2 + d)^(3/2)*d - 15*sqrt(-c^2*d*x 
^2 + d)*d^2)*a + b*integrate((-c^2*d*x^2 + d)^(5/2)*log(c*x + sqrt(c*x + 1 
)*sqrt(c*x - 1))/x, x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x} \, dx=\frac {\sqrt {d}\, d^{2} \left (3 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}-11 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}+23 \sqrt {-c^{2} x^{2}+1}\, a +15 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x}d x \right ) b +15 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) b \,c^{4}-30 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x d x \right ) b \,c^{2}+15 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a -23 a \right )}{15} \] Input:

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

Output:

(sqrt(d)*d**2*(3*sqrt( - c**2*x**2 + 1)*a*c**4*x**4 - 11*sqrt( - c**2*x**2 
 + 1)*a*c**2*x**2 + 23*sqrt( - c**2*x**2 + 1)*a + 15*int((sqrt( - c**2*x** 
2 + 1)*acosh(c*x))/x,x)*b + 15*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**3, 
x)*b*c**4 - 30*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x,x)*b*c**2 + 15*log( 
tan(asin(c*x)/2))*a - 23*a))/15