\(\int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) [183]

Optimal result

Integrand size = 29, antiderivative size = 365 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {4144 b^2 \sqrt {d-c^2 d x^2}}{3375 c^6 d}-\frac {272 b^2 x^2 \sqrt {d-c^2 d x^2}}{3375 c^4 d}-\frac {2 b^2 x^4 \sqrt {d-c^2 d x^2}}{125 c^2 d}+\frac {16 b x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 c^5 d \sqrt {-1+c x} \sqrt {1+c x}}+\frac {8 b x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 c^3 d \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 c d \sqrt {-1+c x} \sqrt {1+c x}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{5 c^2 d} \] Output:

-4144/3375*b^2*(-c^2*d*x^2+d)^(1/2)/c^6/d-272/3375*b^2*x^2*(-c^2*d*x^2+d)^ 
(1/2)/c^4/d-2/125*b^2*x^4*(-c^2*d*x^2+d)^(1/2)/c^2/d+16/15*b*x*(-c^2*d*x^2 
+d)^(1/2)*(a+b*arccosh(c*x))/c^5/d/(c*x-1)^(1/2)/(c*x+1)^(1/2)+8/45*b*x^3* 
(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^3/d/(c*x-1)^(1/2)/(c*x+1)^(1/2)+ 
2/25*b*x^5*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c/d/(c*x-1)^(1/2)/(c*x+ 
1)^(1/2)-8/15*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/c^6/d-4/15*x^2*(-c 
^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/c^4/d-1/5*x^4*(-c^2*d*x^2+d)^(1/2)* 
(a+b*arccosh(c*x))^2/c^2/d
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.70 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (30 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (120+20 c^2 x^2+9 c^4 x^4\right )-225 a^2 \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right )-2 b^2 \left (-2072+1936 c^2 x^2+109 c^4 x^4+27 c^6 x^6\right )+30 b \left (b c x \sqrt {-1+c x} \sqrt {1+c x} \left (120+20 c^2 x^2+9 c^4 x^4\right )-15 a \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right )\right ) \text {arccosh}(c x)-225 b^2 \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right ) \text {arccosh}(c x)^2\right )}{3375 c^6 d (-1+c x) (1+c x)} \] Input:

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

Output:

(Sqrt[d - c^2*d*x^2]*(30*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(120 + 20*c^ 
2*x^2 + 9*c^4*x^4) - 225*a^2*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6) - 2*b^ 
2*(-2072 + 1936*c^2*x^2 + 109*c^4*x^4 + 27*c^6*x^6) + 30*b*(b*c*x*Sqrt[-1 
+ c*x]*Sqrt[1 + c*x]*(120 + 20*c^2*x^2 + 9*c^4*x^4) - 15*a*(-8 + 4*c^2*x^2 
 + c^4*x^4 + 3*c^6*x^6))*ArcCosh[c*x] - 225*b^2*(-8 + 4*c^2*x^2 + c^4*x^4 
+ 3*c^6*x^6)*ArcCosh[c*x]^2))/(3375*c^6*d*(-1 + c*x)*(1 + c*x))
 

Rubi [A] (verified)

Time = 1.66 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.29, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {6353, 6298, 111, 27, 111, 27, 83, 6353, 6298, 111, 27, 83, 6329, 2009}

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

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 83

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 83

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

\(\Big \downarrow \) 6329

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/5*(x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(c^2*d) - (2*b*Sqrt[ 
-1 + c*x]*Sqrt[1 + c*x]*(-1/5*(b*c*((x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5* 
c^2) + (4*((2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^4) + (x^2*Sqrt[-1 + c*x]* 
Sqrt[1 + c*x])/(3*c^2)))/(5*c^2))) + (x^5*(a + b*ArcCosh[c*x]))/5))/(5*c*S 
qrt[d - c^2*d*x^2]) + (4*(-1/3*(x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x 
])^2)/(c^2*d) - (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1/3*(b*c*((2*Sqrt[-1 + 
 c*x]*Sqrt[1 + c*x])/(3*c^4) + (x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^2)) 
) + (x^3*(a + b*ArcCosh[c*x]))/3))/(3*c*Sqrt[d - c^2*d*x^2]) + (2*(-((Sqrt 
[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(c^2*d)) - (2*b*Sqrt[-1 + c*x]*Sqr 
t[1 + c*x]*(a*x - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c + b*x*ArcCosh[c*x]))/ 
(c*Sqrt[d - c^2*d*x^2])))/(3*c^2)))/(5*c^2)
 

Defintions of rubi rules used

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 83

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

rule 111

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

rule 2009

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

rule 6298

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

rule 6329

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

rule 6353

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

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.54

method	result	size
orering	\(\frac {\left (1647 c^{8} x^{8}+1684 c^{6} x^{6}+34306 c^{4} x^{4}-102032 c^{2} x^{2}+62160\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{3375 c^{8} x^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {2 \left (c x -1\right ) \left (c x +1\right ) \left (162 c^{6} x^{6}+491 c^{4} x^{4}+7472 c^{2} x^{2}-10360\right ) \left (\frac {5 x^{4} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{\sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 x^{5} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {-c^{2} d \,x^{2}+d}\, \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {x^{6} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )}{3375 x^{6} c^{8}}+\frac {\left (27 c^{4} x^{4}+136 c^{2} x^{2}+2072\right ) \left (c x +1\right )^{2} \left (c x -1\right )^{2} \left (\frac {20 x^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{\sqrt {-c^{2} d \,x^{2}+d}}+\frac {20 x^{4} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {-c^{2} d \,x^{2}+d}\, \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {11 x^{5} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x^{5} b^{2} c^{2}}{\left (c x -1\right ) \left (c x +1\right ) \sqrt {-c^{2} d \,x^{2}+d}}+\frac {4 x^{6} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{3} d}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {x^{5} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\sqrt {-c^{2} d \,x^{2}+d}\, \left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {x^{5} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\sqrt {-c^{2} d \,x^{2}+d}\, \sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}+\frac {3 x^{7} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{4} d^{2}}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}\right )}{3375 c^{8} x^{5}}\)	\(561\)
default	\(\text {Expression too large to display}\)	\(1314\)
parts	\(\text {Expression too large to display}\)	\(1314\)

Input:

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

Output:

1/3375*(1647*c^8*x^8+1684*c^6*x^6+34306*c^4*x^4-102032*c^2*x^2+62160)/c^8/ 
x^2*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2)-2/3375*(c*x-1)*(c*x+1)*(162* 
c^6*x^6+491*c^4*x^4+7472*c^2*x^2-10360)/x^6/c^8*(5*x^4*(a+b*arccosh(c*x))^ 
2/(-c^2*d*x^2+d)^(1/2)+2*x^5*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2)*b*c/( 
c*x-1)^(1/2)/(c*x+1)^(1/2)+x^6*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(3/2)*c 
^2*d)+1/3375*(27*c^4*x^4+136*c^2*x^2+2072)/c^8*(c*x+1)^2*(c*x-1)^2/x^5*(20 
*x^3*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2)+20*x^4*(a+b*arccosh(c*x))/( 
-c^2*d*x^2+d)^(1/2)*b*c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+11*x^5*(a+b*arccosh(c* 
x))^2/(-c^2*d*x^2+d)^(3/2)*c^2*d+2*x^5*b^2*c^2/(c*x-1)/(c*x+1)/(-c^2*d*x^2 
+d)^(1/2)+4*x^6*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2)*b*c^3/(c*x-1)^(1/2 
)/(c*x+1)^(1/2)*d-x^5*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2)*b*c^2/(c*x-1 
)^(3/2)/(c*x+1)^(1/2)-x^5*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2)*b*c^2/(c 
*x-1)^(1/2)/(c*x+1)^(3/2)+3*x^7*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2)* 
c^4*d^2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.95 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {225 \, {\left (3 \, b^{2} c^{6} x^{6} + b^{2} c^{4} x^{4} + 4 \, b^{2} c^{2} x^{2} - 8 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 30 \, {\left (9 \, a b c^{5} x^{5} + 20 \, a b c^{3} x^{3} + 120 \, a b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 30 \, {\left ({\left (9 \, b^{2} c^{5} x^{5} + 20 \, b^{2} c^{3} x^{3} + 120 \, b^{2} c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 15 \, {\left (3 \, a b c^{6} x^{6} + a b c^{4} x^{4} + 4 \, a b c^{2} x^{2} - 8 \, a b\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} x^{6} + {\left (225 \, a^{2} + 218 \, b^{2}\right )} c^{4} x^{4} + 4 \, {\left (225 \, a^{2} + 968 \, b^{2}\right )} c^{2} x^{2} - 1800 \, a^{2} - 4144 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{3375 \, {\left (c^{8} d x^{2} - c^{6} d\right )}} \] Input:

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

Output:

-1/3375*(225*(3*b^2*c^6*x^6 + b^2*c^4*x^4 + 4*b^2*c^2*x^2 - 8*b^2)*sqrt(-c 
^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 30*(9*a*b*c^5*x^5 + 20*a*b* 
c^3*x^3 + 120*a*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 30*((9*b^2 
*c^5*x^5 + 20*b^2*c^3*x^3 + 120*b^2*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 
 - 1) - 15*(3*a*b*c^6*x^6 + a*b*c^4*x^4 + 4*a*b*c^2*x^2 - 8*a*b)*sqrt(-c^2 
*d*x^2 + d))*log(c*x + sqrt(c^2*x^2 - 1)) + (27*(25*a^2 + 2*b^2)*c^6*x^6 + 
 (225*a^2 + 218*b^2)*c^4*x^4 + 4*(225*a^2 + 968*b^2)*c^2*x^2 - 1800*a^2 - 
4144*b^2)*sqrt(-c^2*d*x^2 + d))/(c^8*d*x^2 - c^6*d)
 

Sympy [F]

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

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

Output:

Integral(x**5*(a + b*acosh(c*x))**2/sqrt(-d*(c*x - 1)*(c*x + 1)), x)
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.12 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} b^{2} \operatorname {arcosh}\left (c x\right )^{2} - \frac {2}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a b \operatorname {arcosh}\left (c x\right ) - \frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a^{2} - \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} x^{4} + 136 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} x^{2} + \frac {2072 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d}}{c^{2}}}{c^{4} d} - \frac {15 \, {\left (9 \, c^{4} \sqrt {-d} x^{5} + 20 \, c^{2} \sqrt {-d} x^{3} + 120 \, \sqrt {-d} x\right )} \operatorname {arcosh}\left (c x\right )}{c^{5} d}\right )} + \frac {2 \, {\left (9 \, c^{4} \sqrt {-d} x^{5} + 20 \, c^{2} \sqrt {-d} x^{3} + 120 \, \sqrt {-d} x\right )} a b}{225 \, c^{5} d} \] Input:

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

Output:

-1/15*(3*sqrt(-c^2*d*x^2 + d)*x^4/(c^2*d) + 4*sqrt(-c^2*d*x^2 + d)*x^2/(c^ 
4*d) + 8*sqrt(-c^2*d*x^2 + d)/(c^6*d))*b^2*arccosh(c*x)^2 - 2/15*(3*sqrt(- 
c^2*d*x^2 + d)*x^4/(c^2*d) + 4*sqrt(-c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt(- 
c^2*d*x^2 + d)/(c^6*d))*a*b*arccosh(c*x) - 1/15*(3*sqrt(-c^2*d*x^2 + d)*x^ 
4/(c^2*d) + 4*sqrt(-c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt(-c^2*d*x^2 + d)/(c 
^6*d))*a^2 - 2/3375*b^2*((27*sqrt(c^2*x^2 - 1)*c^2*sqrt(-d)*x^4 + 136*sqrt 
(c^2*x^2 - 1)*sqrt(-d)*x^2 + 2072*sqrt(c^2*x^2 - 1)*sqrt(-d)/c^2)/(c^4*d) 
- 15*(9*c^4*sqrt(-d)*x^5 + 20*c^2*sqrt(-d)*x^3 + 120*sqrt(-d)*x)*arccosh(c 
*x)/(c^5*d)) + 2/225*(9*c^4*sqrt(-d)*x^5 + 20*c^2*sqrt(-d)*x^3 + 120*sqrt( 
-d)*x)*a*b/(c^5*d)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-3 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}-4 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-8 \sqrt {-c^{2} x^{2}+1}\, a^{2}+30 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{5}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{6}+15 \left (\int \frac {\mathit {acosh} \left (c x \right )^{2} x^{5}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{6}}{15 \sqrt {d}\, c^{6}} \] Input:

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

Output:

( - 3*sqrt( - c**2*x**2 + 1)*a**2*c**4*x**4 - 4*sqrt( - c**2*x**2 + 1)*a** 
2*c**2*x**2 - 8*sqrt( - c**2*x**2 + 1)*a**2 + 30*int((acosh(c*x)*x**5)/sqr 
t( - c**2*x**2 + 1),x)*a*b*c**6 + 15*int((acosh(c*x)**2*x**5)/sqrt( - c**2 
*x**2 + 1),x)*b**2*c**6)/(15*sqrt(d)*c**6)